1611:, and, indeed, there is another definition that explicitly works with the volume. Several, actually; one has both strong mixing and weak mixing; they are inequivalent, although a strong mixing system is always weakly mixing. The measure-based definitions are not compatible with the definition of topological mixing: there are systems which are one, but not the other. The general situation remains cloudy: for example, given three sets
36:
70:
7207:
4415:
1684:
of colored dye that is being mixed into a cup of some sort of sticky liquid, say, corn syrup, or shampoo, or the like. Practical experience shows that mixing sticky fluids can be quite hard: there is usually some corner of the container where it is hard to get the dye mixed into. Pick as set
8639:
6072:
642:
A formal difficulty arises when one tries to reconcile the volume of sets with the need to preserve their size under a map. The problem arises because, in general, several different points in the domain of a function can map to the same point in its range; that is, there may be
4120:
3088:
in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is not true: there exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing. The
2799:
4785:
8284:
8112:
7028:
7984:
1657:, one can define 3-mixing. As of 2020, it is not known if 2-mixing implies 3-mixing. (If one thinks of ergodicity as "1-mixing", then it is clear that 1-mixing does not imply 2-mixing; there are systems that are ergodic but not mixing.)
5724:
6859:
Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.
3642:
in the covering family, to all open sets by disjoint union, to all closed sets by taking the complement, to all measurable sets by using the regularity of
Lebesgue measure to approximate any set with open and closed sets. Thus,
4249:
3775:
The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von
Neumann's ergodic theorem, ergodicity of a dynamical system
2577:
1867:
3347:
Given a topological space, such as the unit interval (whether it has its end points or not), we can construct a measure on it by taking the open sets, then take their unions, complements, unions, complements, and so on to
2213:
8523:
4866:
3231:
3000:
1271:, never to be returned to. An example would be water running downhill—once it's run down, it will never come back up again. The lake that forms at the bottom of this river can, however, become well-mixed. The
3086:
5949:
2350:, one wants to look at where that dye "came from" (presumably, it was poured in at the top, at some time in the past). One must be sure that every place it might have "come from" eventually gets mixed into
3732:
3562:
136:
to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies
7853:
3309:
1567:
The above definition of topological mixing should be enough to provide an informal idea of mixing (it is equivalent to the formal definition, given below). However, it made no mention of the volume of
3971:
2903:
793:
5410:
5181:
6971:
7463:
6921:
5570:
5513:
7322:
6739:
6116:
912:
7403:
6420:
1514:
6592:
8785:
8423:
5267:
2660:
872:
4466:
4171:
3822:
2424:
205:
7247:
4669:
4570:
3922:
1653:
1174:
1005:
6265:
6234:
2476:
1388:
8185:
7897:
7733:
5870:
4662:
4242:
1097:
6841:
5942:
4516:
3870:
4609:
3961:
3410:
7502:
8012:
6183:, we can impose a discrete probability distribution on it, then consider the probability distribution on the "coin flip" space, where each "coin flip" can take results from
3437:
2011:
447:
391:
303:
8814:
8454:
7755:
7695:
7628:
7530:
7202:{\displaystyle \alpha (s)=\sup \left\{|\mathbb {P} (A\cap B)-\mathbb {P} (A)\mathbb {P} (B)|:-\infty <t<\infty ,A\in X_{-\infty }^{t},B\in X_{t+s}^{\infty }\right\}}
6801:
6365:
4998:
4901:
1558:
5125:
5456:
5352:
5312:
5084:
821:
527:
334:
7281:
1044:
713:
6326:
501:
7669:
7016:
6203:
6179:
2328:
2265:
667:
639:; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).
365:
7905:
6683:
6532:
2614:
2295:
1464:
940:
741:
6769:
6450:
2641:
1534:
8732:
8370:
5595:
3374:
617:
556:
257:
7598:
3760:
3640:
3588:
3337:
3131:
1436:
1304:
77:
to points colored red and blue, initially separated. The baker's map is mixing, shown by the red and blue points being completely mixed after several iterations.
7572:
7552:
6649:
6627:
5892:
5786:
of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the
Lebesgue measure.
5038:
5018:
4963:
4941:
4921:
4191:
3608:
3471:
2822:
2368:
2348:
2131:
2111:
2091:
2071:
2051:
2031:
1970:
1950:
1930:
1910:
1890:
1763:
1743:
1723:
1703:
1682:
1607:
1585:
1410:
1348:
1326:
1266:
1238:
1214:
1194:
1117:
1064:
637:
576:
417:
277:
230:
6152:
4410:{\displaystyle \lim _{N\to \infty }{1 \over N}\sum _{n=0}^{N-1}\left|\int _{X}f\circ T^{n}\cdot g\,d\mu -\int _{X}f\,d\mu \cdot \int _{X}g\,d\mu \right|=0.}
2484:
1774:
2136:
149:
The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, drinks, cooking ingredients,
8634:{\displaystyle \beta _{t}=\int \sup _{0\leq \varphi \leq 1}\left|{\mathcal {E}}_{t}\varphi (x)-\int \varphi \,d\mathbb {Q} \right|\,d\mathbb {Q} .}
4798:
3412:, and so it is a Lebesgue measure space. Verifying strong-mixing can be simplified if we only need to check a smaller set of measurable sets.
8974:
8891:
8877:
3140:
2911:
6067:{\displaystyle \left\{\left({\tfrac {k}{n^{s}}},{\tfrac {k+1}{n^{s}}}\right)\smallsetminus \mathbb {Q} :s\geq 0,\leq k<n^{s}\right\}}
2427:
154:
117:
141:: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" condition than ergodicity).
3012:
876:. It has the important property of not "losing track" of where things came from. More strongly, it has the important property that
3646:
3476:
7763:
4115:{\displaystyle \lim _{N\to \infty }\left\|{1 \over N}\sum _{n=0}^{N-1}f\circ T^{n}-\int _{X}f\,d\mu \right\|_{L^{2}(X,\mu )}=0.}
3240:
153:, smoke in a smoke-filled room, and so on. To provide the mathematical rigor, such descriptions begin with the definition of a
2827:
748:
5359:
5130:
6932:
7413:
6871:
5518:
5461:
150:
102:
3383:
In most applications of ergodic theory, the underlying space is almost-everywhere isomorphic to an open subset of some
7286:
6693:
6273:(one letter to the left) is strongly mixing, since it is strongly mixing on the covering family of cylinder sets. The
6081:
883:
7337:
6377:
2794:{\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=0}^{n-1}\left|\mu (A\cap T^{-k}B)-\mu (A)\mu (B)\right|=0.}
1471:
6541:
8740:
8378:
5186:
2330:
was used to define the concept of a measure-preserving map. When looking at how much dye got mixed into the corner
1278:
Mixing is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets
1272:
422:
4780:{\displaystyle \lim _{n\to \infty }\int _{X}f\circ T^{n}\cdot g\,d\mu =\int _{X}f\,d\mu \cdot \int _{X}g\,d\mu .}
3349:
828:
54:
7601:
6472:
4423:
4128:
3779:
2381:
1275:
states that every ergodic system can be split into two parts: the conservative part, and the dissipative part.
162:
7217:
4523:
3875:
3376:
on the Borel sets, then add in all the subsets of measure-zero ("negligible sets"). This is how we obtain the
8279:{\displaystyle \alpha _{t}=\sup _{\varphi \in Z:\|\varphi \|_{\infty }=1}\|{\mathcal {E}}_{t}\varphi \|_{1}.}
1616:
1122:
947:
9008:
6468:
6241:
6210:
5802:
5789:
Many maps considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the
2445:
1357:
8906:
7861:
7324:
denotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times
7700:
5831:
4616:
4196:
1072:
6853:
6814:
6494:
6286:
5899:
4473:
3827:
237:
133:
94:
8107:{\displaystyle \rho _{t}=\sup _{\varphi \in Z:\,\|\varphi \|_{2}=1}\|{\mathcal {E}}_{t}\varphi \|_{2}.}
4577:
3929:
7735:
denote the space of Borel-measurable functions that are square-integrable with respect to the measure
4790:
Since the system is assumed to be measure preserving, this last line is equivalent to saying that the
3386:
8897:
Chen, Xiaohong; Hansen, Lars Peter; Carrasco, Marine (2010). "Nonlinearity and temporal dependence".
7472:
6476:
8911:
3418:
1975:
428:
372:
284:
6992:
5794:
5783:
3090:
1241:
8793:
8433:
7738:
7678:
7607:
7534:. That is to say, a strongly mixing process is such that, in a way that is uniform over all times
7509:
6779:
6338:
4970:
4873:
1543:
8924:
6924:
6772:
6506:
5822:
5089:
2228:
2224:
1245:
449:
consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a
113:
6809:
being the continuous parameter, with the requirement that a non-empty intersection hold for all
5423:
5319:
5279:
5051:
3093:
was historically the first example given of a system that is weak-mixing but not strong-mixing.
800:
506:
313:
7979:{\displaystyle Z=\left\{\varphi \in L^{2}(\mathbb {Q} ):\int \varphi \,d\mathbb {Q} =0\right\}}
7254:
1010:
674:
8970:
8887:
8873:
6453:
6299:
474:
469:
44:
7641:
7001:
6188:
6164:
5719:{\displaystyle \lim _{m,n\to \infty }\mu (A\cap T^{-m}B\cap T^{-m-n}C)=\mu (A)\mu (B)\mu (C)}
2300:
2237:
646:
341:
8916:
6976:
6662:
6511:
5818:
3377:
2589:
2270:
1537:
1443:
919:
720:
578:– it is squashed or stretched, folded or cut into pieces. Mathematical examples include the
454:
98:
49:
6747:
6428:
2619:
1519:
8717:
8355:
6464:
3444:
3359:
593:
532:
242:
232:
is understood to be the total space to be filled: the mixing bowl, the smoke-filled room,
7577:
3739:
3619:
3567:
3316:
3110:
1415:
1283:
5767:
It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong
8946:
7672:
7557:
7537:
6634:
6612:
6498:
6294:
6274:
5877:
5814:
5023:
5003:
4948:
4926:
4906:
4176:
3593:
3456:
3006:
2807:
2435:
2353:
2333:
2220:
2116:
2096:
2076:
2056:
2036:
2016:
1955:
1935:
1915:
1895:
1875:
1748:
1728:
1708:
1688:
1667:
1592:
1570:
1395:
1333:
1311:
1251:
1223:
1217:
1199:
1179:
1102:
1049:
622:
579:
561:
462:
402:
262:
215:
109:
74:
6125:
421:; this doesn't quite work, as not all subsets of a space have a volume (famously, the
9002:
6988:
6975:. The sequence space into which the process maps can be endowed with a topology, the
6480:
5810:
5798:
2216:
1268:
583:
8928:
69:
8824:
6984:
1664:
is made in reference to the volume of a pair of sets. Consider, for example, a set
8920:
2572:{\displaystyle \lim _{n\to \infty }\mu \left(A\cap T^{-n}B\right)=\mu (A)\mu (B).}
2093:, which itself is 10% of the total, and so, in the end, after mixing, the part of
1862:{\displaystyle \lim _{n\to \infty }\mu \left(T^{-n}A\cap B\right)=\mu (A)\mu (B).}
8954:
5806:
82:
2582:
For shifts parametrized by a continuous variable instead of a discrete integer
2208:{\displaystyle \mu \left({\mbox{after-mixing}}(A)\cap B\right)=\mu (A)\mu (B).}
743:
has no size. These difficulties can be avoided by working with the inverse map
5790:
5772:
4791:
138:
2297:
in the definition is a bit subtle, but it follows from the same reasons why
1069:
One is now interested in studying the time evolution of the system. If a set
17:
6595:
3353:
1561:
458:
450:
394:
7630:; more colloquially, the process, in a strong sense, forgets its history.
4967:, one can informally see mixing as the property that the random variables
4861:{\displaystyle \lim _{n\to \infty }\operatorname {Cov} (f\circ T^{n},g)=0}
6980:
6333:
6290:
2219:
in probabilities; this is not an accident, but rather a consequence that
1392:, a system is said to be (topologically) mixing if there is an integer
90:
3226:{\displaystyle \mu (T^{-n}(A)\cap B)=\mu (A\cap B)=\mu (\emptyset )=0}
6856:(with respect to the topology) eigenfunctions of the shift operator.
2995:{\displaystyle \left|\mu (A\cap T^{-n}B)-\mu (A)\mu (B)\right|\to 0,}
1768:
One phrases the definition of strong mixing as the requirement that
308:
8162:
may either decay at an exponential rate, or be always equal to one.
944:. The proper definition of a volume-preserving map is one for which
5314:
is weakly mixing if and only if, for any ergodic dynamical system
587:
5828:
The dyadic map is "shift to left in binary". In general, for any
279:
and of its subspaces. The collection of subspaces is denoted by
7989:
denote the space of square-integrable functions with mean zero.
3081:{\displaystyle \mu \left(A\cap T^{-n}B\right)\to \mu (A)\mu (B)}
1705:
that hard-to-reach corner. The question of mixing is then, can
3447:, which is less restrictive as it allows non-disjoint unions.
1725:, after a long enough period of time, not only penetrate into
29:
6995:; this is the smallest σ-algebra that contains the topology.
5925:
5269:
We then have the following characterizations of weak mixing:
3727:{\displaystyle \lim _{n}\mu (T^{-n}(A)\cap B)=\mu (A)\mu (B)}
3557:{\displaystyle \lim _{n}\mu (T^{-n}(A)\cap B)=\mu (A)\mu (B)}
2215:
This product-of-volumes has more than passing resemblance to
823:
to the parts that were assembled to make it: these parts are
453:—the collection of subsets that can be constructed by taking
8573:
8249:
8077:
7770:
6947:
4438:
4143:
3794:
3424:
2463:
2396:
1640:
1375:
1084:
899:
889:
859:
780:
770:
434:
378:
290:
177:
120:. Several different definitions for mixing exist, including
7848:{\displaystyle {\mathcal {E}}_{t}\varphi (x)=\mathbb {E} }
6505:
is topologically transitive if and only if there exists a
3304:{\displaystyle \vert \mu (A\cap B)-\mu (A)\mu (B)\vert =0}
3439:
is a set of measurable sets, such that any open set is a
101:
in the everyday world: e.g. mixing paint, mixing drinks,
6277:
is isomorphic to a shift map, so it is strongly mixing.
3105:
If the action of the map decomposes into two components
259:
is understood to define the natural volume of the space
2898:{\displaystyle \mu (A\cap T^{-n}B)-\mu (A)\mu (B)\to 0}
2033:
is 10% of the total volume, and that the volume of dye
788:{\displaystyle T^{-1}:{\mathcal {A}}\to {\mathcal {A}}}
5986:
5964:
5405:{\displaystyle (X\times Y,\mu \otimes \nu ,T\times S)}
5176:{\displaystyle (X\times Y,\mu \otimes \nu ,T\times S)}
3339:
has zero measure, and the other one has full measure.
2149:
2073:
is uniformly distributed, then it is occupying 10% of
369:; the size is its volume. Naively, one could imagine
8796:
8743:
8720:
8526:
8436:
8381:
8358:
8188:
8015:
7908:
7864:
7766:
7741:
7703:
7681:
7644:
7610:
7580:
7560:
7540:
7512:
7475:
7416:
7340:
7289:
7257:
7220:
7031:
7004:
6966:{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
6935:
6874:
6817:
6782:
6750:
6696:
6665:
6637:
6615:
6544:
6514:
6431:
6380:
6341:
6302:
6244:
6213:
6191:
6167:
6128:
6084:
5952:
5902:
5880:
5834:
5598:
5585:, to distinguish it from higher orders of mixing. A
5521:
5464:
5426:
5362:
5322:
5282:
5189:
5133:
5092:
5054:
5026:
5006:
4973:
4951:
4929:
4909:
4876:
4801:
4672:
4619:
4580:
4526:
4476:
4426:
4252:
4199:
4179:
4131:
3974:
3932:
3878:
3830:
3824:
is equivalent to the property that, for any function
3782:
3742:
3649:
3622:
3596:
3570:
3479:
3459:
3421:
3389:
3362:
3319:
3243:
3143:
3113:
3015:
2914:
2830:
2810:
2663:
2622:
2592:
2487:
2448:
2384:
2356:
2336:
2303:
2273:
2240:
2227:
are the same theory: they share the same axioms (the
2139:
2119:
2099:
2079:
2059:
2039:
2019:
1978:
1958:
1938:
1918:
1898:
1878:
1777:
1751:
1731:
1711:
1691:
1670:
1619:
1595:
1573:
1546:
1522:
1474:
1446:
1418:
1398:
1360:
1336:
1314:
1286:
1254:
1226:
1202:
1182:
1125:
1105:
1075:
1052:
1013:
950:
922:
886:
831:
803:
751:
723:
677:
649:
625:
596:
564:
535:
509:
477:
431:
405:
375:
344:
316:
287:
265:
245:
218:
165:
8823:-mixing if and only if it is an aperiodic recurrent
8155:. For a stationary Markov process, the coefficients
6285:
A form of mixing may be defined without appeal to a
6207:. We can either construct the singly-infinite space
4943:grows. Actually, since this works for any function
468:The time evolution of the system is described by a
8990:
8965:Theorem 2.36, Manfred Einsiedler and Thomas Ward,
8808:
8779:
8726:
8633:
8448:
8417:
8364:
8278:
8106:
7978:
7891:
7847:
7749:
7727:
7689:
7663:
7622:
7592:
7566:
7546:
7524:
7496:
7458:{\displaystyle (X_{t})_{-\infty <t<\infty }}
7457:
7397:
7316:
7275:
7241:
7201:
7010:
6965:
6916:{\displaystyle (X_{t})_{-\infty <t<\infty }}
6915:
6835:
6795:
6763:
6733:
6677:
6643:
6621:
6586:
6526:
6444:
6414:
6359:
6320:
6259:
6228:
6197:
6173:
6146:
6110:
6066:
5936:
5886:
5864:
5718:
5565:{\displaystyle (X^{2},\mu \otimes \mu ,T\times T)}
5564:
5508:{\displaystyle (X^{2},\mu \otimes \mu ,T\times T)}
5507:
5450:
5404:
5346:
5306:
5261:
5175:
5119:
5078:
5032:
5012:
4992:
4957:
4935:
4915:
4895:
4860:
4779:
4656:
4603:
4564:
4510:
4460:
4409:
4236:
4185:
4165:
4114:
3955:
3916:
3864:
3816:
3754:
3726:
3634:
3602:
3582:
3556:
3465:
3431:
3404:
3368:
3331:
3303:
3225:
3125:
3080:
2994:
2897:
2816:
2793:
2635:
2608:
2571:
2470:
2418:
2362:
2342:
2322:
2289:
2259:
2207:
2125:
2105:
2085:
2065:
2045:
2025:
2005:
1964:
1944:
1924:
1904:
1884:
1861:
1757:
1737:
1717:
1697:
1676:
1647:
1601:
1579:
1552:
1528:
1508:
1458:
1430:
1404:
1382:
1342:
1320:
1298:
1260:
1232:
1208:
1188:
1168:
1111:
1091:
1058:
1038:
999:
934:
906:
866:
815:
787:
735:
707:
661:
631:
611:
570:
550:
521:
495:
441:
411:
385:
359:
328:
297:
271:
251:
224:
199:
8843:-mixing. There is no direct relationship between
8465:-mixing coefficients are always smaller than the
3924:converges strongly and in the sense of Cesàro to
8967:Ergodic theory with a view towards number theory
8947:Ergodic Theory: Basic Examples and Constructions
8870:Ergodic theory with a view towards number theory
8831:-mixing coefficients are always bigger than the
8544:
8203:
8030:
7317:{\displaystyle -\infty \leq a\leq b\leq \infty }
7047:
6734:{\displaystyle f^{n}(A)\cap B\neq \varnothing .}
6159:Similarly, for any finite or countable alphabet
6111:{\displaystyle (0,1)\smallsetminus \mathbb {Q} }
5600:
4803:
4674:
4254:
3976:
3651:
3481:
2665:
2489:
1779:
907:{\displaystyle {\mathcal {A}}\to {\mathcal {A}}}
7858:denote the conditional expectation operator on
7398:{\displaystyle \{X_{a},X_{a+1},\ldots ,X_{b}\}}
6415:{\displaystyle f^{n}(A)\cap B\neq \varnothing }
5581:The definition given above is sometimes called
1765:with the same proportion as it does elsewhere?
1509:{\displaystyle T^{n}(A)\cap B\neq \varnothing }
465:; these can always be taken to be measurable.
8951:Encyclopedia of Complexity and Systems Science
6587:{\displaystyle \{f^{n}(x):n\in \mathbb {N} \}}
2013:is a bit more subtle. Imagine that the volume
8955:https://doi.org/10.1007/978-0-387-30440-3_177
8780:{\displaystyle {\frac {\ln \xi (t)}{t}}\to 0}
8418:{\displaystyle {\frac {\ln \xi (t)}{t}}\to 0}
5262:{\displaystyle (T\times S)(x,y)=(T(x),S(y)).}
27:Mathematical description of mixing substances
8:
8986:
8984:
8982:
8851:-mixing: neither of them implies the other.
8264:
8243:
8226:
8219:
8092:
8071:
8054:
8047:
7392:
7341:
6824:
6818:
6581:
6545:
5859:
5841:
3292:
3244:
5515:is also ergodic. If this is the case, then
867:{\displaystyle T^{-1}(A)\in {\mathcal {A}}}
93:: the attempt to describe the irreversible
43:It has been suggested that this article be
8651:if these coefficients converge to zero as
8507:-mixing, with sub-exponential decay rate.
8296:if these coefficients converge to zero as
8124:if these coefficients converge to zero as
5944:is strongly mixing on the covering family
8945:Matthew Nicol and Karl Petersen, (2009) "
8910:
8795:
8744:
8742:
8719:
8690:-mixing with a sub-exponential decay rate
8624:
8623:
8619:
8610:
8609:
8605:
8578:
8572:
8571:
8547:
8531:
8525:
8435:
8382:
8380:
8357:
8332:-mixing with a sub-exponential decay rate
8267:
8254:
8248:
8247:
8229:
8206:
8193:
8187:
8095:
8082:
8076:
8075:
8057:
8046:
8033:
8020:
8014:
7961:
7960:
7956:
7940:
7939:
7930:
7907:
7879:
7878:
7869:
7863:
7830:
7814:
7797:
7796:
7775:
7769:
7768:
7765:
7743:
7742:
7740:
7718:
7717:
7708:
7702:
7683:
7682:
7680:
7652:
7643:
7609:
7579:
7559:
7539:
7511:
7474:
7434:
7424:
7415:
7386:
7361:
7348:
7339:
7288:
7267:
7262:
7256:
7219:
7188:
7177:
7158:
7150:
7111:
7098:
7097:
7084:
7083:
7061:
7060:
7055:
7030:
7003:
6956:
6955:
6946:
6945:
6934:
6892:
6882:
6873:
6816:
6787:
6781:
6755:
6749:
6701:
6695:
6664:
6636:
6614:
6577:
6576:
6552:
6543:
6513:
6436:
6430:
6385:
6379:
6340:
6301:
6251:
6250:
6249:
6243:
6220:
6219:
6218:
6212:
6190:
6166:
6127:
6120:, and therefore it is strongly mixing on
6104:
6103:
6083:
6053:
6021:
6020:
6003:
5985:
5973:
5963:
5951:
5928:
5924:
5901:
5879:
5833:
5656:
5637:
5603:
5597:
5529:
5520:
5472:
5463:
5425:
5361:
5321:
5281:
5188:
5132:
5091:
5053:
5025:
5005:
4984:
4972:
4950:
4928:
4908:
4887:
4875:
4837:
4806:
4800:
4767:
4758:
4744:
4735:
4721:
4709:
4693:
4677:
4671:
4630:
4618:
4594:
4585:
4579:
4550:
4540:
4525:
4487:
4475:
4461:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
4437:
4436:
4425:
4389:
4380:
4366:
4357:
4343:
4331:
4315:
4294:
4283:
4269:
4257:
4251:
4210:
4198:
4178:
4166:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
4142:
4141:
4130:
4083:
4078:
4066:
4057:
4044:
4022:
4011:
3997:
3979:
3973:
3946:
3937:
3931:
3902:
3892:
3877:
3841:
3829:
3817:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
3793:
3792:
3781:
3741:
3670:
3654:
3648:
3621:
3595:
3569:
3500:
3484:
3478:
3458:
3423:
3422:
3420:
3396:
3392:
3391:
3388:
3361:
3318:
3242:
3154:
3142:
3112:
3034:
3014:
2936:
2913:
2847:
2829:
2809:
2738:
2705:
2694:
2680:
2668:
2662:
2627:
2621:
2597:
2591:
2522:
2492:
2486:
2462:
2461:
2447:
2419:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
2395:
2394:
2383:
2355:
2335:
2308:
2302:
2278:
2272:
2245:
2239:
2148:
2138:
2118:
2098:
2078:
2058:
2038:
2018:
1977:
1957:
1937:
1917:
1897:
1877:
1806:
1782:
1776:
1750:
1730:
1710:
1690:
1669:
1639:
1638:
1618:
1594:
1572:
1545:
1521:
1479:
1473:
1445:
1417:
1397:
1374:
1373:
1359:
1335:
1313:
1285:
1253:
1225:
1201:
1181:
1151:
1141:
1130:
1124:
1104:
1083:
1082:
1074:
1051:
1018:
1012:
976:
949:
921:
898:
897:
888:
887:
885:
858:
857:
836:
830:
802:
779:
778:
769:
768:
756:
750:
722:
676:
648:
624:
595:
563:
558:will in general be a deformed version of
534:
508:
476:
433:
432:
430:
404:
377:
376:
374:
343:
315:
289:
288:
286:
264:
244:
217:
200:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
176:
175:
164:
8819:A strictly stationary Markov process is
8307:-mixing with exponential decay rate" if
8135:-mixing with exponential decay rate” if
7242:{\displaystyle -\infty <s<\infty }
4565:{\displaystyle (f\circ T^{n})_{n\geq 0}}
4468:is strongly mixing if, for any function
3917:{\displaystyle (f\circ T^{n})_{n\geq 0}}
2231:), even as they use different notation.
2053:will also be 10% of the grand total. If
1119:over a long period of time (that is, if
89:is an abstract concept originating from
68:
8938:
8863:Ergodic Problems of Classical Mechanics
7554:and all events, the events before time
6725:
6409:
4173:is weakly mixing if, for any functions
3443:union of sets in it. Compare this with
1648:{\displaystyle A,B,C\in {\mathcal {A}}}
1547:
1503:
1169:{\displaystyle \cup _{k=1}^{n}T^{k}(A)}
1000:{\displaystyle \mu (A)=\mu (T^{-1}(A))}
8663:-mixing with an exponential decay rate
6260:{\displaystyle \Sigma ^{\mathbb {Z} }}
6229:{\displaystyle \Sigma ^{\mathbb {N} }}
6479:. A related idea is expressed by the
6076:, therefore it is strongly mixing on
5589:may be defined as a system for which
5183:on the Cartesian product by defining
5127:one can construct a dynamical system
5048:Given two measured dynamical systems
2647:being the continuous-time parameter.
2471:{\displaystyle A,B\in {\mathcal {A}}}
1383:{\displaystyle A,B\in {\mathcal {A}}}
1240:behaves in this way, the system is a
7:
8868:Manfred Einsiedler and Thomas Ward,
7892:{\displaystyle L^{2}(\mathbb {Q} ).}
3616:Extend the mixing equation from all
2586:, the same definition applies, with
2133:is 1% of the total volume. That is,
1046:describes all the pieces-parts that
118:measure-preserving dynamical systems
7728:{\displaystyle L^{2}(\mathbb {Q} )}
5865:{\displaystyle n\in \{2,3,\dots \}}
4657:{\displaystyle g\in L^{2}(X,\mu ),}
4237:{\displaystyle g\in L^{2}(X,\mu ),}
2905:in the usual sense, weak mixing if
2428:measure-preserving dynamical system
1092:{\displaystyle A\in {\mathcal {A}}}
155:measure-preserving dynamical system
8991:Chen, Hansen & Carrasco (2010)
8803:
8443:
8230:
7617:
7519:
7450:
7438:
7332:, i.e. the σ-algebra generated by
7311:
7293:
7236:
7224:
7189:
7154:
7134:
7122:
6939:
6908:
6896:
6836:{\displaystyle \Vert g\Vert >N}
6246:
6215:
6192:
6168:
5937:{\displaystyle T(x)=nx{\bmod {1}}}
5616:
4813:
4684:
4511:{\displaystyle f\in L^{2}(X,\mu )}
4264:
3986:
3865:{\displaystyle f\in L^{2}(X,\mu )}
3380:and the Lebesgue measurable sets.
3211:
2675:
2499:
1789:
25:
8835:-mixing ones, so if a process is
8714:for some non-increasing function
8352:for some non-increasing function
6987:. These cylinder sets generate a
5760: = 2,3,4,... is called
4604:{\displaystyle \int _{X}f\,d\mu }
3956:{\displaystyle \int _{X}f\,d\mu }
2650:A dynamical system is said to be
8489:-mixing, it will necessarily be
6852:is one that has no non-constant
6332:if, for every pair of non-empty
5458:is weakly mixing if and only if
3453:For Lebesgue measure spaces, if
3405:{\displaystyle \mathbb {R} ^{n}}
3099:Weak mixing implies ergodicity.
1308:, and not just between some set
132:, with the last not requiring a
34:
7497:{\displaystyle \alpha (s)\to 0}
4870:, so that the random variables
1932:in time, so that one is mixing
797:; it will map any given subset
8800:
8771:
8762:
8756:
8593:
8587:
8485:, therefore if the process is
8440:
8409:
8400:
8394:
7944:
7936:
7883:
7875:
7842:
7820:
7807:
7801:
7790:
7784:
7722:
7714:
7658:
7645:
7614:
7516:
7488:
7485:
7479:
7431:
7417:
7112:
7108:
7102:
7094:
7088:
7077:
7065:
7056:
7041:
7035:
6960:
6936:
6889:
6875:
6864:Mixing in stochastic processes
6744:For a continuous-time system,
6713:
6707:
6564:
6558:
6471:(a continuous linear map on a
6397:
6391:
6312:
6141:
6129:
6097:
6085:
5912:
5906:
5745:similarly. A system which is
5729:holds for all measurable sets
5713:
5707:
5701:
5695:
5689:
5683:
5674:
5624:
5613:
5559:
5522:
5502:
5465:
5445:
5427:
5399:
5363:
5341:
5323:
5301:
5283:
5253:
5250:
5244:
5235:
5229:
5223:
5217:
5205:
5202:
5190:
5170:
5134:
5111:
5093:
5073:
5055:
4849:
4824:
4810:
4681:
4648:
4636:
4547:
4527:
4505:
4493:
4455:
4427:
4261:
4228:
4216:
4160:
4132:
4101:
4089:
4074:
3993:
3983:
3899:
3879:
3859:
3847:
3811:
3783:
3721:
3715:
3709:
3703:
3694:
3685:
3679:
3663:
3551:
3545:
3539:
3533:
3524:
3515:
3509:
3493:
3432:{\displaystyle {\mathcal {C}}}
3289:
3283:
3277:
3271:
3262:
3250:
3214:
3208:
3199:
3187:
3178:
3169:
3163:
3147:
3075:
3069:
3063:
3057:
3051:
2983:
2975:
2969:
2963:
2957:
2948:
2923:
2889:
2886:
2880:
2874:
2868:
2859:
2834:
2777:
2771:
2765:
2759:
2750:
2725:
2672:
2563:
2557:
2551:
2545:
2496:
2413:
2385:
2199:
2193:
2187:
2181:
2161:
2155:
2006:{\displaystyle \mu (A)\mu (B)}
2000:
1994:
1988:
1982:
1952:while holding the test volume
1853:
1847:
1841:
1835:
1786:
1491:
1485:
1352:. That is, given any two sets
1163:
1157:
1033:
1027:
994:
991:
985:
969:
960:
954:
926:
894:
851:
845:
775:
702:
696:
687:
681:
606:
600:
545:
539:
487:
442:{\displaystyle {\mathcal {A}}}
386:{\displaystyle {\mathcal {A}}}
354:
348:
298:{\displaystyle {\mathcal {A}}}
194:
166:
1:
8921:10.1016/j.jeconom.2009.10.001
8865:, (1968) W. A. Benjamin, Inc.
7675:with stationary distribution
6467:, a topologically transitive
6236:or the doubly-infinite space
5872:, the "shift to left in base
5044:Products of dynamical systems
1273:ergodic decomposition theorem
619:must have the same volume as
8809:{\displaystyle t\to \infty }
8449:{\displaystyle t\to \infty }
7750:{\displaystyle \mathbb {Q} }
7690:{\displaystyle \mathbb {Q} }
7623:{\displaystyle s\to \infty }
7525:{\displaystyle s\to \infty }
6983:of this topology are called
6796:{\displaystyle \varphi _{g}}
6360:{\displaystyle A,B\subset X}
4993:{\displaystyle f\circ T^{n}}
4896:{\displaystyle f\circ T^{n}}
3356:. Next, we define a measure
2434:being the time-evolution or
1553:{\displaystyle \varnothing }
1216:), the system is said to be
307:, and the size of any given
73:Repeated application of the
8503:, the process may still be
8493:-mixing too. However, when
5120:{\displaystyle (Y,\nu ,S),}
3590:in a covering family, then
3473:is measure-preserving, and
2438:. The system is said to be
2374:Mixing in dynamical systems
914:is the inverse of some map
9025:
8861:V. I. Arnold and A. Avez,
7634:Mixing in Markov processes
7574:and the events after time
6653:, there exists an integer
6367:, there exists an integer
5451:{\displaystyle (X,\mu ,T)}
5347:{\displaystyle (Y,\nu ,S)}
5307:{\displaystyle (X,\mu ,T)}
5079:{\displaystyle (X,\mu ,T)}
1244:, placed in contrast to a
816:{\displaystyle A\subset X}
522:{\displaystyle A\subset X}
329:{\displaystyle A\subset X}
7276:{\displaystyle X_{a}^{b}}
7020:strong mixing coefficient
4613:, i.e., for any function
3235:, so weak mixing implies
1099:eventually visits all of
1039:{\displaystyle T^{-1}(A)}
880:(measure-preserving) map
708:{\displaystyle T(x)=T(y)}
425:). Thus, conventionally,
151:industrial process mixing
60:Proposed since July 2024.
8839:-mixing it will also be
6473:topological vector space
6330:topologically transitive
6321:{\displaystyle f:X\to X}
5803:Kolmogorov automorphisms
717:. Worse, a single point
496:{\displaystyle T:X\to X}
8899:Journal of Econometrics
7664:{\displaystyle (X_{t})}
7011:{\displaystyle \alpha }
6927:on a probability space
6850:weak topological mixing
6605:A system is said to be
6469:bounded linear operator
6198:{\displaystyle \Sigma }
6174:{\displaystyle \Sigma }
2323:{\displaystyle T^{-1}A}
2260:{\displaystyle T^{-n}A}
662:{\displaystyle x\neq y}
360:{\displaystyle \mu (A)}
108:The concept appears in
8810:
8781:
8728:
8644:The process is called
8635:
8450:
8419:
8366:
8289:The process is called
8280:
8117:The process is called
8108:
7980:
7893:
7849:
7751:
7729:
7691:
7665:
7624:
7594:
7568:
7548:
7526:
7498:
7459:
7399:
7318:
7277:
7243:
7203:
7012:
6967:
6917:
6837:
6797:
6765:
6735:
6679:
6678:{\displaystyle n>N}
6645:
6623:
6588:
6528:
6527:{\displaystyle x\in X}
6446:
6416:
6361:
6322:
6261:
6230:
6199:
6175:
6148:
6112:
6068:
5938:
5888:
5866:
5720:
5587:strong 3-mixing system
5572:is also weakly mixing.
5566:
5509:
5452:
5406:
5348:
5308:
5263:
5177:
5121:
5080:
5034:
5020:become independent as
5014:
4994:
4959:
4937:
4917:
4897:
4862:
4781:
4658:
4605:
4566:
4512:
4462:
4411:
4305:
4238:
4187:
4167:
4116:
4033:
3957:
3918:
3866:
3818:
3756:
3728:
3636:
3604:
3584:
3558:
3467:
3433:
3406:
3370:
3333:
3305:
3227:
3127:
3082:
3009:sense, and ergodic if
2996:
2899:
2818:
2795:
2716:
2637:
2610:
2609:{\displaystyle T^{-n}}
2573:
2472:
2420:
2364:
2344:
2324:
2291:
2290:{\displaystyle T^{n}A}
2261:
2209:
2127:
2107:
2087:
2067:
2047:
2027:
2007:
1966:
1946:
1926:
1906:
1886:
1863:
1759:
1739:
1719:
1699:
1678:
1649:
1603:
1581:
1554:
1530:
1510:
1460:
1459:{\displaystyle n>N}
1432:
1406:
1384:
1344:
1322:
1300:
1262:
1234:
1210:
1190:
1170:
1113:
1093:
1060:
1040:
1001:
936:
935:{\displaystyle X\to X}
908:
868:
817:
789:
737:
736:{\displaystyle x\in X}
709:
663:
633:
613:
572:
552:
523:
497:
443:
413:
387:
361:
330:
299:
273:
253:
226:
201:
78:
8811:
8782:
8729:
8636:
8451:
8420:
8367:
8281:
8109:
7981:
7894:
7850:
7752:
7730:
7692:
7666:
7625:
7595:
7569:
7549:
7527:
7499:
7460:
7400:
7319:
7278:
7244:
7204:
7013:
6968:
6918:
6838:
6798:
6766:
6764:{\displaystyle f^{n}}
6736:
6680:
6657:, such that, for all
6646:
6624:
6589:
6529:
6495:complete metric space
6447:
6445:{\displaystyle f^{n}}
6417:
6362:
6323:
6269:. In both cases, the
6262:
6231:
6200:
6176:
6149:
6113:
6069:
5939:
5889:
5867:
5721:
5567:
5510:
5453:
5407:
5349:
5309:
5264:
5178:
5122:
5081:
5035:
5015:
4995:
4960:
4938:
4923:become orthogonal as
4918:
4898:
4863:
4782:
4659:
4606:
4567:
4513:
4463:
4412:
4279:
4239:
4188:
4168:
4117:
4007:
3958:
3919:
3867:
3819:
3757:
3729:
3637:
3605:
3585:
3559:
3468:
3434:
3407:
3371:
3334:
3306:
3228:
3128:
3083:
2997:
2900:
2819:
2796:
2690:
2638:
2636:{\displaystyle T_{g}}
2611:
2574:
2473:
2421:
2365:
2345:
2325:
2292:
2262:
2234:The reason for using
2210:
2128:
2108:
2088:
2068:
2048:
2028:
2008:
1967:
1947:
1927:
1907:
1887:
1864:
1760:
1740:
1720:
1700:
1679:
1650:
1604:
1582:
1555:
1531:
1529:{\displaystyle \cap }
1511:
1461:
1433:
1407:
1385:
1345:
1323:
1301:
1263:
1248:, where some subsets
1235:
1211:
1191:
1171:
1114:
1094:
1061:
1041:
1002:
937:
909:
869:
818:
790:
738:
710:
664:
634:
614:
573:
553:
524:
498:
444:
423:Banach–Tarski paradox
414:
388:
362:
331:
300:
274:
254:
227:
202:
95:thermodynamic process
72:
8794:
8741:
8727:{\displaystyle \xi }
8718:
8524:
8515:-mixing coefficients
8434:
8379:
8365:{\displaystyle \xi }
8356:
8186:
8170:-mixing coefficients
8013:
7997:-mixing coefficients
7906:
7862:
7764:
7739:
7701:
7679:
7642:
7608:
7578:
7558:
7538:
7510:
7473:
7414:
7338:
7287:
7255:
7218:
7029:
7002:
6933:
6872:
6815:
6780:
6748:
6694:
6663:
6635:
6613:
6609:if, given open sets
6607:topologically mixing
6542:
6538:such that its orbit
6512:
6477:hypercyclic operator
6475:) is usually called
6429:
6378:
6339:
6300:
6242:
6211:
6189:
6165:
6126:
6082:
5950:
5900:
5878:
5832:
5784:Irrational rotations
5762:mixing of all orders
5596:
5519:
5462:
5424:
5360:
5320:
5280:
5187:
5131:
5090:
5052:
5024:
5004:
4971:
4949:
4927:
4907:
4874:
4799:
4670:
4617:
4578:
4572:converges weakly to
4524:
4474:
4424:
4250:
4197:
4177:
4129:
3972:
3930:
3876:
3828:
3780:
3740:
3647:
3620:
3594:
3568:
3477:
3457:
3419:
3387:
3369:{\displaystyle \mu }
3360:
3352:, to obtain all the
3317:
3241:
3141:
3111:
3013:
2912:
2828:
2824:is strong mixing if
2808:
2661:
2620:
2590:
2485:
2446:
2382:
2354:
2334:
2301:
2271:
2238:
2137:
2117:
2097:
2077:
2057:
2037:
2017:
1976:
1972:fixed. The product
1956:
1936:
1916:
1896:
1876:
1775:
1749:
1729:
1709:
1689:
1668:
1617:
1593:
1571:
1544:
1520:
1472:
1444:
1416:
1396:
1358:
1334:
1312:
1284:
1252:
1224:
1200:
1180:
1123:
1103:
1073:
1050:
1011:
948:
920:
884:
829:
801:
749:
721:
675:
647:
623:
612:{\displaystyle T(A)}
594:
562:
551:{\displaystyle T(A)}
533:
507:
503:. Given some subset
475:
429:
403:
373:
342:
314:
285:
263:
252:{\displaystyle \mu }
243:
216:
163:
145:Informal explanation
114:stochastic processes
7600:tend towards being
7593:{\displaystyle t+s}
7272:
7193:
7163:
6771:is replaced by the
6534:, that is, a point
5420:A dynamical system
5276:A dynamical system
4420:A dynamical system
4125:A dynamical system
3755:{\displaystyle A,B}
3734:for all measurable
3635:{\displaystyle A,B}
3583:{\displaystyle A,B}
3332:{\displaystyle A,B}
3126:{\displaystyle A,B}
1892:serves to separate
1872:The time parameter
1431:{\displaystyle A,B}
1412:such that, for all
1299:{\displaystyle A,B}
1242:conservative system
1146:
586:, both inspired by
8969:, (2011) Springer
8886:, (2006) Springer
8884:Probability Theory
8872:, (2011) Springer
8806:
8777:
8724:
8631:
8564:
8446:
8415:
8362:
8276:
8242:
8104:
8070:
7976:
7889:
7845:
7747:
7725:
7687:
7671:were a stationary
7661:
7620:
7590:
7564:
7544:
7522:
7494:
7455:
7395:
7314:
7273:
7258:
7239:
7199:
7173:
7146:
7008:
6998:Define a function
6963:
6925:stochastic process
6913:
6833:
6793:
6761:
6731:
6675:
6641:
6619:
6584:
6524:
6442:
6412:
6357:
6318:
6293:of the system. A
6281:Topological mixing
6257:
6226:
6195:
6171:
6144:
6108:
6064:
6010:
5980:
5934:
5884:
5862:
5823:negative curvature
5716:
5620:
5562:
5505:
5448:
5402:
5344:
5304:
5259:
5173:
5117:
5076:
5030:
5010:
4990:
4955:
4933:
4913:
4893:
4858:
4817:
4777:
4688:
4654:
4601:
4562:
4508:
4458:
4407:
4268:
4234:
4183:
4163:
4112:
3990:
3953:
3914:
3862:
3814:
3752:
3724:
3659:
3632:
3610:is strong mixing.
3600:
3580:
3554:
3489:
3463:
3429:
3415:A covering family
3402:
3366:
3329:
3301:
3223:
3123:
3078:
2992:
2895:
2814:
2791:
2679:
2633:
2606:
2569:
2503:
2468:
2416:
2360:
2340:
2320:
2287:
2257:
2225:probability theory
2205:
2153:
2123:
2103:
2083:
2063:
2043:
2023:
2003:
1962:
1942:
1922:
1902:
1882:
1859:
1793:
1755:
1735:
1715:
1695:
1674:
1645:
1599:
1577:
1550:
1526:
1506:
1456:
1428:
1402:
1380:
1340:
1318:
1296:
1258:
1246:dissipative system
1230:
1206:
1186:
1176:approaches all of
1166:
1126:
1109:
1089:
1056:
1036:
997:
932:
904:
864:
813:
785:
733:
705:
659:
629:
609:
568:
548:
519:
493:
439:
409:
383:
357:
326:
295:
269:
249:
222:
197:
130:topological mixing
79:
8975:978-0-85729-020-5
8892:978-1-84800-047-6
8878:978-0-85729-020-5
8769:
8543:
8407:
8202:
8029:
7567:{\displaystyle t}
7547:{\displaystyle t}
6644:{\displaystyle B}
6622:{\displaystyle A}
6507:hypercyclic point
6289:, using only the
6009:
5979:
5887:{\displaystyle n}
5819:compact manifolds
5741:. We can define
5599:
5033:{\displaystyle n}
5013:{\displaystyle g}
4958:{\displaystyle g}
4936:{\displaystyle n}
4916:{\displaystyle g}
4802:
4673:
4277:
4253:
4186:{\displaystyle f}
4005:
3975:
3650:
3603:{\displaystyle T}
3480:
3466:{\displaystyle T}
3343:Covering families
2817:{\displaystyle T}
2688:
2664:
2488:
2363:{\displaystyle B}
2343:{\displaystyle B}
2229:Kolmogorov axioms
2152:
2126:{\displaystyle B}
2106:{\displaystyle A}
2086:{\displaystyle B}
2066:{\displaystyle A}
2046:{\displaystyle A}
2026:{\displaystyle B}
1965:{\displaystyle B}
1945:{\displaystyle A}
1925:{\displaystyle B}
1905:{\displaystyle A}
1885:{\displaystyle n}
1778:
1758:{\displaystyle B}
1738:{\displaystyle B}
1718:{\displaystyle A}
1698:{\displaystyle B}
1677:{\displaystyle A}
1602:{\displaystyle B}
1580:{\displaystyle A}
1405:{\displaystyle N}
1343:{\displaystyle X}
1321:{\displaystyle A}
1261:{\displaystyle A}
1233:{\displaystyle A}
1209:{\displaystyle n}
1189:{\displaystyle X}
1112:{\displaystyle X}
1059:{\displaystyle A}
632:{\displaystyle A}
590:-making. The set
571:{\displaystyle A}
412:{\displaystyle X}
272:{\displaystyle X}
225:{\displaystyle X}
103:industrial mixing
67:
66:
62:
16:(Redirected from
9016:
8993:
8988:
8977:
8963:
8957:
8943:
8932:
8914:
8815:
8813:
8812:
8807:
8786:
8784:
8783:
8778:
8770:
8765:
8745:
8733:
8731:
8730:
8725:
8713:
8706:
8684:
8677:
8657:
8640:
8638:
8637:
8632:
8627:
8618:
8614:
8613:
8583:
8582:
8577:
8576:
8563:
8536:
8535:
8502:
8484:
8457:
8455:
8453:
8452:
8447:
8424:
8422:
8421:
8416:
8408:
8403:
8383:
8371:
8369:
8368:
8363:
8351:
8326:
8319:
8302:
8285:
8283:
8282:
8277:
8272:
8271:
8259:
8258:
8253:
8252:
8241:
8234:
8233:
8198:
8197:
8172:of the process {
8154:
8147:
8130:
8113:
8111:
8110:
8105:
8100:
8099:
8087:
8086:
8081:
8080:
8069:
8062:
8061:
8025:
8024:
7999:of the process {
7985:
7983:
7982:
7977:
7975:
7971:
7964:
7943:
7935:
7934:
7898:
7896:
7895:
7890:
7882:
7874:
7873:
7854:
7852:
7851:
7846:
7835:
7834:
7819:
7818:
7800:
7780:
7779:
7774:
7773:
7756:
7754:
7753:
7748:
7746:
7734:
7732:
7731:
7726:
7721:
7713:
7712:
7696:
7694:
7693:
7688:
7686:
7670:
7668:
7667:
7662:
7657:
7656:
7629:
7627:
7626:
7621:
7599:
7597:
7596:
7591:
7573:
7571:
7570:
7565:
7553:
7551:
7550:
7545:
7533:
7531:
7529:
7528:
7523:
7503:
7501:
7500:
7495:
7464:
7462:
7461:
7456:
7454:
7453:
7429:
7428:
7406:
7404:
7402:
7401:
7396:
7391:
7390:
7372:
7371:
7353:
7352:
7323:
7321:
7320:
7315:
7282:
7280:
7279:
7274:
7271:
7266:
7250:
7248:
7246:
7245:
7240:
7208:
7206:
7205:
7200:
7198:
7194:
7192:
7187:
7162:
7157:
7115:
7101:
7087:
7064:
7059:
7017:
7015:
7014:
7009:
6977:product topology
6974:
6972:
6970:
6969:
6964:
6959:
6951:
6950:
6922:
6920:
6919:
6914:
6912:
6911:
6887:
6886:
6844:
6842:
6840:
6839:
6834:
6804:
6802:
6800:
6799:
6794:
6792:
6791:
6770:
6768:
6767:
6762:
6760:
6759:
6740:
6738:
6737:
6732:
6706:
6705:
6686:
6684:
6682:
6681:
6676:
6652:
6650:
6648:
6647:
6642:
6628:
6626:
6625:
6620:
6593:
6591:
6590:
6585:
6580:
6557:
6556:
6533:
6531:
6530:
6525:
6451:
6449:
6448:
6443:
6441:
6440:
6421:
6419:
6418:
6413:
6390:
6389:
6366:
6364:
6363:
6358:
6327:
6325:
6324:
6319:
6268:
6266:
6264:
6263:
6258:
6256:
6255:
6254:
6235:
6233:
6232:
6227:
6225:
6224:
6223:
6206:
6204:
6202:
6201:
6196:
6182:
6180:
6178:
6177:
6172:
6155:
6153:
6151:
6150:
6147:{\displaystyle }
6145:
6119:
6117:
6115:
6114:
6109:
6107:
6075:
6073:
6071:
6070:
6065:
6063:
6059:
6058:
6057:
6024:
6016:
6012:
6011:
6008:
6007:
5998:
5987:
5981:
5978:
5977:
5965:
5943:
5941:
5940:
5935:
5933:
5932:
5895:
5893:
5891:
5890:
5885:
5871:
5869:
5868:
5863:
5795:Arnold's cat map
5771:-mixing implies
5725:
5723:
5722:
5717:
5670:
5669:
5645:
5644:
5619:
5571:
5569:
5568:
5563:
5534:
5533:
5514:
5512:
5511:
5506:
5477:
5476:
5457:
5455:
5454:
5449:
5412:is also ergodic.
5411:
5409:
5408:
5403:
5355:
5353:
5351:
5350:
5345:
5313:
5311:
5310:
5305:
5268:
5266:
5265:
5260:
5182:
5180:
5179:
5174:
5126:
5124:
5123:
5118:
5085:
5083:
5082:
5077:
5039:
5037:
5036:
5031:
5019:
5017:
5016:
5011:
4999:
4997:
4996:
4991:
4989:
4988:
4966:
4964:
4962:
4961:
4956:
4942:
4940:
4939:
4934:
4922:
4920:
4919:
4914:
4902:
4900:
4899:
4894:
4892:
4891:
4869:
4867:
4865:
4864:
4859:
4842:
4841:
4816:
4786:
4784:
4783:
4778:
4763:
4762:
4740:
4739:
4714:
4713:
4698:
4697:
4687:
4663:
4661:
4660:
4655:
4635:
4634:
4612:
4610:
4608:
4607:
4602:
4590:
4589:
4571:
4569:
4568:
4563:
4561:
4560:
4545:
4544:
4519:
4517:
4515:
4514:
4509:
4492:
4491:
4467:
4465:
4464:
4459:
4442:
4441:
4416:
4414:
4413:
4408:
4400:
4396:
4385:
4384:
4362:
4361:
4336:
4335:
4320:
4319:
4304:
4293:
4278:
4270:
4267:
4243:
4241:
4240:
4235:
4215:
4214:
4192:
4190:
4189:
4184:
4172:
4170:
4169:
4164:
4147:
4146:
4121:
4119:
4118:
4113:
4105:
4104:
4088:
4087:
4077:
4073:
4062:
4061:
4049:
4048:
4032:
4021:
4006:
3998:
3989:
3964:
3962:
3960:
3959:
3954:
3942:
3941:
3923:
3921:
3920:
3915:
3913:
3912:
3897:
3896:
3871:
3869:
3868:
3863:
3846:
3845:
3823:
3821:
3820:
3815:
3798:
3797:
3763:
3761:
3759:
3758:
3753:
3733:
3731:
3730:
3725:
3678:
3677:
3658:
3641:
3639:
3638:
3633:
3609:
3607:
3606:
3601:
3589:
3587:
3586:
3581:
3563:
3561:
3560:
3555:
3508:
3507:
3488:
3472:
3470:
3469:
3464:
3445:base in topology
3438:
3436:
3435:
3430:
3428:
3427:
3411:
3409:
3408:
3403:
3401:
3400:
3395:
3378:Lebesgue measure
3375:
3373:
3372:
3367:
3338:
3336:
3335:
3330:
3312:
3310:
3308:
3307:
3302:
3234:
3232:
3230:
3229:
3224:
3162:
3161:
3134:
3132:
3130:
3129:
3124:
3087:
3085:
3084:
3079:
3050:
3046:
3042:
3041:
3001:
2999:
2998:
2993:
2982:
2978:
2944:
2943:
2904:
2902:
2901:
2896:
2855:
2854:
2823:
2821:
2820:
2815:
2804:In other words,
2800:
2798:
2797:
2792:
2784:
2780:
2746:
2745:
2715:
2704:
2689:
2681:
2678:
2642:
2640:
2639:
2634:
2632:
2631:
2615:
2613:
2612:
2607:
2605:
2604:
2578:
2576:
2575:
2570:
2538:
2534:
2530:
2529:
2502:
2477:
2475:
2474:
2469:
2467:
2466:
2425:
2423:
2422:
2417:
2400:
2399:
2369:
2367:
2366:
2361:
2349:
2347:
2346:
2341:
2329:
2327:
2326:
2321:
2316:
2315:
2296:
2294:
2293:
2288:
2283:
2282:
2266:
2264:
2263:
2258:
2253:
2252:
2214:
2212:
2211:
2206:
2174:
2170:
2154:
2150:
2132:
2130:
2129:
2124:
2112:
2110:
2109:
2104:
2092:
2090:
2089:
2084:
2072:
2070:
2069:
2064:
2052:
2050:
2049:
2044:
2032:
2030:
2029:
2024:
2012:
2010:
2009:
2004:
1971:
1969:
1968:
1963:
1951:
1949:
1948:
1943:
1931:
1929:
1928:
1923:
1911:
1909:
1908:
1903:
1891:
1889:
1888:
1883:
1868:
1866:
1865:
1860:
1828:
1824:
1814:
1813:
1792:
1764:
1762:
1761:
1756:
1744:
1742:
1741:
1736:
1724:
1722:
1721:
1716:
1704:
1702:
1701:
1696:
1683:
1681:
1680:
1675:
1656:
1654:
1652:
1651:
1646:
1644:
1643:
1610:
1608:
1606:
1605:
1600:
1586:
1584:
1583:
1578:
1559:
1557:
1556:
1551:
1538:set intersection
1535:
1533:
1532:
1527:
1515:
1513:
1512:
1507:
1484:
1483:
1467:
1465:
1463:
1462:
1457:
1437:
1435:
1434:
1429:
1411:
1409:
1408:
1403:
1391:
1389:
1387:
1386:
1381:
1379:
1378:
1351:
1349:
1347:
1346:
1341:
1327:
1325:
1324:
1319:
1307:
1305:
1303:
1302:
1297:
1267:
1265:
1264:
1259:
1239:
1237:
1236:
1231:
1215:
1213:
1212:
1207:
1195:
1193:
1192:
1187:
1175:
1173:
1172:
1167:
1156:
1155:
1145:
1140:
1118:
1116:
1115:
1110:
1098:
1096:
1095:
1090:
1088:
1087:
1065:
1063:
1062:
1057:
1045:
1043:
1042:
1037:
1026:
1025:
1006:
1004:
1003:
998:
984:
983:
943:
941:
939:
938:
933:
913:
911:
910:
905:
903:
902:
893:
892:
875:
873:
871:
870:
865:
863:
862:
844:
843:
822:
820:
819:
814:
796:
794:
792:
791:
786:
784:
783:
774:
773:
764:
763:
742:
740:
739:
734:
716:
714:
712:
711:
706:
668:
666:
665:
660:
638:
636:
635:
630:
618:
616:
615:
610:
577:
575:
574:
569:
557:
555:
554:
549:
528:
526:
525:
520:
502:
500:
499:
494:
448:
446:
445:
440:
438:
437:
420:
418:
416:
415:
410:
392:
390:
389:
384:
382:
381:
368:
366:
364:
363:
358:
335:
333:
332:
327:
306:
304:
302:
301:
296:
294:
293:
278:
276:
275:
270:
258:
256:
255:
250:
231:
229:
228:
223:
208:
206:
204:
203:
198:
181:
180:
58:
50:Mixing (physics)
38:
37:
30:
21:
9024:
9023:
9019:
9018:
9017:
9015:
9014:
9013:
8999:
8998:
8997:
8996:
8989:
8980:
8964:
8960:
8944:
8940:
8935:
8912:10.1.1.597.8777
8896:
8857:
8792:
8791:
8746:
8739:
8738:
8716:
8715:
8708:
8698:
8693:
8679:
8671:
8666:
8652:
8570:
8569:
8565:
8527:
8522:
8521:
8499:
8494:
8482:
8475:
8470:
8432:
8431:
8429:
8384:
8377:
8376:
8354:
8353:
8340:
8335:
8321:
8313:
8308:
8297:
8263:
8246:
8225:
8189:
8184:
8183:
8177:
8160:
8149:
8141:
8136:
8125:
8091:
8074:
8053:
8016:
8011:
8010:
8004:
7926:
7919:
7915:
7904:
7903:
7865:
7860:
7859:
7826:
7810:
7767:
7762:
7761:
7737:
7736:
7704:
7699:
7698:
7677:
7676:
7648:
7640:
7639:
7636:
7606:
7605:
7576:
7575:
7556:
7555:
7536:
7535:
7508:
7507:
7505:
7471:
7470:
7467:strongly mixing
7430:
7420:
7412:
7411:
7382:
7357:
7344:
7336:
7335:
7333:
7285:
7284:
7253:
7252:
7216:
7215:
7213:
7054:
7050:
7027:
7026:
7000:
6999:
6993:Borel σ-algebra
6931:
6930:
6928:
6888:
6878:
6870:
6869:
6866:
6813:
6812:
6810:
6783:
6778:
6777:
6775:
6751:
6746:
6745:
6697:
6692:
6691:
6661:
6660:
6658:
6633:
6632:
6630:
6611:
6610:
6548:
6540:
6539:
6510:
6509:
6465:operator theory
6432:
6427:
6426:
6381:
6376:
6375:
6337:
6336:
6298:
6297:
6283:
6245:
6240:
6239:
6237:
6214:
6209:
6208:
6187:
6186:
6184:
6163:
6162:
6160:
6124:
6123:
6121:
6080:
6079:
6077:
6049:
5999:
5988:
5969:
5962:
5958:
5957:
5953:
5948:
5947:
5945:
5898:
5897:
5876:
5875:
5873:
5830:
5829:
5781:
5743:strong k-mixing
5652:
5633:
5594:
5593:
5583:strong 2-mixing
5579:
5577:Generalizations
5525:
5517:
5516:
5468:
5460:
5459:
5422:
5421:
5358:
5357:
5318:
5317:
5315:
5278:
5277:
5185:
5184:
5129:
5128:
5088:
5087:
5050:
5049:
5046:
5022:
5021:
5002:
5001:
4980:
4969:
4968:
4947:
4946:
4944:
4925:
4924:
4905:
4904:
4883:
4872:
4871:
4833:
4797:
4796:
4794:
4754:
4731:
4705:
4689:
4668:
4667:
4626:
4615:
4614:
4581:
4576:
4575:
4573:
4546:
4536:
4522:
4521:
4520:, the sequence
4483:
4472:
4471:
4469:
4422:
4421:
4376:
4353:
4327:
4311:
4310:
4306:
4248:
4247:
4206:
4195:
4194:
4175:
4174:
4127:
4126:
4079:
4053:
4040:
3996:
3992:
3991:
3970:
3969:
3933:
3928:
3927:
3925:
3898:
3888:
3874:
3873:
3872:, the sequence
3837:
3826:
3825:
3778:
3777:
3773:
3738:
3737:
3735:
3666:
3645:
3644:
3618:
3617:
3592:
3591:
3566:
3565:
3496:
3475:
3474:
3455:
3454:
3417:
3416:
3390:
3385:
3384:
3358:
3357:
3345:
3315:
3314:
3239:
3238:
3236:
3150:
3139:
3138:
3136:
3135:, then we have
3109:
3108:
3106:
3030:
3023:
3019:
3011:
3010:
2932:
2919:
2915:
2910:
2909:
2843:
2826:
2825:
2806:
2805:
2734:
2721:
2717:
2659:
2658:
2623:
2618:
2617:
2593:
2588:
2587:
2518:
2511:
2507:
2483:
2482:
2444:
2443:
2380:
2379:
2376:
2352:
2351:
2332:
2331:
2304:
2299:
2298:
2274:
2269:
2268:
2241:
2236:
2235:
2147:
2143:
2135:
2134:
2115:
2114:
2095:
2094:
2075:
2074:
2055:
2054:
2035:
2034:
2015:
2014:
1974:
1973:
1954:
1953:
1934:
1933:
1914:
1913:
1894:
1893:
1874:
1873:
1802:
1801:
1797:
1773:
1772:
1747:
1746:
1727:
1726:
1707:
1706:
1687:
1686:
1666:
1665:
1660:The concept of
1615:
1614:
1612:
1591:
1590:
1588:
1569:
1568:
1542:
1541:
1518:
1517:
1475:
1470:
1469:
1468:, one has that
1442:
1441:
1439:
1414:
1413:
1394:
1393:
1356:
1355:
1353:
1332:
1331:
1329:
1310:
1309:
1282:
1281:
1279:
1250:
1249:
1222:
1221:
1220:. If every set
1198:
1197:
1178:
1177:
1147:
1121:
1120:
1101:
1100:
1071:
1070:
1048:
1047:
1014:
1009:
1008:
972:
946:
945:
918:
917:
915:
882:
881:
832:
827:
826:
824:
799:
798:
752:
747:
746:
744:
719:
718:
673:
672:
670:
645:
644:
621:
620:
592:
591:
560:
559:
531:
530:
505:
504:
473:
472:
463:set complements
427:
426:
401:
400:
398:
371:
370:
340:
339:
337:
312:
311:
283:
282:
280:
261:
260:
241:
240:
214:
213:
161:
160:
158:
147:
63:
39:
35:
28:
23:
22:
15:
12:
11:
5:
9022:
9020:
9012:
9011:
9009:Ergodic theory
9001:
9000:
8995:
8994:
8978:
8958:
8937:
8936:
8934:
8933:
8905:(2): 155–169.
8894:
8882:Achim Klenke,
8880:
8866:
8858:
8856:
8853:
8805:
8802:
8799:
8788:
8787:
8776:
8773:
8768:
8764:
8761:
8758:
8755:
8752:
8749:
8723:
8696:
8669:
8642:
8641:
8630:
8626:
8622:
8617:
8612:
8608:
8604:
8601:
8598:
8595:
8592:
8589:
8586:
8581:
8575:
8568:
8562:
8559:
8556:
8553:
8550:
8546:
8542:
8539:
8534:
8530:
8497:
8480:
8473:
8469:-mixing ones:
8445:
8442:
8439:
8426:
8425:
8414:
8411:
8406:
8402:
8399:
8396:
8393:
8390:
8387:
8361:
8338:
8311:
8287:
8286:
8275:
8270:
8266:
8262:
8257:
8251:
8245:
8240:
8237:
8232:
8228:
8224:
8221:
8218:
8215:
8212:
8209:
8205:
8201:
8196:
8192:
8175:
8158:
8139:
8115:
8114:
8103:
8098:
8094:
8090:
8085:
8079:
8073:
8068:
8065:
8060:
8056:
8052:
8049:
8045:
8042:
8039:
8036:
8032:
8028:
8023:
8019:
8002:
7987:
7986:
7974:
7970:
7967:
7963:
7959:
7955:
7952:
7949:
7946:
7942:
7938:
7933:
7929:
7925:
7922:
7918:
7914:
7911:
7888:
7885:
7881:
7877:
7872:
7868:
7856:
7855:
7844:
7841:
7838:
7833:
7829:
7825:
7822:
7817:
7813:
7809:
7806:
7803:
7799:
7795:
7792:
7789:
7786:
7783:
7778:
7772:
7745:
7724:
7720:
7716:
7711:
7707:
7685:
7673:Markov process
7660:
7655:
7651:
7647:
7635:
7632:
7619:
7616:
7613:
7589:
7586:
7583:
7563:
7543:
7521:
7518:
7515:
7493:
7490:
7487:
7484:
7481:
7478:
7465:is said to be
7452:
7449:
7446:
7443:
7440:
7437:
7433:
7427:
7423:
7419:
7394:
7389:
7385:
7381:
7378:
7375:
7370:
7367:
7364:
7360:
7356:
7351:
7347:
7343:
7313:
7310:
7307:
7304:
7301:
7298:
7295:
7292:
7270:
7265:
7261:
7238:
7235:
7232:
7229:
7226:
7223:
7210:
7209:
7197:
7191:
7186:
7183:
7180:
7176:
7172:
7169:
7166:
7161:
7156:
7153:
7149:
7145:
7142:
7139:
7136:
7133:
7130:
7127:
7124:
7121:
7118:
7114:
7110:
7107:
7104:
7100:
7096:
7093:
7090:
7086:
7082:
7079:
7076:
7073:
7070:
7067:
7063:
7058:
7053:
7049:
7046:
7043:
7040:
7037:
7034:
7007:
6962:
6958:
6954:
6949:
6944:
6941:
6938:
6910:
6907:
6904:
6901:
6898:
6895:
6891:
6885:
6881:
6877:
6865:
6862:
6832:
6829:
6826:
6823:
6820:
6790:
6786:
6758:
6754:
6742:
6741:
6730:
6727:
6724:
6721:
6718:
6715:
6712:
6709:
6704:
6700:
6674:
6671:
6668:
6640:
6618:
6583:
6579:
6575:
6572:
6569:
6566:
6563:
6560:
6555:
6551:
6547:
6523:
6520:
6517:
6499:isolated point
6439:
6435:
6423:
6422:
6411:
6408:
6405:
6402:
6399:
6396:
6393:
6388:
6384:
6356:
6353:
6350:
6347:
6344:
6328:is said to be
6317:
6314:
6311:
6308:
6305:
6295:continuous map
6282:
6279:
6253:
6248:
6222:
6217:
6194:
6170:
6143:
6140:
6137:
6134:
6131:
6106:
6102:
6099:
6096:
6093:
6090:
6087:
6062:
6056:
6052:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6027:
6023:
6019:
6015:
6006:
6002:
5997:
5994:
5991:
5984:
5976:
5972:
5968:
5961:
5956:
5931:
5927:
5923:
5920:
5917:
5914:
5911:
5908:
5905:
5883:
5861:
5858:
5855:
5852:
5849:
5846:
5843:
5840:
5837:
5815:tangent bundle
5799:horseshoe maps
5780:
5777:
5727:
5726:
5715:
5712:
5709:
5706:
5703:
5700:
5697:
5694:
5691:
5688:
5685:
5682:
5679:
5676:
5673:
5668:
5665:
5662:
5659:
5655:
5651:
5648:
5643:
5640:
5636:
5632:
5629:
5626:
5623:
5618:
5615:
5612:
5609:
5606:
5602:
5578:
5575:
5574:
5573:
5561:
5558:
5555:
5552:
5549:
5546:
5543:
5540:
5537:
5532:
5528:
5524:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5475:
5471:
5467:
5447:
5444:
5441:
5438:
5435:
5432:
5429:
5414:
5413:
5401:
5398:
5395:
5392:
5389:
5386:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5303:
5300:
5297:
5294:
5291:
5288:
5285:
5258:
5255:
5252:
5249:
5246:
5243:
5240:
5237:
5234:
5231:
5228:
5225:
5222:
5219:
5216:
5213:
5210:
5207:
5204:
5201:
5198:
5195:
5192:
5172:
5169:
5166:
5163:
5160:
5157:
5154:
5151:
5148:
5145:
5142:
5139:
5136:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5045:
5042:
5029:
5009:
4987:
4983:
4979:
4976:
4954:
4932:
4912:
4890:
4886:
4882:
4879:
4857:
4854:
4851:
4848:
4845:
4840:
4836:
4832:
4829:
4826:
4823:
4820:
4815:
4812:
4809:
4805:
4788:
4787:
4776:
4773:
4770:
4766:
4761:
4757:
4753:
4750:
4747:
4743:
4738:
4734:
4730:
4727:
4724:
4720:
4717:
4712:
4708:
4704:
4701:
4696:
4692:
4686:
4683:
4680:
4676:
4653:
4650:
4647:
4644:
4641:
4638:
4633:
4629:
4625:
4622:
4600:
4597:
4593:
4588:
4584:
4559:
4556:
4553:
4549:
4543:
4539:
4535:
4532:
4529:
4507:
4504:
4501:
4498:
4495:
4490:
4486:
4482:
4479:
4457:
4454:
4451:
4448:
4445:
4440:
4435:
4432:
4429:
4418:
4417:
4406:
4403:
4399:
4395:
4392:
4388:
4383:
4379:
4375:
4372:
4369:
4365:
4360:
4356:
4352:
4349:
4346:
4342:
4339:
4334:
4330:
4326:
4323:
4318:
4314:
4309:
4303:
4300:
4297:
4292:
4289:
4286:
4282:
4276:
4273:
4266:
4263:
4260:
4256:
4233:
4230:
4227:
4224:
4221:
4218:
4213:
4209:
4205:
4202:
4182:
4162:
4159:
4156:
4153:
4150:
4145:
4140:
4137:
4134:
4123:
4122:
4111:
4108:
4103:
4100:
4097:
4094:
4091:
4086:
4082:
4076:
4072:
4069:
4065:
4060:
4056:
4052:
4047:
4043:
4039:
4036:
4031:
4028:
4025:
4020:
4017:
4014:
4010:
4004:
4001:
3995:
3988:
3985:
3982:
3978:
3952:
3949:
3945:
3940:
3936:
3911:
3908:
3905:
3901:
3895:
3891:
3887:
3884:
3881:
3861:
3858:
3855:
3852:
3849:
3844:
3840:
3836:
3833:
3813:
3810:
3807:
3804:
3801:
3796:
3791:
3788:
3785:
3772:
3766:
3751:
3748:
3745:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3676:
3673:
3669:
3665:
3662:
3657:
3653:
3631:
3628:
3625:
3599:
3579:
3576:
3573:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3506:
3503:
3499:
3495:
3492:
3487:
3483:
3462:
3426:
3399:
3394:
3365:
3344:
3341:
3328:
3325:
3322:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3160:
3157:
3153:
3149:
3146:
3122:
3119:
3116:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3049:
3045:
3040:
3037:
3033:
3029:
3026:
3022:
3018:
3003:
3002:
2991:
2988:
2985:
2981:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2942:
2939:
2935:
2931:
2928:
2925:
2922:
2918:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2853:
2850:
2846:
2842:
2839:
2836:
2833:
2813:
2802:
2801:
2790:
2787:
2783:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2744:
2741:
2737:
2733:
2730:
2727:
2724:
2720:
2714:
2711:
2708:
2703:
2700:
2697:
2693:
2687:
2684:
2677:
2674:
2671:
2667:
2630:
2626:
2603:
2600:
2596:
2580:
2579:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2537:
2533:
2528:
2525:
2521:
2517:
2514:
2510:
2506:
2501:
2498:
2495:
2491:
2465:
2460:
2457:
2454:
2451:
2436:shift operator
2415:
2412:
2409:
2406:
2403:
2398:
2393:
2390:
2387:
2375:
2372:
2359:
2339:
2319:
2314:
2311:
2307:
2286:
2281:
2277:
2256:
2251:
2248:
2244:
2221:measure theory
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2173:
2169:
2166:
2163:
2160:
2157:
2146:
2142:
2122:
2102:
2082:
2062:
2042:
2022:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1961:
1941:
1921:
1901:
1881:
1870:
1869:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1827:
1823:
1820:
1817:
1812:
1809:
1805:
1800:
1796:
1791:
1788:
1785:
1781:
1754:
1745:but also fill
1734:
1714:
1694:
1673:
1642:
1637:
1634:
1631:
1628:
1625:
1622:
1598:
1576:
1549:
1525:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1482:
1478:
1455:
1452:
1449:
1427:
1424:
1421:
1401:
1377:
1372:
1369:
1366:
1363:
1339:
1317:
1295:
1292:
1289:
1257:
1229:
1205:
1185:
1165:
1162:
1159:
1154:
1150:
1144:
1139:
1136:
1133:
1129:
1108:
1086:
1081:
1078:
1055:
1035:
1032:
1029:
1024:
1021:
1017:
996:
993:
990:
987:
982:
979:
975:
971:
968:
965:
962:
959:
956:
953:
931:
928:
925:
901:
896:
891:
861:
856:
853:
850:
847:
842:
839:
835:
812:
809:
806:
782:
777:
772:
767:
762:
759:
755:
732:
729:
726:
704:
701:
698:
695:
692:
689:
686:
683:
680:
658:
655:
652:
628:
608:
605:
602:
599:
567:
547:
544:
541:
538:
518:
515:
512:
492:
489:
486:
483:
480:
436:
408:
380:
356:
353:
350:
347:
325:
322:
319:
292:
268:
248:
221:
196:
193:
190:
187:
184:
179:
174:
171:
168:
146:
143:
112:—the study of
110:ergodic theory
65:
64:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
9021:
9010:
9007:
9006:
9004:
8992:
8987:
8985:
8983:
8979:
8976:
8972:
8968:
8962:
8959:
8956:
8952:
8948:
8942:
8939:
8930:
8926:
8922:
8918:
8913:
8908:
8904:
8900:
8895:
8893:
8889:
8885:
8881:
8879:
8875:
8871:
8867:
8864:
8860:
8859:
8854:
8852:
8850:
8846:
8842:
8838:
8834:
8830:
8826:
8822:
8817:
8797:
8774:
8766:
8759:
8753:
8750:
8747:
8737:
8736:
8735:
8721:
8711:
8704:
8700:
8691:
8689:
8682:
8676:
8672:
8664:
8662:
8655:
8650:
8648:
8628:
8620:
8615:
8606:
8602:
8599:
8596:
8590:
8584:
8579:
8566:
8560:
8557:
8554:
8551:
8548:
8540:
8537:
8532:
8528:
8520:
8519:
8518:
8517:are given by
8516:
8514:
8508:
8506:
8500:
8492:
8488:
8483:
8476:
8468:
8464:
8459:
8437:
8412:
8404:
8397:
8391:
8388:
8385:
8375:
8374:
8373:
8359:
8349:
8345:
8341:
8333:
8331:
8324:
8318:
8314:
8306:
8300:
8295:
8293:
8273:
8268:
8260:
8255:
8238:
8235:
8222:
8216:
8213:
8210:
8207:
8199:
8194:
8190:
8182:
8181:
8180:
8178:
8171:
8169:
8163:
8161:
8152:
8146:
8142:
8134:
8128:
8123:
8121:
8101:
8096:
8088:
8083:
8066:
8063:
8058:
8050:
8043:
8040:
8037:
8034:
8026:
8021:
8017:
8009:
8008:
8007:
8005:
7998:
7996:
7990:
7972:
7968:
7965:
7957:
7953:
7950:
7947:
7931:
7927:
7923:
7920:
7916:
7912:
7909:
7902:
7901:
7900:
7899:Finally, let
7886:
7870:
7866:
7839:
7836:
7831:
7827:
7823:
7815:
7811:
7804:
7793:
7787:
7781:
7776:
7760:
7759:
7758:
7709:
7705:
7674:
7653:
7649:
7633:
7631:
7611:
7603:
7587:
7584:
7581:
7561:
7541:
7513:
7491:
7482:
7476:
7468:
7447:
7444:
7441:
7435:
7425:
7421:
7408:
7387:
7383:
7379:
7376:
7373:
7368:
7365:
7362:
7358:
7354:
7349:
7345:
7331:
7327:
7308:
7305:
7302:
7299:
7296:
7290:
7268:
7263:
7259:
7251:. The symbol
7233:
7230:
7227:
7221:
7195:
7184:
7181:
7178:
7174:
7170:
7167:
7164:
7159:
7151:
7147:
7143:
7140:
7137:
7131:
7128:
7125:
7119:
7116:
7105:
7091:
7080:
7074:
7071:
7068:
7051:
7044:
7038:
7032:
7025:
7024:
7023:
7021:
7018:, called the
7005:
6996:
6994:
6990:
6986:
6985:cylinder sets
6982:
6978:
6952:
6942:
6926:
6905:
6902:
6899:
6893:
6883:
6879:
6863:
6861:
6857:
6855:
6851:
6846:
6830:
6827:
6821:
6808:
6788:
6784:
6774:
6756:
6752:
6728:
6722:
6719:
6716:
6710:
6702:
6698:
6690:
6689:
6688:
6672:
6669:
6666:
6656:
6638:
6616:
6608:
6603:
6601:
6597:
6573:
6570:
6567:
6561:
6553:
6549:
6537:
6521:
6518:
6515:
6508:
6504:
6500:
6496:
6492:
6488:
6484:
6482:
6481:wandering set
6478:
6474:
6470:
6466:
6462:
6458:
6456:
6437:
6433:
6406:
6403:
6400:
6394:
6386:
6382:
6374:
6373:
6372:
6370:
6354:
6351:
6348:
6345:
6342:
6335:
6331:
6315:
6309:
6306:
6303:
6296:
6292:
6288:
6280:
6278:
6276:
6272:
6157:
6138:
6135:
6132:
6100:
6094:
6091:
6088:
6060:
6054:
6050:
6046:
6043:
6040:
6037:
6034:
6031:
6028:
6025:
6017:
6013:
6004:
6000:
5995:
5992:
5989:
5982:
5974:
5970:
5966:
5959:
5954:
5929:
5921:
5918:
5915:
5909:
5903:
5881:
5856:
5853:
5850:
5847:
5844:
5838:
5835:
5826:
5824:
5820:
5816:
5812:
5811:geodesic flow
5808:
5804:
5800:
5796:
5792:
5787:
5785:
5778:
5776:
5774:
5770:
5765:
5763:
5759:
5755:
5751:
5748:
5744:
5740:
5736:
5732:
5710:
5704:
5698:
5692:
5686:
5680:
5677:
5671:
5666:
5663:
5660:
5657:
5653:
5649:
5646:
5641:
5638:
5634:
5630:
5627:
5621:
5610:
5607:
5604:
5592:
5591:
5590:
5588:
5584:
5576:
5556:
5553:
5550:
5547:
5544:
5541:
5538:
5535:
5530:
5526:
5499:
5496:
5493:
5490:
5487:
5484:
5481:
5478:
5473:
5469:
5442:
5439:
5436:
5433:
5430:
5419:
5416:
5415:
5396:
5393:
5390:
5387:
5384:
5381:
5378:
5375:
5372:
5369:
5366:
5356:, the system
5338:
5335:
5332:
5329:
5326:
5298:
5295:
5292:
5289:
5286:
5275:
5272:
5271:
5270:
5256:
5247:
5241:
5238:
5232:
5226:
5220:
5214:
5211:
5208:
5199:
5196:
5193:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5140:
5137:
5114:
5108:
5105:
5102:
5099:
5096:
5070:
5067:
5064:
5061:
5058:
5043:
5041:
5027:
5007:
4985:
4981:
4977:
4974:
4952:
4930:
4910:
4888:
4884:
4880:
4877:
4855:
4852:
4846:
4843:
4838:
4834:
4830:
4827:
4821:
4818:
4807:
4793:
4774:
4771:
4768:
4764:
4759:
4755:
4751:
4748:
4745:
4741:
4736:
4732:
4728:
4725:
4722:
4718:
4715:
4710:
4706:
4702:
4699:
4694:
4690:
4678:
4666:
4665:
4664:
4651:
4645:
4642:
4639:
4631:
4627:
4623:
4620:
4598:
4595:
4591:
4586:
4582:
4557:
4554:
4551:
4541:
4537:
4533:
4530:
4502:
4499:
4496:
4488:
4484:
4480:
4477:
4452:
4449:
4446:
4443:
4433:
4430:
4404:
4401:
4397:
4393:
4390:
4386:
4381:
4377:
4373:
4370:
4367:
4363:
4358:
4354:
4350:
4347:
4344:
4340:
4337:
4332:
4328:
4324:
4321:
4316:
4312:
4307:
4301:
4298:
4295:
4290:
4287:
4284:
4280:
4274:
4271:
4258:
4246:
4245:
4244:
4231:
4225:
4222:
4219:
4211:
4207:
4203:
4200:
4180:
4157:
4154:
4151:
4148:
4138:
4135:
4109:
4106:
4098:
4095:
4092:
4084:
4080:
4070:
4067:
4063:
4058:
4054:
4050:
4045:
4041:
4037:
4034:
4029:
4026:
4023:
4018:
4015:
4012:
4008:
4002:
3999:
3980:
3968:
3967:
3966:
3950:
3947:
3943:
3938:
3934:
3909:
3906:
3903:
3893:
3889:
3885:
3882:
3856:
3853:
3850:
3842:
3838:
3834:
3831:
3808:
3805:
3802:
3799:
3789:
3786:
3770:
3767:
3765:
3749:
3746:
3743:
3718:
3712:
3706:
3700:
3697:
3691:
3688:
3682:
3674:
3671:
3667:
3660:
3655:
3629:
3626:
3623:
3615:
3611:
3597:
3577:
3574:
3571:
3548:
3542:
3536:
3530:
3527:
3521:
3518:
3512:
3504:
3501:
3497:
3490:
3485:
3460:
3452:
3448:
3446:
3442:
3413:
3397:
3381:
3379:
3363:
3355:
3351:
3342:
3340:
3326:
3323:
3320:
3298:
3295:
3286:
3280:
3274:
3268:
3265:
3259:
3256:
3253:
3247:
3220:
3217:
3205:
3202:
3196:
3193:
3190:
3184:
3181:
3175:
3172:
3166:
3158:
3155:
3151:
3144:
3120:
3117:
3114:
3104:
3100:
3098:
3094:
3092:
3091:Chacon system
3072:
3066:
3060:
3054:
3047:
3043:
3038:
3035:
3031:
3027:
3024:
3020:
3016:
3008:
2989:
2986:
2979:
2972:
2966:
2960:
2954:
2951:
2945:
2940:
2937:
2933:
2929:
2926:
2920:
2916:
2908:
2907:
2906:
2892:
2883:
2877:
2871:
2865:
2862:
2856:
2851:
2848:
2844:
2840:
2837:
2831:
2811:
2788:
2785:
2781:
2774:
2768:
2762:
2756:
2753:
2747:
2742:
2739:
2735:
2731:
2728:
2722:
2718:
2712:
2709:
2706:
2701:
2698:
2695:
2691:
2685:
2682:
2669:
2657:
2656:
2655:
2653:
2648:
2646:
2628:
2624:
2601:
2598:
2594:
2585:
2566:
2560:
2554:
2548:
2542:
2539:
2535:
2531:
2526:
2523:
2519:
2515:
2512:
2508:
2504:
2493:
2481:
2480:
2479:
2458:
2455:
2452:
2449:
2441:
2440:strong mixing
2437:
2433:
2429:
2410:
2407:
2404:
2401:
2391:
2388:
2373:
2371:
2357:
2337:
2317:
2312:
2309:
2305:
2284:
2279:
2275:
2254:
2249:
2246:
2242:
2232:
2230:
2226:
2222:
2218:
2217:Bayes theorem
2202:
2196:
2190:
2184:
2178:
2175:
2171:
2167:
2164:
2158:
2144:
2140:
2120:
2100:
2080:
2060:
2040:
2020:
1997:
1991:
1985:
1979:
1959:
1939:
1919:
1899:
1879:
1856:
1850:
1844:
1838:
1832:
1829:
1825:
1821:
1818:
1815:
1810:
1807:
1803:
1798:
1794:
1783:
1771:
1770:
1769:
1766:
1752:
1732:
1712:
1692:
1671:
1663:
1662:strong mixing
1658:
1635:
1632:
1629:
1626:
1623:
1620:
1596:
1574:
1565:
1563:
1539:
1523:
1500:
1497:
1494:
1488:
1480:
1476:
1453:
1450:
1447:
1425:
1422:
1419:
1399:
1370:
1367:
1364:
1361:
1337:
1315:
1293:
1290:
1287:
1276:
1274:
1270:
1255:
1247:
1243:
1227:
1219:
1203:
1183:
1160:
1152:
1148:
1142:
1137:
1134:
1131:
1127:
1106:
1079:
1076:
1067:
1053:
1030:
1022:
1019:
1015:
988:
980:
977:
973:
966:
963:
957:
951:
929:
923:
879:
854:
848:
840:
837:
833:
810:
807:
804:
765:
760:
757:
753:
730:
727:
724:
699:
693:
690:
684:
678:
656:
653:
650:
640:
626:
603:
597:
589:
585:
584:horseshoe map
581:
565:
542:
536:
516:
513:
510:
490:
484:
481:
478:
471:
466:
464:
460:
456:
455:intersections
452:
424:
406:
396:
351:
345:
323:
320:
317:
310:
266:
246:
239:
235:
219:
210:
191:
188:
185:
182:
172:
169:
157:, written as
156:
152:
144:
142:
140:
135:
131:
127:
123:
122:strong mixing
119:
115:
111:
106:
104:
100:
96:
92:
88:
84:
76:
71:
61:
56:
52:
51:
46:
41:
32:
31:
19:
18:Strong mixing
8966:
8961:
8950:
8941:
8902:
8898:
8883:
8869:
8862:
8848:
8847:-mixing and
8844:
8840:
8836:
8832:
8828:
8825:Harris chain
8820:
8818:
8789:
8709:
8702:
8694:
8687:
8686:
8685:, and it is
8680:
8674:
8667:
8660:
8659:
8653:
8646:
8645:
8643:
8512:
8511:
8509:
8504:
8495:
8490:
8486:
8478:
8471:
8466:
8462:
8460:
8427:
8347:
8343:
8336:
8329:
8328:
8327:, and it is
8322:
8316:
8309:
8304:
8298:
8291:
8290:
8288:
8173:
8167:
8166:
8164:
8156:
8150:
8144:
8137:
8132:
8126:
8119:
8118:
8116:
8000:
7994:
7993:
7991:
7988:
7857:
7637:
7466:
7410:The process
7409:
7329:
7325:
7211:
7019:
6997:
6867:
6858:
6849:
6847:
6806:
6743:
6654:
6606:
6604:
6599:
6535:
6502:
6490:
6486:
6485:
6460:
6454:
6424:
6368:
6329:
6284:
6270:
6158:
5827:
5813:on the unit
5788:
5782:
5768:
5766:
5761:
5757:
5753:
5749:
5746:
5742:
5738:
5734:
5730:
5728:
5586:
5582:
5580:
5418:Proposition.
5417:
5274:Proposition.
5273:
5047:
4789:
4419:
4124:
3774:
3768:
3613:
3612:
3450:
3449:
3440:
3414:
3382:
3346:
3313:, so one of
3102:
3101:
3096:
3095:
3004:
2803:
2651:
2649:
2644:
2616:replaced by
2583:
2581:
2442:if, for any
2439:
2431:
2377:
2233:
2151:after-mixing
1871:
1767:
1661:
1659:
1566:
1277:
1068:
877:
641:
467:
233:
211:
148:
129:
125:
121:
107:
86:
80:
59:
48:
8953:, Springer
8734:satisfying
8372:satisfying
7757:. Also let
7602:independent
6275:Baker's map
5807:Anosov flow
3771:formulation
2654:if one has
2652:weak mixing
2267:instead of
2113:that is in
1269:wander away
1066:came from.
580:baker's map
126:weak mixing
83:mathematics
75:baker's map
8855:References
6854:continuous
6687:, one has
6457:th iterate
6371:such that
5805:, and the
5791:dyadic map
5773:ergodicity
4792:covariance
3354:Borel sets
2478:, one has
1196:for large
529:, its map
393:to be the
139:ergodicity
8907:CiteSeerX
8804:∞
8801:→
8772:→
8754:ξ
8751:
8722:ξ
8678:for some
8603:φ
8600:∫
8597:−
8585:φ
8558:≤
8555:φ
8552:≤
8541:∫
8529:β
8444:∞
8441:→
8410:→
8392:ξ
8389:
8360:ξ
8320:for some
8303:, it is "
8265:‖
8261:φ
8244:‖
8231:∞
8227:‖
8223:φ
8220:‖
8211:∈
8208:φ
8191:α
8148:for some
8093:‖
8089:φ
8072:‖
8055:‖
8051:φ
8048:‖
8038:∈
8035:φ
8018:ρ
7954:φ
7951:∫
7924:∈
7921:φ
7824:∣
7805:φ
7782:φ
7618:∞
7615:→
7520:∞
7517:→
7489:→
7477:α
7451:∞
7439:∞
7436:−
7377:…
7312:∞
7309:≤
7303:≤
7297:≤
7294:∞
7291:−
7237:∞
7225:∞
7222:−
7190:∞
7171:∈
7155:∞
7152:−
7144:∈
7135:∞
7123:∞
7120:−
7081:−
7072:∩
7033:α
7006:α
6989:σ-algebra
6981:open sets
6940:Ω
6909:∞
6897:∞
6894:−
6825:‖
6819:‖
6785:φ
6726:∅
6723:≠
6717:∩
6574:∈
6519:∈
6463:. In the
6410:∅
6407:≠
6401:∩
6352:⊂
6334:open sets
6313:→
6271:shift map
6247:Σ
6216:Σ
6193:Σ
6169:Σ
6101:∖
6041:≤
6032:≥
6018:∖
5857:…
5839:∈
5705:μ
5693:μ
5681:μ
5664:−
5658:−
5650:∩
5639:−
5631:∩
5622:μ
5617:∞
5614:→
5554:×
5545:μ
5542:⊗
5539:μ
5497:×
5488:μ
5485:⊗
5482:μ
5437:μ
5394:×
5385:ν
5382:⊗
5379:μ
5370:×
5333:ν
5293:μ
5197:×
5165:×
5156:ν
5153:⊗
5150:μ
5141:×
5103:ν
5065:μ
4978:∘
4881:∘
4831:∘
4822:
4814:∞
4811:→
4772:μ
4756:∫
4752:⋅
4749:μ
4733:∫
4726:μ
4716:⋅
4703:∘
4691:∫
4685:∞
4682:→
4646:μ
4624:∈
4599:μ
4583:∫
4555:≥
4534:∘
4503:μ
4481:∈
4447:μ
4394:μ
4378:∫
4374:⋅
4371:μ
4355:∫
4351:−
4348:μ
4338:⋅
4325:∘
4313:∫
4299:−
4281:∑
4265:∞
4262:→
4226:μ
4204:∈
4152:μ
4099:μ
4071:μ
4055:∫
4051:−
4038:∘
4027:−
4009:∑
3987:∞
3984:→
3951:μ
3935:∫
3907:≥
3886:∘
3857:μ
3835:∈
3803:μ
3713:μ
3701:μ
3689:∩
3672:−
3661:μ
3543:μ
3531:μ
3519:∩
3502:−
3491:μ
3364:μ
3281:μ
3269:μ
3266:−
3257:∩
3248:μ
3212:∅
3206:μ
3194:∩
3185:μ
3173:∩
3156:−
3145:μ
3067:μ
3055:μ
3052:→
3036:−
3028:∩
3017:μ
2984:→
2967:μ
2955:μ
2952:−
2938:−
2930:∩
2921:μ
2890:→
2878:μ
2866:μ
2863:−
2849:−
2841:∩
2832:μ
2769:μ
2757:μ
2754:−
2740:−
2732:∩
2723:μ
2710:−
2692:∑
2676:∞
2673:→
2599:−
2555:μ
2543:μ
2524:−
2516:∩
2505:μ
2500:∞
2497:→
2459:∈
2405:μ
2310:−
2247:−
2191:μ
2179:μ
2165:∩
2141:μ
1992:μ
1980:μ
1845:μ
1833:μ
1819:∩
1808:−
1795:μ
1790:∞
1787:→
1636:∈
1562:empty set
1548:∅
1524:∩
1516:. Here,
1504:∅
1501:≠
1495:∩
1371:∈
1128:∪
1080:∈
1020:−
978:−
967:μ
952:μ
927:→
895:→
855:∈
838:−
808:⊂
776:→
758:−
728:∈
654:≠
514:⊂
488:→
451:Borel set
395:power set
346:μ
321:⊂
247:μ
186:μ
9003:Category
8929:10567129
8658:, it is
7697:and let
7638:Suppose
7212:for all
6497:with no
6291:topology
5779:Examples
5756:for all
4075:‖
3994:‖
3965:, i.e.,
3564:for all
3451:Theorem.
3441:disjoint
3350:infinity
3097:Theorem.
1536:denotes
1007:because
582:and the
212:The set
8649:-mixing
8456:
8430:
8294:-mixing
8131:, and “
8122:-mixing
7532:
7506:
7405:
7334:
7283:, with
7249:
7214:
6973:
6929:
6843:
6811:
6805:, with
6803:
6776:
6685:
6659:
6651:
6631:
6501:, then
6452:is the
6287:measure
6267:
6238:
6205:
6185:
6181:
6161:
6154:
6122:
6118:
6078:
6074:
5946:
5894:
5874:
5354:
5316:
5040:grows.
4965:
4945:
4868:
4795:
4611:
4574:
4518:
4470:
3963:
3926:
3762:
3736:
3311:
3237:
3233:
3137:
3133:
3107:
3005:in the
2430:, with
1655:
1613:
1609:
1589:
1560:is the
1466:
1440:
1390:
1354:
1350:
1330:
1306:
1280:
1218:ergodic
942:
916:
874:
825:
795:
745:
715:
671:
419:
399:
367:
338:
305:
281:
238:measure
207:
159:
134:measure
91:physics
55:Discuss
8973:
8927:
8909:
8890:
8876:
8827:. The
8683:> 0
8325:> 0
8179:} are
8153:> 0
8006:} are
6991:, the
6979:. The
6487:Lemma:
6425:where
5896:" map
5754:mixing
5747:strong
3614:Proof.
3103:Proof.
3007:Cesàro
459:unions
309:subset
99:mixing
87:mixing
45:merged
8925:S2CID
8705:) → 0
8673:<
8342:<
8315:<
8143:<
7022:, as
6923:be a
6596:dense
6493:is a
5809:(the
2643:with
2426:be a
669:with
588:bread
47:with
8971:ISBN
8888:ISBN
8874:ISBN
8510:The
8461:The
8165:The
7992:The
7448:<
7442:<
7328:and
7234:<
7228:<
7132:<
7126:<
6906:<
6900:<
6868:Let
6828:>
6773:flow
6670:>
6629:and
6047:<
5086:and
5000:and
4903:and
4193:and
2378:Let
2223:and
1912:and
1587:and
1540:and
1451:>
1438:and
1328:and
461:and
236:The
234:etc.
128:and
116:and
8949:",
8917:doi
8903:155
8790:as
8712:→ ∞
8707:as
8692:if
8665:if
8656:→ ∞
8545:sup
8501:= 1
8428:as
8334:if
8301:→ ∞
8204:sup
8129:→ ∞
8031:sup
7604:as
7504:as
7469:if
7048:sup
6598:in
6594:is
6489:If
6459:of
5926:mod
5825:.)
5821:of
5817:of
5601:lim
4819:Cov
4804:lim
4675:lim
4255:lim
3977:lim
3652:lim
3482:lim
2666:lim
2490:lim
1780:lim
878:any
470:map
397:of
336:is
97:of
81:In
53:. (
9005::
8981:^
8923:.
8915:.
8901:.
8816:.
8748:ln
8675:γe
8477:≤
8458:.
8386:ln
8317:γe
7407:.
6848:A
6845:.
6602:.
6483:.
6156:.
5801:,
5797:,
5793:,
5775:.
5764:.
5737:,
5733:,
4405:0.
4110:0.
3764:.
2789:0.
2370:.
1564:.
457:,
209:.
124:,
105:.
85:,
8931:.
8919::
8849:ρ
8845:β
8841:α
8837:β
8833:α
8829:β
8821:β
8798:t
8775:0
8767:t
8763:)
8760:t
8757:(
8710:t
8703:t
8701:(
8699:ξ
8697:t
8695:β
8688:β
8681:δ
8670:t
8668:β
8661:β
8654:t
8647:β
8629:.
8625:Q
8621:d
8616:|
8611:Q
8607:d
8594:)
8591:x
8588:(
8580:t
8574:E
8567:|
8561:1
8549:0
8538:=
8533:t
8513:β
8505:α
8498:t
8496:ρ
8491:α
8487:ρ
8481:t
8479:ρ
8474:t
8472:α
8467:ρ
8463:α
8438:t
8413:0
8405:t
8401:)
8398:t
8395:(
8350:)
8348:t
8346:(
8344:ξ
8339:t
8337:α
8330:α
8323:δ
8312:t
8310:α
8305:α
8299:t
8292:α
8274:.
8269:1
8256:t
8250:E
8239:1
8236:=
8217::
8214:Z
8200:=
8195:t
8176:t
8174:x
8168:α
8159:t
8157:ρ
8151:δ
8145:e
8140:t
8138:ρ
8133:ρ
8127:t
8120:ρ
8102:.
8097:2
8084:t
8078:E
8067:1
8064:=
8059:2
8044::
8041:Z
8027:=
8022:t
8003:t
8001:x
7995:ρ
7973:}
7969:0
7966:=
7962:Q
7958:d
7948::
7945:)
7941:Q
7937:(
7932:2
7928:L
7917:{
7913:=
7910:Z
7887:.
7884:)
7880:Q
7876:(
7871:2
7867:L
7843:]
7840:x
7837:=
7832:0
7828:X
7821:)
7816:t
7812:X
7808:(
7802:[
7798:E
7794:=
7791:)
7788:x
7785:(
7777:t
7771:E
7744:Q
7723:)
7719:Q
7715:(
7710:2
7706:L
7684:Q
7659:)
7654:t
7650:X
7646:(
7612:s
7588:s
7585:+
7582:t
7562:t
7542:t
7514:s
7492:0
7486:)
7483:s
7480:(
7445:t
7432:)
7426:t
7422:X
7418:(
7393:}
7388:b
7384:X
7380:,
7374:,
7369:1
7366:+
7363:a
7359:X
7355:,
7350:a
7346:X
7342:{
7330:b
7326:a
7306:b
7300:a
7269:b
7264:a
7260:X
7231:s
7196:}
7185:s
7182:+
7179:t
7175:X
7168:B
7165:,
7160:t
7148:X
7141:A
7138:,
7129:t
7117::
7113:|
7109:)
7106:B
7103:(
7099:P
7095:)
7092:A
7089:(
7085:P
7078:)
7075:B
7069:A
7066:(
7062:P
7057:|
7052:{
7045:=
7042:)
7039:s
7036:(
6961:)
6957:P
6953:,
6948:F
6943:,
6937:(
6903:t
6890:)
6884:t
6880:X
6876:(
6831:N
6822:g
6807:g
6789:g
6757:n
6753:f
6729:.
6720:B
6714:)
6711:A
6708:(
6703:n
6699:f
6673:N
6667:n
6655:N
6639:B
6617:A
6600:X
6582:}
6578:N
6571:n
6568::
6565:)
6562:x
6559:(
6554:n
6550:f
6546:{
6536:x
6522:X
6516:x
6503:f
6491:X
6461:f
6455:n
6438:n
6434:f
6404:B
6398:)
6395:A
6392:(
6387:n
6383:f
6369:n
6355:X
6349:B
6346:,
6343:A
6316:X
6310:X
6307::
6304:f
6252:Z
6221:N
6142:]
6139:1
6136:,
6133:0
6130:[
6105:Q
6098:)
6095:1
6092:,
6089:0
6086:(
6061:}
6055:s
6051:n
6044:k
6038:,
6035:0
6029:s
6026::
6022:Q
6014:)
6005:s
6001:n
5996:1
5993:+
5990:k
5983:,
5975:s
5971:n
5967:k
5960:(
5955:{
5930:1
5922:x
5919:n
5916:=
5913:)
5910:x
5907:(
5904:T
5882:n
5860:}
5854:,
5851:3
5848:,
5845:2
5842:{
5836:n
5769:m
5758:k
5752:-
5750:k
5739:C
5735:B
5731:A
5714:)
5711:C
5708:(
5702:)
5699:B
5696:(
5690:)
5687:A
5684:(
5678:=
5675:)
5672:C
5667:n
5661:m
5654:T
5647:B
5642:m
5635:T
5628:A
5625:(
5611:n
5608:,
5605:m
5560:)
5557:T
5551:T
5548:,
5536:,
5531:2
5527:X
5523:(
5503:)
5500:T
5494:T
5491:,
5479:,
5474:2
5470:X
5466:(
5446:)
5443:T
5440:,
5434:,
5431:X
5428:(
5400:)
5397:S
5391:T
5388:,
5376:,
5373:Y
5367:X
5364:(
5342:)
5339:S
5336:,
5330:,
5327:Y
5324:(
5302:)
5299:T
5296:,
5290:,
5287:X
5284:(
5257:.
5254:)
5251:)
5248:y
5245:(
5242:S
5239:,
5236:)
5233:x
5230:(
5227:T
5224:(
5221:=
5218:)
5215:y
5212:,
5209:x
5206:(
5203:)
5200:S
5194:T
5191:(
5171:)
5168:S
5162:T
5159:,
5147:,
5144:Y
5138:X
5135:(
5115:,
5112:)
5109:S
5106:,
5100:,
5097:Y
5094:(
5074:)
5071:T
5068:,
5062:,
5059:X
5056:(
5028:n
5008:g
4986:n
4982:T
4975:f
4953:g
4931:n
4911:g
4889:n
4885:T
4878:f
4856:0
4853:=
4850:)
4847:g
4844:,
4839:n
4835:T
4828:f
4825:(
4808:n
4775:.
4769:d
4765:g
4760:X
4746:d
4742:f
4737:X
4729:=
4723:d
4719:g
4711:n
4707:T
4700:f
4695:X
4679:n
4652:,
4649:)
4643:,
4640:X
4637:(
4632:2
4628:L
4621:g
4596:d
4592:f
4587:X
4558:0
4552:n
4548:)
4542:n
4538:T
4531:f
4528:(
4506:)
4500:,
4497:X
4494:(
4489:2
4485:L
4478:f
4456:)
4453:T
4450:,
4444:,
4439:A
4434:,
4431:X
4428:(
4402:=
4398:|
4391:d
4387:g
4382:X
4368:d
4364:f
4359:X
4345:d
4341:g
4333:n
4329:T
4322:f
4317:X
4308:|
4302:1
4296:N
4291:0
4288:=
4285:n
4275:N
4272:1
4259:N
4232:,
4229:)
4223:,
4220:X
4217:(
4212:2
4208:L
4201:g
4181:f
4161:)
4158:T
4155:,
4149:,
4144:A
4139:,
4136:X
4133:(
4107:=
4102:)
4096:,
4093:X
4090:(
4085:2
4081:L
4068:d
4064:f
4059:X
4046:n
4042:T
4035:f
4030:1
4024:N
4019:0
4016:=
4013:n
4003:N
4000:1
3981:N
3948:d
3944:f
3939:X
3910:0
3904:n
3900:)
3894:n
3890:T
3883:f
3880:(
3860:)
3854:,
3851:X
3848:(
3843:2
3839:L
3832:f
3812:)
3809:T
3806:,
3800:,
3795:A
3790:,
3787:X
3784:(
3769:L
3750:B
3747:,
3744:A
3722:)
3719:B
3716:(
3710:)
3707:A
3704:(
3698:=
3695:)
3692:B
3686:)
3683:A
3680:(
3675:n
3668:T
3664:(
3656:n
3630:B
3627:,
3624:A
3598:T
3578:B
3575:,
3572:A
3552:)
3549:B
3546:(
3540:)
3537:A
3534:(
3528:=
3525:)
3522:B
3516:)
3513:A
3510:(
3505:n
3498:T
3494:(
3486:n
3461:T
3425:C
3398:n
3393:R
3327:B
3324:,
3321:A
3299:0
3296:=
3293:|
3290:)
3287:B
3284:(
3278:)
3275:A
3272:(
3263:)
3260:B
3254:A
3251:(
3245:|
3221:0
3218:=
3215:)
3209:(
3203:=
3200:)
3197:B
3191:A
3188:(
3182:=
3179:)
3176:B
3170:)
3167:A
3164:(
3159:n
3152:T
3148:(
3121:B
3118:,
3115:A
3076:)
3073:B
3070:(
3064:)
3061:A
3058:(
3048:)
3044:B
3039:n
3032:T
3025:A
3021:(
2990:,
2987:0
2980:|
2976:)
2973:B
2970:(
2964:)
2961:A
2958:(
2949:)
2946:B
2941:n
2934:T
2927:A
2924:(
2917:|
2893:0
2887:)
2884:B
2881:(
2875:)
2872:A
2869:(
2860:)
2857:B
2852:n
2845:T
2838:A
2835:(
2812:T
2786:=
2782:|
2778:)
2775:B
2772:(
2766:)
2763:A
2760:(
2751:)
2748:B
2743:k
2736:T
2729:A
2726:(
2719:|
2713:1
2707:n
2702:0
2699:=
2696:k
2686:n
2683:1
2670:n
2645:g
2629:g
2625:T
2602:n
2595:T
2584:n
2567:.
2564:)
2561:B
2558:(
2552:)
2549:A
2546:(
2540:=
2536:)
2532:B
2527:n
2520:T
2513:A
2509:(
2494:n
2464:A
2456:B
2453:,
2450:A
2432:T
2414:)
2411:T
2408:,
2402:,
2397:A
2392:,
2389:X
2386:(
2358:B
2338:B
2318:A
2313:1
2306:T
2285:A
2280:n
2276:T
2255:A
2250:n
2243:T
2203:.
2200:)
2197:B
2194:(
2188:)
2185:A
2182:(
2176:=
2172:)
2168:B
2162:)
2159:A
2156:(
2145:(
2121:B
2101:A
2081:B
2061:A
2041:A
2021:B
2001:)
1998:B
1995:(
1989:)
1986:A
1983:(
1960:B
1940:A
1920:B
1900:A
1880:n
1857:.
1854:)
1851:B
1848:(
1842:)
1839:A
1836:(
1830:=
1826:)
1822:B
1816:A
1811:n
1804:T
1799:(
1784:n
1753:B
1733:B
1713:A
1693:B
1672:A
1641:A
1633:C
1630:,
1627:B
1624:,
1621:A
1597:B
1575:A
1498:B
1492:)
1489:A
1486:(
1481:n
1477:T
1454:N
1448:n
1426:B
1423:,
1420:A
1400:N
1376:A
1368:B
1365:,
1362:A
1338:X
1316:A
1294:B
1291:,
1288:A
1256:A
1228:A
1204:n
1184:X
1164:)
1161:A
1158:(
1153:k
1149:T
1143:n
1138:1
1135:=
1132:k
1107:X
1085:A
1077:A
1054:A
1034:)
1031:A
1028:(
1023:1
1016:T
995:)
992:)
989:A
986:(
981:1
974:T
970:(
964:=
961:)
958:A
955:(
930:X
924:X
900:A
890:A
860:A
852:)
849:A
846:(
841:1
834:T
811:X
805:A
781:A
771:A
766::
761:1
754:T
731:X
725:x
703:)
700:y
697:(
694:T
691:=
688:)
685:x
682:(
679:T
657:y
651:x
627:A
607:)
604:A
601:(
598:T
566:A
546:)
543:A
540:(
537:T
517:X
511:A
491:X
485:X
482::
479:T
435:A
407:X
379:A
355:)
352:A
349:(
324:X
318:A
291:A
267:X
220:X
195:)
192:T
189:,
183:,
178:A
173:,
170:X
167:(
57:)
20:)
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