4640:
2518:
4288:
5888:. A number of classification theorems have been obtained; but quite interestingly, a number of anti-classification theorems have been found as well. The anti-classification theorems state that there are more than a countable number of isomorphism classes, and that a countable amount of information is not sufficient to classify isomorphisms.
4635:{\displaystyle {\begin{aligned}\bigvee _{n=0}^{N}T^{-n}Q&=\{Q_{i_{0}}\cap T^{-1}Q_{i_{1}}\cap \cdots \cap T^{-N}Q_{i_{N}}\\&{}\qquad {\mbox{ where }}i_{\ell }=1,\ldots ,k,\ \ell =0,\ldots ,N,\ \\&{}\qquad \qquad \mu \left(Q_{i_{0}}\cap T^{-1}Q_{i_{1}}\cap \cdots \cap T^{-N}Q_{i_{N}}\right)>0\}\\\end{aligned}}}
2508:
This system does exhibit one key idea from the classification of measure-preserving dynamical systems: two ensembles, having different temperatures, are inequivalent. The entropy for a given canonical ensemble depends on its temperature; as physical systems, it is "obvious" that when the temperatures
4986:
2425:
of these. This measure is understood to apply to the ensemble. So, for example, one of the possible boxes in the ensemble has all of the atoms on one side of the box. One can compute the likelihood of this, in the
MaxwellâBoltzmann measure. It will be enormously tiny, of order
4273:
766:
4780:
5131:
481:
1895:
is all that is left, after all of the transient modes have decayed away. The transient modes are precisely those eigenvectors of the transfer operator that have eigenvalue less than one; the invariant measure
4859:
2121:
2123:
The "ensemble" is the collection of all such points, that is, the collection of all such possible boxes (of which there are an uncountably-infinite number). This ensemble of all-possible-boxes is the space
2292:
2504:
or some other interaction suitable for a liquid or a plasma; in such cases, the invariant measure is no longer the
MaxwellâBoltzmann distribution. The art of physics is finding reasonable approximations.
4066:
2031:
2765:
5341:
4293:
2475:
5862:
2500:
is difficult, and, even if written down, it is hard to perform practical computations with it. Difficulties are compounded if there are interactions between the particles themselves, like a
264:
5588:
1916:
is the one mode that does not decay away. The rate of decay of the transient modes are given by (the logarithm of) their eigenvalues; the eigenvalue one corresponds to infinite half-life.
3192:
1314:
1145:
5768:
5506:
5045:
4831:
3955:
3730:
3397:
3345:
3293:
3029:
2981:
1893:
1809:
138:
5650:
2834:
5700:
2383:
1654:
1611:
of the transfer operator (recall, the FP eigenvector is the largest eigenvector of a matrix; in this case it is the eigenvector which has the eigenvalue one: the invariant measure.)
1461:
1057:
553:
1362:
5231:
7385:
3860:
338:
3135:
4122:
3638:
3551:
2565:
1961:
1714:
606:
2891:
646:
372:
6039:
3064:
1511:
5937:
5886:
5459:
4685:
2621:
the flow of a
Hamiltonian vector field on the tangent bundle of a closed connected smooth manifold is measure-preserving (using the measure induced on the Borel sets by the
1410:
1386:
1172:
1084:
190:
7463:
3460:
7480:
5393:
3421:
3098:
1553:
1756:
that are measure-like. Measure-like, in that they preserve the Borel properties, but are no longer invariant; they are in general dissipative and so give insights into
994:
955:
303:
1256:
5794:
3600:
3513:
3243:
6108:
5533:
5251:
3574:
3487:
3216:
2169:
1914:
1754:
1601:
400:
5820:
6128:
6079:
6059:
5909:
5720:
5433:
5413:
5203:
2498:
2423:
2403:
2312:
2142:
2051:
1981:
1845:
1734:
1674:
1581:
1216:
1196:
214:
164:
914:
874:
842:
802:
641:
6325:
5997:
6470:
6197:
4693:
2570:
Unlike the informal example above, the examples below are sufficiently well-defined and tractable that explicit, formal computations can be performed.
6788:
6647:
5053:
7303:
5944:
5660:
One of the primary activities in the study of measure-preserving systems is their classification according to their properties. That is, let
7134:
6551:
6258:
6248:
405:
6674:
5947:, it can be shown that there are uncountably many non-Kakutani equivalent ergodic measure-preserving transformations of each entropy type.
2611:
5256:
4981:{\displaystyle h_{\mu }(T,{\mathcal {Q}})=\lim _{N\rightarrow \infty }{\frac {1}{N}}H\left(\bigvee _{n=0}^{N}T^{-n}{\mathcal {Q}}\right).}
2626:
7295:
7475:
2172:
7081:
2060:
7432:
6607:
6579:
6417:
6179:
2182:
3971:
7551:
7422:
2597:
6234:
5253:
is absolutely continuous with respect to the
Lebesgue measure, then we have the Rokhlin formula (section 4.3 and section 12.3 ):
7232:
6905:
5144:
in 1959 shows that the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the
1986:
1928:
from physics provides an informal example. Consider, for example, a fluid, gas or plasma in a box of width, length and height
7470:
6761:
6146:
6140:
2713:
2637:
1983:
atoms. A single atom in that box might be anywhere, having arbitrary velocity; it would be represented by a single point in
54:
61:. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from
7417:
7311:
7217:
2429:
1608:
7336:
7316:
7280:
7204:
6924:
6640:
7458:
7556:
7237:
7199:
7151:
5828:
7363:
1559:. Almost all properties and behaviors of dynamical systems are defined in terms of the pushforward. For example, the
221:
7541:
7331:
7321:
7242:
7209:
6840:
6749:
5538:
996:. For the paint thickness to remain unchanged (measure-preserving), the mass of incoming paint should be the same:
7380:
3140:
7285:
7061:
6989:
1812:
1261:
1096:
70:
7370:
7453:
6899:
6830:
6602:, Andreas Greven, Gerhard Keller, and Gerald Warnecke, eds. Princeton University Press, Princeton, NJ (2003).
5725:
5002:
4788:
3912:
3687:
3354:
3302:
3248:
2986:
2938:
2583:
1850:
1766:
95:
6766:
5593:
2780:
5663:
2317:
1617:
1415:
999:
495:
7561:
7222:
6980:
6940:
6633:
6498:
6274:
Foreman, Matthew; Weiss, Benjamin (2019). "From
Odometers to Circular Systems: A Global Structure Theorem".
2615:
2604:
1925:
1319:
5208:
4268:{\displaystyle Q\vee R=\{Q_{i}\cap R_{j}\mid i=1,\ldots ,k,\ j=1,\ldots ,m,\ \mu (Q_{i}\cap R_{j})>0\}.}
7505:
7405:
7227:
6949:
6795:
6489:
6405:
5464:
3812:
3645:
761:{\displaystyle Tx=2x\mod 1={\begin{cases}2x{\text{ if }}x<1/2\\2x-1{\text{ if }}x>1/2\\\end{cases}}}
308:
3110:
7141:
7066:
7019:
7014:
7009:
6851:
6734:
6692:
6217:
3605:
3518:
2532:
1931:
1683:
916:
half as well. The two layers of thin paint, layered together, recreates the exact same paint thickness.
558:
2845:
345:
7375:
7341:
7249:
6959:
6914:
6756:
6679:
6005:
5823:
4645:
which plays crucial role in the construction of the measure-theoretic entropy of a dynamical system.
4113:
3037:
2653:
is not a single transformation that is iterated to give the dynamics of the system, but instead is a
2509:
differ, so do the systems. This holds in general: systems with different entropy are not isomorphic.
1816:
1468:
5943:. There are a variety of other anti-classification results. For example, replacing isomorphism with
5918:
5867:
5438:
5415:
has full measure or zero measure. Piecewise expanding and Markov means that there is a partition of
4666:
1391:
1367:
1153:
1065:
682:
171:
7358:
7348:
7194:
7158:
7036:
6984:
6713:
6670:
6570:
Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as
Chapter 2 in
5179:
4660:
4654:
3101:
2658:
2501:
2480:
The only reason that this is an "informal example" is because writing down the transition function
1514:
1219:
375:
267:
62:
58:
3433:
7510:
7270:
7255:
6954:
6835:
6813:
6450:
6387:
6334:
6301:
6283:
5353:
4072:
3751:
1757:
66:
6149: â Certain dynamical systems will eventually return to (or approximate) their initial state
3405:
3080:
2649:
The definition of a measure-preserving dynamical system can be generalized to the case in which
1520:
6574:, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991).
960:
922:
273:
7427:
7163:
7124:
7119:
7026:
6944:
6729:
6702:
6603:
6575:
6547:
6413:
6254:
6175:
5145:
3866:
3198:
2768:
2640:
establishes the existence of a suitable measure to form a measure-preserving dynamical system.
1677:
1604:
1560:
1229:
379:
86:
82:
46:
6527:
Katok, A.; Hasselblatt, B. (1995). "Introduction to the modern theory of dynamical systems".
5773:
7546:
7444:
7353:
7129:
7114:
7104:
7089:
7056:
7051:
7041:
6919:
6894:
6709:
6507:
6442:
6377:
6344:
6293:
5156:
3669:
3579:
3492:
3222:
2590:
2522:
6087:
5954:
Ergodic measure-preserving transformations with a pure point spectrum have been classified.
5511:
7520:
7500:
7275:
7173:
7168:
7146:
7004:
6969:
6889:
6783:
5961:
5957:
5236:
3673:
3559:
3472:
3201:
2176:
2154:
1899:
1739:
1586:
385:
5799:
492:
One may ask why the measure preserving transformation is defined in terms of the inverse
17:
7410:
7265:
7260:
7071:
7046:
6999:
6929:
6909:
6869:
6859:
6656:
6587:
6113:
6064:
6044:
5894:
5705:
5418:
5398:
5188:
2483:
2408:
2388:
2297:
2127:
2036:
1966:
1830:
1824:
1719:
1659:
1566:
1201:
1181:
199:
149:
50:
31:
6110:, then such a generator exists iff the system is isomorphic to the Bernoulli shift on
879:
847:
807:
775:
614:
7535:
7515:
7178:
7099:
7094:
6994:
6964:
6934:
6884:
6879:
6874:
6864:
6778:
6697:
6512:
6493:
6305:
5912:
5160:
769:
193:
6391:
6253:. London Mathematical Society Student Texts. Cambridge: Cambridge University Press.
7109:
7031:
6771:
6546:. New Mathematical Monographs. Cambridge: Cambridge University Press. p. 106.
6468:
Sinai, Ya. (1962). "A weak isomorphism of transformations with invariant measure".
6433:
Halmos, P.; von
Neumann, J. (1942). "Operator methods in classical mechanics. II".
5982:
5344:
5149:
2925:
2837:
2662:
2633:
6808:
6591:
4775:{\displaystyle H({\mathcal {Q}})=-\sum _{Q\in {\mathcal {Q}}}\mu (Q)\log \mu (Q).}
6595:
6974:
5152:
5141:
2929:
2622:
2575:
2517:
38:
1614:
There are two classification problems of interest. One, discussed below, fixes
6818:
2477:
Of all possible boxes in the ensemble, this is a ridiculously small fraction.
1820:
1223:
6583:(Provides expository introduction, with exercises, and extensive references.)
1174:
which preserve intersections, unions and complements (so that it is a map of
6800:
6744:
6739:
5940:
2148:
1815:. One might ask: how did it get that way? Often, the answer is by stirring,
1175:
1087:
5167:
is either less than 1/2 or not; and likewise so is the fractional part of 2
5126:{\displaystyle h_{\mu }(T)=\sup _{\mathcal {Q}}h_{\mu }(T,{\mathcal {Q}}).}
1222:). Every such conservative, Borel-preserving map can be specified by some
30:"Area-preserving map" redirects here. For the map projection concept, see
6825:
6684:
5137:
1556:
6349:
6320:
6297:
6195:
Sinai, Ya. G. (1959). "On the Notion of
Entropy of a Dynamical System".
6621:(gives a more involved example of measure-preserving dynamical system.)
6454:
5950:
These stand in contrast to the classification theorems. These include:
5178:
is compact and endowed with a topology, or is a metric space, then the
1811:
often describes a physical system that is in equilibrium, for example,
6382:
6365:
5343:
This allows calculation of entropy of many interval maps, such as the
476:{\displaystyle \forall A\in {\mathcal {B}}\;\;\mu (T^{-1}(A))=\mu (A)}
6617:
5891:
The first anti-classification theorem, due to Hjorth, states that if
2654:
6446:
6321:"Measure preserving Diffeomorphisms of the Torus are unclassifiable"
3865:
The set of symbolic names with respect to a partition is called the
6339:
6288:
2516:
6625:
5979:
Given a dynamical system on a
Lebesgue space of measure 1, where
1563:
is defined in terms of the pushforward of the transformation map
2116:{\displaystyle (w\times l\times h)^{N}\times \mathbb {R} ^{3N}.}
6629:
2897:
The earlier, simpler case fits into this framework by defining
1656:
and asks about the isomorphism classes of a transformation map
2287:{\displaystyle p_{i}(x,y,z,v_{x},v_{y},v_{z})\,d^{3}x\,d^{3}p}
772:. Now, distribute an even layer of paint on the unit interval
5140:
is taken over all finite measurable partitions. A theorem of
1763:
In terms of physics, the measure-preserving dynamical system
5924:
5873:
5834:
5740:
5678:
5112:
5085:
5017:
4965:
4884:
4803:
4732:
4705:
4672:
4061:{\displaystyle T^{-1}Q=\{T^{-1}Q_{1},\ldots ,T^{-1}Q_{k}\}.}
3927:
3702:
3369:
3317:
3122:
3001:
2953:
2435:
1865:
1781:
1698:
1632:
1421:
1397:
1373:
1325:
1267:
1159:
1102:
1071:
420:
233:
177:
110:
1364:, but this is not enough to specify all such possible maps
754:
2707:
above. In particular, the transformations obey the rules:
2026:{\displaystyle w\times l\times h\times \mathbb {R} ^{3}.}
6572:
Ergodic Theory, Symbolic
Dynamics and Hyperbolic Spaces
6494:"Bernoulli shifts with the same entropy are isomorphic"
6366:"On invariants for measure preserving transformations"
5985:
5435:
into finitely many open intervals, such that for some
4441:
2760:{\displaystyle T_{0}=\mathrm {id} _{X}:X\rightarrow X}
1847:
describes this stirring, mixing, etc. then the system
919:
More generally, the paint that would arrive at subset
81:
A measure-preserving dynamical system is defined as a
6116:
6090:
6067:
6047:
6008:
5921:
5897:
5870:
5831:
5802:
5776:
5728:
5708:
5666:
5596:
5541:
5514:
5467:
5441:
5421:
5401:
5356:
5259:
5239:
5211:
5191:
5056:
5005:
4862:
4791:
4696:
4669:
4291:
4125:
3974:
3915:
3815:
3690:
3608:
3582:
3562:
3521:
3495:
3475:
3436:
3408:
3357:
3305:
3251:
3225:
3204:
3143:
3113:
3083:
3040:
2989:
2941:
2848:
2783:
2716:
2665:
upon the given probability space) of transformations
2574:ÎŒ could be the normalized angle measure dΞ/2Ï on the
2535:
2486:
2432:
2411:
2391:
2320:
2300:
2185:
2157:
2130:
2063:
2039:
1989:
1969:
1934:
1902:
1853:
1833:
1769:
1742:
1722:
1686:
1662:
1620:
1589:
1569:
1523:
1471:
1418:
1394:
1370:
1322:
1264:
1232:
1204:
1184:
1156:
1099:
1068:
1002:
963:
925:
882:
850:
810:
778:
649:
617:
561:
498:
408:
388:
348:
311:
276:
224:
202:
174:
152:
98:
89:
transformation on it. In more detail, it is a system
45:
is an object of study in the abstract formulation of
6614:
The entropy of strange billiards inside n-simplexes.
5336:{\displaystyle h_{\mu }(T)=\int \ln |dT/dx|\mu (dx)}
4785:
The measure-theoretic entropy of a dynamical system
7493:
7441:
7394:
7294:
7187:
7080:
6849:
6722:
6663:
3762:measurable pair-wise disjoint sets. Given a point
2470:{\displaystyle {\mathcal {O}}\left(2^{-3N}\right).}
804:, and then map the paint forward. The paint on the
53:in particular. Measure-preserving systems obey the
6122:
6102:
6073:
6053:
6033:
5991:
5931:
5903:
5880:
5856:
5814:
5788:
5762:
5714:
5694:
5644:
5582:
5527:
5508:on each open interval. Markov means that for each
5500:
5453:
5427:
5407:
5387:
5335:
5245:
5225:
5205:is an ergodic, piecewise expanding, and Markov on
5197:
5125:
5039:
4980:
4825:
4774:
4679:
4634:
4267:
4060:
3949:
3854:
3724:
3632:
3594:
3568:
3545:
3507:
3481:
3454:
3415:
3391:
3339:
3287:
3237:
3210:
3186:
3129:
3092:
3058:
3023:
2975:
2885:
2828:
2759:
2559:
2492:
2469:
2417:
2397:
2377:
2306:
2286:
2163:
2136:
2115:
2045:
2025:
1975:
1955:
1908:
1887:
1839:
1803:
1748:
1728:
1708:
1668:
1648:
1595:
1575:
1547:
1505:
1455:
1404:
1380:
1356:
1308:
1250:
1210:
1190:
1166:
1139:
1078:
1051:
988:
949:
908:
868:
836:
796:
760:
635:
611:Consider the typical measure on the unit interval
600:
547:
475:
394:
366:
332:
297:
258:
208:
184:
158:
132:
6410:Equivalence of measure preserving transformations
3785:can belong to only one of the parts as well. The
2603:with the definition of an appropriate measure, a
1827:or other such processes. If a transformation map
6529:Encyclopedia of Mathematics and its Applications
6412:. Mem. American Mathematical Soc. Vol. 37.
5999:is invertible, measure preserving, and ergodic.
5857:{\displaystyle {\mathcal {R}}\subset U\times U.}
5080:
4896:
3073:if it satisfies the following three properties:
5656:Classification and anti-classification theorems
1388:. That is, conservative, Borel-preserving maps
259:{\displaystyle \mu :{\mathcal {B}}\rightarrow }
3648:of dynamical systems and their homomorphisms.
3427:if, in addition, there exists another mapping
2696:âȘ {0}, or [0, +â)), where each transformation
6641:
6165:
6163:
5722:be the set of all measure preserving systems
5583:{\displaystyle T(I_{i})\cap I_{i}=\emptyset }
3465:that is also a homomorphism, which satisfies
8:
7386:RieszâMarkovâKakutani representation theorem
6326:Journal of the European Mathematical Society
5960:are classified by their metric entropy. See
4625:
4340:
4259:
4138:
4052:
3994:
3187:{\displaystyle \mu (\varphi ^{-1}B)=\nu (B)}
6531:. Vol. 54. Cambridge University Press.
5969:
1412:cannot, in general, be written in the form
1309:{\displaystyle {\mathcal {T}}(A)=T^{-1}(A)}
1140:{\displaystyle {\mathcal {T}}:P(X)\to P(X)}
7481:Vitale's random BrunnâMinkowski inequality
7398:
6648:
6634:
6626:
6319:Foreman, Matthew; Weiss, Benjamin (2022).
5864:The goal is then to describe the relation
3412:
2893:, whenever all the terms are well-defined.
426:
425:
6511:
6381:
6348:
6338:
6287:
6115:
6089:
6066:
6046:
6013:
6007:
5984:
5923:
5922:
5920:
5896:
5872:
5871:
5869:
5833:
5832:
5830:
5801:
5775:
5763:{\displaystyle (X,{\mathcal {B}},\mu ,T)}
5739:
5738:
5727:
5707:
5677:
5676:
5665:
5636:
5623:
5607:
5595:
5568:
5552:
5540:
5519:
5513:
5481:
5468:
5466:
5440:
5420:
5400:
5361:
5355:
5313:
5302:
5291:
5264:
5258:
5238:
5219:
5218:
5210:
5190:
5111:
5110:
5095:
5084:
5083:
5061:
5055:
5040:{\displaystyle (X,{\mathcal {B}},T,\mu )}
5016:
5015:
5004:
4964:
4963:
4954:
4944:
4933:
4911:
4899:
4883:
4882:
4867:
4861:
4826:{\displaystyle (X,{\mathcal {B}},T,\mu )}
4802:
4801:
4790:
4731:
4730:
4723:
4704:
4703:
4695:
4671:
4670:
4668:
4606:
4601:
4588:
4567:
4562:
4549:
4534:
4529:
4513:
4451:
4440:
4437:
4424:
4419:
4406:
4385:
4380:
4367:
4352:
4347:
4321:
4311:
4300:
4292:
4290:
4244:
4231:
4158:
4145:
4124:
4046:
4033:
4014:
4001:
3979:
3973:
3950:{\displaystyle (X,{\mathcal {B}},T,\mu )}
3926:
3925:
3914:
3841:
3836:
3820:
3814:
3725:{\displaystyle (X,{\mathcal {B}},T,\mu )}
3701:
3700:
3689:
3607:
3581:
3561:
3520:
3494:
3474:
3435:
3407:
3392:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
3368:
3367:
3356:
3340:{\displaystyle (Y,{\mathcal {B}},\nu ,S)}
3316:
3315:
3304:
3288:{\displaystyle \varphi (Tx)=S(\varphi x)}
3250:
3224:
3203:
3154:
3142:
3121:
3120:
3112:
3082:
3039:
3024:{\displaystyle (Y,{\mathcal {B}},\nu ,S)}
3000:
2999:
2988:
2976:{\displaystyle (X,{\mathcal {A}},\mu ,T)}
2952:
2951:
2940:
2874:
2858:
2853:
2847:
2814:
2801:
2788:
2782:
2739:
2731:
2721:
2715:
2553:
2552:
2534:
2485:
2448:
2434:
2433:
2431:
2410:
2405:atoms, the probability is the product of
2390:
2369:
2356:
2343:
2319:
2299:
2275:
2270:
2261:
2256:
2247:
2234:
2221:
2190:
2184:
2156:
2129:
2101:
2097:
2096:
2086:
2062:
2038:
2014:
2010:
2009:
1988:
1968:
1933:
1901:
1888:{\displaystyle (X,{\mathcal {B}},\mu ,T)}
1864:
1863:
1852:
1832:
1804:{\displaystyle (X,{\mathcal {B}},\mu ,T)}
1780:
1779:
1768:
1741:
1721:
1697:
1696:
1685:
1661:
1631:
1630:
1619:
1588:
1568:
1522:
1482:
1470:
1420:
1419:
1417:
1396:
1395:
1393:
1372:
1371:
1369:
1324:
1323:
1321:
1288:
1266:
1265:
1263:
1231:
1203:
1183:
1158:
1157:
1155:
1101:
1100:
1098:
1070:
1069:
1067:
1028:
1001:
968:
962:
924:
889:
881:
849:
823:
809:
777:
743:
729:
705:
691:
677:
670:
669:
648:
616:
560:
509:
497:
437:
419:
418:
407:
387:
347:
310:
275:
232:
231:
223:
201:
176:
175:
173:
151:
133:{\displaystyle (X,{\mathcal {B}},\mu ,T)}
109:
108:
97:
5645:{\displaystyle T(I_{i})\cap I_{i}=I_{i}}
2829:{\displaystyle T_{s}\circ T_{t}=T_{t+s}}
6247:Pollicott, Mark; Yuri, Michiko (1998).
6159:
5695:{\displaystyle (X,{\mathcal {B}},\mu )}
5163:into the intervals . Every real number
2378:{\displaystyle x,y,z,v_{x},v_{y},v_{z}}
1649:{\displaystyle (X,{\mathcal {B}},\mu )}
1456:{\displaystyle {\mathcal {T}}(A)=T(A);}
1052:{\displaystyle \mu (A)=\mu (T^{-1}(A))}
608:. This can be understood intuitively.
548:{\displaystyle \mu (T^{-1}(A))=\mu (A)}
318:
6143:on the existence of invariant measures
3869:of the dynamical system. A partition
1357:{\displaystyle {\mathcal {T}}(A)=T(A)}
1150:Consider now the special case of maps
555:instead of the forward transformation
5226:{\displaystyle X\subset \mathbb {R} }
7:
7494:Applications & related
6616:J. Phys. A 28(17), page 5033, 1995.
6250:Dynamical Systems and Ergodic Theory
6235:The Shannon-McMillan-Breiman Theorem
6218:"Metric Entropy of Dynamical System"
5501:{\displaystyle |T'|\geq 1+\epsilon }
3855:{\displaystyle T^{n}x\in Q_{a_{n}}.}
2645:Generalization to groups and monoids
333:{\displaystyle \mu (\varnothing )=0}
6084:If the entropy is exactly equal to
3130:{\displaystyle B\in {\mathcal {B}}}
2703:satisfies the same requirements as
1316:. Of course, one could also define
43:measure-preserving dynamical system
6590:, "Entropy in Dynamical Systems" (
5577:
5535:from those open intervals, either
4906:
4280:refinement of an iterated pullback
3633:{\displaystyle y=\varphi (\psi y)}
3546:{\displaystyle x=\psi (\varphi x)}
2735:
2732:
2560:{\displaystyle x\mapsto 2x\mod 1.}
1956:{\displaystyle w\times l\times h,}
1709:{\displaystyle (X,{\mathcal {B}})}
844:half is spread thinly over all of
601:{\displaystyle \mu (T(A))=\mu (A)}
409:
27:Subject of study in ergodic theory
25:
6408:; Rudolph, D.; Weiss, B. (1982).
6172:An Introduction to Ergodic Theory
5159:. That is, one may partition the
3781:. Similarly, the iterated point
3071:homomorphism of dynamical systems
2886:{\displaystyle T_{s}^{-1}=T_{-s}}
2627:Liouville's theorem (Hamiltonian)
7423:Lebesgue differentiation theorem
7304:Carathéodory's extension theorem
5970:Krieger finite generator theorem
3793:, with regards to the partition
3425:isomorphism of dynamical systems
2598:interval exchange transformation
367:{\displaystyle T:X\rightarrow X}
6034:{\displaystyle h_{T}\leq \ln k}
4516:
4515:
4439:
4278:With these two constructs, the
3797:, is the sequence of integers {
3059:{\displaystyle \varphi :X\to Y}
2935:Consider two dynamical systems
2548:
1506:{\displaystyle \mu (T^{-1}(A))}
665:
69:systems) as well as systems in
6612:T. SchĂŒrmann and I. Hoffmann,
6598:), appearing as Chapter 16 in
5932:{\displaystyle {\mathcal {R}}}
5881:{\displaystyle {\mathcal {R}}}
5757:
5729:
5689:
5667:
5613:
5600:
5558:
5545:
5482:
5469:
5454:{\displaystyle \epsilon >0}
5376:
5370:
5330:
5321:
5314:
5292:
5276:
5270:
5117:
5101:
5073:
5067:
5034:
5006:
4903:
4889:
4873:
4820:
4792:
4766:
4760:
4748:
4742:
4710:
4700:
4680:{\displaystyle {\mathcal {Q}}}
4250:
4224:
3944:
3916:
3719:
3691:
3627:
3618:
3540:
3531:
3446:
3386:
3358:
3334:
3306:
3282:
3273:
3264:
3255:
3181:
3175:
3166:
3147:
3050:
3018:
2990:
2970:
2942:
2751:
2539:
2253:
2196:
2173:MaxwellâBoltzmann distribution
2083:
2064:
1882:
1854:
1798:
1770:
1760:and the route to equilibrium.
1703:
1687:
1643:
1621:
1542:
1539:
1533:
1527:
1500:
1497:
1491:
1475:
1447:
1441:
1432:
1426:
1405:{\displaystyle {\mathcal {T}}}
1381:{\displaystyle {\mathcal {T}}}
1351:
1345:
1336:
1330:
1303:
1297:
1278:
1272:
1242:
1167:{\displaystyle {\mathcal {T}}}
1134:
1128:
1122:
1119:
1113:
1079:{\displaystyle {\mathcal {T}}}
1046:
1043:
1037:
1021:
1012:
1006:
983:
977:
944:
932:
903:
883:
863:
851:
831:
811:
791:
779:
630:
618:
595:
589:
580:
577:
571:
565:
542:
536:
527:
524:
518:
502:
470:
464:
455:
452:
446:
430:
358:
321:
315:
286:
280:
253:
241:
238:
185:{\displaystyle {\mathcal {B}}}
143:with the following structure:
127:
99:
1:
6130:symbols with equal measures.
6061:, then the system has a size-
3680:Symbolic names and generators
2836:, whenever all the terms are
2314:having position and velocity
6544:Entropy in dynamical systems
6542:Downarowicz, Tomasz (2011).
6513:10.1016/0001-8708(70)90029-0
5702:be a measure space, and let
4833:with respect to a partition
3885:has a unique symbolic name.
3684:Consider a dynamical system
3455:{\displaystyle \psi :Y\to X}
2661:, in which case we have the
1609:FrobeniusâPerron eigenvector
1603:can now be understood as an
57:, and are a special case of
7476:PrĂ©kopaâLeindler inequality
6147:Poincaré recurrence theorem
5388:{\displaystyle T^{-1}(A)=A}
3774:belongs to only one of the
2294:is the probability of atom
1676:. The other, discussed in
55:Poincaré recurrence theorem
7578:
7418:Lebesgue's density theorem
6471:Doklady Akademii Nauk SSSR
6276:Journal of Modern Dynamics
6198:Doklady Akademii Nauk SSSR
4652:
3676:according to the measure.
3416:{\displaystyle \varphi \;}
3093:{\displaystyle \varphi \ }
1548:{\displaystyle \mu (T(A))}
1218:(because we want it to be
29:
7471:MinkowskiâSteiner formula
7401:
7286:Projection-valued measure
6141:KrylovâBogolyubov theorem
4997:measure-theoretic entropy
4649:Measure-theoretic entropy
3909:} and a dynamical system
2638:KrylovâBogolyubov theorem
2502:van der Waals interaction
1813:thermodynamic equilibrium
989:{\displaystyle T^{-1}(A)}
950:{\displaystyle A\subset }
298:{\displaystyle \mu (X)=1}
71:thermodynamic equilibrium
7454:Isoperimetric inequality
7433:VitaliâHahnâSaks theorem
6762:Carathéodory's criterion
3889:Operations on partitions
3881:if Ό-almost every point
2584:equidistribution theorem
1555:is generically called a
1251:{\displaystyle T:X\to X}
18:Kolmogorov-Sinai entropy
7552:Entropy and information
7459:BrunnâMinkowski theorem
7328:Decomposition theorems
6499:Advances in Mathematics
6170:Walters, Peter (2000).
5796:of two transformations
5789:{\displaystyle S\sim T}
4993:KolmogorovâSinai metric
3893:Given a partition Q = {
2616:random dynamical system
2605:subshift of finite type
2529: : [0,1) â [0,1),
2057:somewhere in the space
1926:microcanonical ensemble
876:, and the paint on the
7506:Descriptive set theory
7406:Disintegration theorem
6841:Universally measurable
6216:Sinai, Ya. G. (2007).
6124:
6104:
6075:
6055:
6035:
5993:
5933:
5905:
5882:
5858:
5816:
5790:
5764:
5716:
5696:
5646:
5584:
5529:
5502:
5455:
5429:
5409:
5389:
5337:
5247:
5227:
5199:
5127:
5041:
4999:of a dynamical system
4982:
4949:
4827:
4776:
4681:
4636:
4316:
4269:
4062:
3951:
3856:
3726:
3644:Hence, one may form a
3634:
3596:
3595:{\displaystyle y\in Y}
3570:
3547:
3509:
3508:{\displaystyle x\in X}
3483:
3456:
3417:
3393:
3341:
3289:
3239:
3238:{\displaystyle x\in X}
3212:
3188:
3131:
3094:
3060:
3025:
2977:
2887:
2830:
2761:
2623:symplectic volume form
2567:
2561:
2494:
2471:
2419:
2399:
2379:
2308:
2288:
2165:
2138:
2117:
2053:atoms would then be a
2047:
2033:A given collection of
2027:
1977:
1957:
1910:
1889:
1841:
1805:
1750:
1736:, and asks about maps
1730:
1710:
1670:
1650:
1597:
1577:
1549:
1507:
1457:
1406:
1382:
1358:
1310:
1252:
1212:
1192:
1168:
1141:
1080:
1053:
990:
957:comes from the subset
951:
910:
870:
838:
798:
762:
637:
602:
549:
477:
396:
368:
334:
299:
260:
210:
186:
160:
134:
7308:Convergence theorems
6767:Cylindrical Ï-algebra
6435:Annals of Mathematics
6125:
6105:
6103:{\displaystyle \ln k}
6076:
6056:
6036:
5994:
5934:
5906:
5883:
5859:
5817:
5791:
5765:
5717:
5697:
5647:
5585:
5530:
5528:{\displaystyle I_{i}}
5503:
5456:
5430:
5410:
5390:
5338:
5248:
5228:
5200:
5182:may also be defined.
5148:is log 2, since
5128:
5042:
4983:
4929:
4853:} is then defined as
4828:
4777:
4682:
4637:
4296:
4270:
4063:
3952:
3857:
3727:
3674:distributed uniformly
3635:
3597:
3571:
3548:
3510:
3484:
3457:
3418:
3394:
3342:
3290:
3240:
3213:
3189:
3132:
3095:
3061:
3026:
2978:
2888:
2831:
2762:
2632:for certain maps and
2562:
2520:
2495:
2472:
2420:
2400:
2380:
2309:
2289:
2166:
2139:
2118:
2048:
2028:
1978:
1958:
1911:
1890:
1842:
1806:
1751:
1731:
1711:
1671:
1651:
1598:
1578:
1550:
1508:
1458:
1407:
1383:
1359:
1311:
1253:
1213:
1193:
1169:
1142:
1081:
1054:
991:
952:
911:
871:
839:
799:
763:
638:
603:
550:
478:
397:
378:transformation which
369:
335:
300:
261:
211:
187:
161:
135:
65:(in particular, most
7376:Minkowski inequality
7250:Cylinder set measure
7135:Infinite-dimensional
6750:equivalence relation
6680:Lebesgue integration
6114:
6088:
6065:
6045:
6006:
5983:
5945:Kakutani equivalence
5919:
5911:is endowed with the
5895:
5868:
5829:
5824:equivalence relation
5800:
5774:
5726:
5706:
5664:
5594:
5539:
5512:
5465:
5439:
5419:
5399:
5354:
5257:
5246:{\displaystyle \mu }
5237:
5209:
5189:
5054:
5003:
4860:
4789:
4694:
4667:
4289:
4123:
3972:
3913:
3879:generating partition
3813:
3688:
3606:
3580:
3569:{\displaystyle \nu }
3560:
3519:
3493:
3482:{\displaystyle \mu }
3473:
3434:
3406:
3355:
3303:
3249:
3223:
3211:{\displaystyle \mu }
3202:
3141:
3111:
3081:
3038:
2987:
2939:
2846:
2781:
2714:
2533:
2484:
2430:
2409:
2389:
2318:
2298:
2183:
2164:{\displaystyle \mu }
2155:
2128:
2061:
2037:
1987:
1967:
1932:
1909:{\displaystyle \mu }
1900:
1851:
1831:
1767:
1749:{\displaystyle \mu }
1740:
1720:
1684:
1660:
1618:
1596:{\displaystyle \mu }
1587:
1567:
1521:
1469:
1416:
1392:
1368:
1320:
1262:
1230:
1202:
1182:
1154:
1097:
1066:
1000:
961:
923:
880:
848:
808:
776:
647:
615:
559:
496:
406:
395:{\displaystyle \mu }
386:
346:
309:
274:
222:
200:
172:
150:
96:
59:conservative systems
7371:Hölder's inequality
7233:of random variables
7195:Measurable function
7082:Particular measures
6671:Absolute continuity
6364:Hjorth, G. (2001).
6298:10.3934/jmd.2019024
5977: —
5815:{\displaystyle S,T}
5350:Ergodic means that
5180:topological entropy
4655:approximate entropy
4071:Further, given two
2866:
1758:dissipative systems
1062:Consider a mapping
268:probability measure
63:classical mechanics
7557:Information theory
7511:Probability theory
6836:Transverse measure
6814:Non-measurable set
6796:Locally measurable
6120:
6100:
6071:
6051:
6031:
5989:
5971:
5929:
5901:
5878:
5854:
5812:
5786:
5770:. An isomorphism
5760:
5712:
5692:
5642:
5580:
5525:
5498:
5451:
5425:
5405:
5385:
5333:
5243:
5223:
5195:
5123:
5090:
5037:
4978:
4910:
4823:
4772:
4738:
4677:
4632:
4630:
4445:
4265:
4058:
3947:
3852:
3722:
3630:
3592:
3566:
3543:
3505:
3479:
3452:
3413:
3389:
3337:
3285:
3235:
3208:
3184:
3127:
3090:
3056:
3021:
2973:
2883:
2849:
2826:
2757:
2568:
2557:
2525:) preserving map:
2490:
2467:
2415:
2395:
2375:
2304:
2284:
2161:
2147:In the case of an
2134:
2113:
2043:
2023:
1973:
1953:
1906:
1885:
1837:
1801:
1746:
1726:
1706:
1666:
1646:
1593:
1573:
1545:
1513:has the form of a
1503:
1453:
1402:
1378:
1354:
1306:
1248:
1208:
1188:
1164:
1137:
1076:
1049:
986:
947:
906:
866:
834:
794:
758:
753:
633:
598:
545:
473:
392:
364:
330:
295:
256:
206:
182:
156:
130:
87:measure-preserving
7542:Dynamical systems
7529:
7528:
7489:
7488:
7218:almost everywhere
7164:Spherical measure
7062:Strictly positive
6990:Projection-valued
6730:Almost everywhere
6703:Probability space
6553:978-0-521-88885-1
6383:10.4064/FM169-1-2
6350:10.4171/JEMS/1151
6260:978-0-521-57294-1
6123:{\displaystyle k}
6074:{\displaystyle k}
6054:{\displaystyle k}
6041:for some integer
5904:{\displaystyle U}
5715:{\displaystyle U}
5428:{\displaystyle X}
5408:{\displaystyle A}
5198:{\displaystyle T}
5146:Bernoulli process
5079:
4919:
4895:
4719:
4507:
4480:
4444:
4443: where
4220:
4193:
3867:symbolic dynamics
3347:is then called a
3089:
3031:. Then a mapping
2924:The concept of a
2769:identity function
2663:action of a group
2493:{\displaystyle T}
2418:{\displaystyle N}
2398:{\displaystyle N}
2307:{\displaystyle i}
2137:{\displaystyle X}
2046:{\displaystyle N}
1976:{\displaystyle N}
1840:{\displaystyle T}
1729:{\displaystyle T}
1678:transfer operator
1669:{\displaystyle T}
1607:; it is just the
1605:invariant measure
1576:{\displaystyle T}
1561:transfer operator
1211:{\displaystyle X}
1191:{\displaystyle X}
1178:) and also sends
732:
694:
209:{\displaystyle X}
159:{\displaystyle X}
83:probability space
47:dynamical systems
16:(Redirected from
7569:
7464:Milman's reverse
7447:
7445:Lebesgue measure
7399:
6803:
6789:infimum/supremum
6710:Measurable space
6650:
6643:
6636:
6627:
6558:
6557:
6539:
6533:
6532:
6524:
6518:
6517:
6515:
6486:
6480:
6479:
6465:
6459:
6458:
6430:
6424:
6423:
6402:
6396:
6395:
6385:
6361:
6355:
6354:
6352:
6342:
6333:(8): 2605â2690.
6316:
6310:
6309:
6291:
6271:
6265:
6264:
6244:
6238:
6231:
6225:
6224:
6222:
6213:
6207:
6206:
6192:
6186:
6185:
6167:
6129:
6127:
6126:
6121:
6109:
6107:
6106:
6101:
6080:
6078:
6077:
6072:
6060:
6058:
6057:
6052:
6040:
6038:
6037:
6032:
6018:
6017:
5998:
5996:
5995:
5990:
5978:
5975:
5958:Bernoulli shifts
5938:
5936:
5935:
5930:
5928:
5927:
5910:
5908:
5907:
5902:
5887:
5885:
5884:
5879:
5877:
5876:
5863:
5861:
5860:
5855:
5838:
5837:
5821:
5819:
5818:
5813:
5795:
5793:
5792:
5787:
5769:
5767:
5766:
5761:
5744:
5743:
5721:
5719:
5718:
5713:
5701:
5699:
5698:
5693:
5682:
5681:
5651:
5649:
5648:
5643:
5641:
5640:
5628:
5627:
5612:
5611:
5589:
5587:
5586:
5581:
5573:
5572:
5557:
5556:
5534:
5532:
5531:
5526:
5524:
5523:
5507:
5505:
5504:
5499:
5485:
5480:
5472:
5460:
5458:
5457:
5452:
5434:
5432:
5431:
5426:
5414:
5412:
5411:
5406:
5394:
5392:
5391:
5386:
5369:
5368:
5342:
5340:
5339:
5334:
5317:
5306:
5295:
5269:
5268:
5252:
5250:
5249:
5244:
5232:
5230:
5229:
5224:
5222:
5204:
5202:
5201:
5196:
5157:binary expansion
5132:
5130:
5129:
5124:
5116:
5115:
5100:
5099:
5089:
5088:
5066:
5065:
5046:
5044:
5043:
5038:
5021:
5020:
4987:
4985:
4984:
4979:
4974:
4970:
4969:
4968:
4962:
4961:
4948:
4943:
4920:
4912:
4909:
4888:
4887:
4872:
4871:
4832:
4830:
4829:
4824:
4807:
4806:
4781:
4779:
4778:
4773:
4737:
4736:
4735:
4709:
4708:
4686:
4684:
4683:
4678:
4676:
4675:
4641:
4639:
4638:
4633:
4631:
4618:
4614:
4613:
4612:
4611:
4610:
4596:
4595:
4574:
4573:
4572:
4571:
4557:
4556:
4541:
4540:
4539:
4538:
4514:
4511:
4505:
4478:
4456:
4455:
4446:
4442:
4438:
4435:
4431:
4430:
4429:
4428:
4414:
4413:
4392:
4391:
4390:
4389:
4375:
4374:
4359:
4358:
4357:
4356:
4329:
4328:
4315:
4310:
4274:
4272:
4271:
4266:
4249:
4248:
4236:
4235:
4218:
4191:
4163:
4162:
4150:
4149:
4112:}, define their
4067:
4065:
4064:
4059:
4051:
4050:
4041:
4040:
4019:
4018:
4009:
4008:
3987:
3986:
3956:
3954:
3953:
3948:
3931:
3930:
3861:
3859:
3858:
3853:
3848:
3847:
3846:
3845:
3825:
3824:
3731:
3729:
3728:
3723:
3706:
3705:
3672:of the point is
3639:
3637:
3636:
3631:
3601:
3599:
3598:
3593:
3575:
3573:
3572:
3567:
3552:
3550:
3549:
3544:
3514:
3512:
3511:
3506:
3488:
3486:
3485:
3480:
3461:
3459:
3458:
3453:
3422:
3420:
3419:
3414:
3398:
3396:
3395:
3390:
3373:
3372:
3346:
3344:
3343:
3338:
3321:
3320:
3294:
3292:
3291:
3286:
3244:
3242:
3241:
3236:
3217:
3215:
3214:
3209:
3193:
3191:
3190:
3185:
3162:
3161:
3136:
3134:
3133:
3128:
3126:
3125:
3099:
3097:
3096:
3091:
3087:
3065:
3063:
3062:
3057:
3030:
3028:
3027:
3022:
3005:
3004:
2982:
2980:
2979:
2974:
2957:
2956:
2932:may be defined.
2892:
2890:
2889:
2884:
2882:
2881:
2865:
2857:
2835:
2833:
2832:
2827:
2825:
2824:
2806:
2805:
2793:
2792:
2766:
2764:
2763:
2758:
2744:
2743:
2738:
2726:
2725:
2680:parametrized by
2634:Markov processes
2591:Bernoulli scheme
2582:a rotation. See
2566:
2564:
2563:
2558:
2523:Lebesgue measure
2499:
2497:
2496:
2491:
2476:
2474:
2473:
2468:
2463:
2459:
2458:
2439:
2438:
2424:
2422:
2421:
2416:
2404:
2402:
2401:
2396:
2384:
2382:
2381:
2376:
2374:
2373:
2361:
2360:
2348:
2347:
2313:
2311:
2310:
2305:
2293:
2291:
2290:
2285:
2280:
2279:
2266:
2265:
2252:
2251:
2239:
2238:
2226:
2225:
2195:
2194:
2171:is given by the
2170:
2168:
2167:
2162:
2143:
2141:
2140:
2135:
2122:
2120:
2119:
2114:
2109:
2108:
2100:
2091:
2090:
2052:
2050:
2049:
2044:
2032:
2030:
2029:
2024:
2019:
2018:
2013:
1982:
1980:
1979:
1974:
1962:
1960:
1959:
1954:
1920:Informal example
1915:
1913:
1912:
1907:
1894:
1892:
1891:
1886:
1869:
1868:
1846:
1844:
1843:
1838:
1810:
1808:
1807:
1802:
1785:
1784:
1755:
1753:
1752:
1747:
1735:
1733:
1732:
1727:
1715:
1713:
1712:
1707:
1702:
1701:
1675:
1673:
1672:
1667:
1655:
1653:
1652:
1647:
1636:
1635:
1602:
1600:
1599:
1594:
1582:
1580:
1579:
1574:
1554:
1552:
1551:
1546:
1512:
1510:
1509:
1504:
1490:
1489:
1462:
1460:
1459:
1454:
1425:
1424:
1411:
1409:
1408:
1403:
1401:
1400:
1387:
1385:
1384:
1379:
1377:
1376:
1363:
1361:
1360:
1355:
1329:
1328:
1315:
1313:
1312:
1307:
1296:
1295:
1271:
1270:
1257:
1255:
1254:
1249:
1217:
1215:
1214:
1209:
1197:
1195:
1194:
1189:
1173:
1171:
1170:
1165:
1163:
1162:
1146:
1144:
1143:
1138:
1106:
1105:
1085:
1083:
1082:
1077:
1075:
1074:
1058:
1056:
1055:
1050:
1036:
1035:
995:
993:
992:
987:
976:
975:
956:
954:
953:
948:
915:
913:
912:
909:{\displaystyle }
907:
893:
875:
873:
872:
869:{\displaystyle }
867:
843:
841:
840:
837:{\displaystyle }
835:
827:
803:
801:
800:
797:{\displaystyle }
795:
767:
765:
764:
759:
757:
756:
747:
733:
730:
709:
695:
692:
642:
640:
639:
636:{\displaystyle }
634:
607:
605:
604:
599:
554:
552:
551:
546:
517:
516:
482:
480:
479:
474:
445:
444:
424:
423:
401:
399:
398:
393:
373:
371:
370:
365:
339:
337:
336:
331:
304:
302:
301:
296:
265:
263:
262:
257:
237:
236:
215:
213:
212:
207:
191:
189:
188:
183:
181:
180:
165:
163:
162:
157:
139:
137:
136:
131:
114:
113:
21:
7577:
7576:
7572:
7571:
7570:
7568:
7567:
7566:
7532:
7531:
7530:
7525:
7521:Spectral theory
7501:Convex analysis
7485:
7442:
7437:
7390:
7290:
7238:in distribution
7183:
7076:
6906:Logarithmically
6845:
6801:
6784:Essential range
6718:
6659:
6654:
6567:
6565:Further reading
6562:
6561:
6554:
6541:
6540:
6536:
6526:
6525:
6521:
6488:
6487:
6483:
6467:
6466:
6462:
6447:10.2307/1968872
6432:
6431:
6427:
6420:
6404:
6403:
6399:
6363:
6362:
6358:
6318:
6317:
6313:
6273:
6272:
6268:
6261:
6246:
6245:
6241:
6232:
6228:
6220:
6215:
6214:
6210:
6194:
6193:
6189:
6182:
6169:
6168:
6161:
6156:
6137:
6132:
6112:
6111:
6086:
6085:
6063:
6062:
6043:
6042:
6009:
6004:
6003:
5981:
5980:
5976:
5973:
5962:Ornstein theory
5917:
5916:
5915:, then the set
5893:
5892:
5866:
5865:
5827:
5826:
5798:
5797:
5772:
5771:
5724:
5723:
5704:
5703:
5662:
5661:
5658:
5632:
5619:
5603:
5592:
5591:
5564:
5548:
5537:
5536:
5515:
5510:
5509:
5473:
5463:
5462:
5437:
5436:
5417:
5416:
5397:
5396:
5357:
5352:
5351:
5260:
5255:
5254:
5235:
5234:
5207:
5206:
5187:
5186:
5091:
5057:
5052:
5051:
5001:
5000:
4950:
4928:
4924:
4863:
4858:
4857:
4852:
4843:
4787:
4786:
4692:
4691:
4665:
4664:
4663:of a partition
4657:
4651:
4629:
4628:
4602:
4597:
4584:
4563:
4558:
4545:
4530:
4525:
4524:
4520:
4509:
4508:
4447:
4433:
4432:
4420:
4415:
4402:
4381:
4376:
4363:
4348:
4343:
4333:
4317:
4287:
4286:
4240:
4227:
4154:
4141:
4121:
4120:
4111:
4102:
4090:
4084:
4042:
4029:
4010:
3997:
3975:
3970:
3969:
3911:
3910:
3908:
3899:
3891:
3837:
3832:
3816:
3811:
3810:
3805:
3779:
3748:
3742:
3686:
3685:
3682:
3654:
3604:
3603:
3578:
3577:
3558:
3557:
3517:
3516:
3491:
3490:
3471:
3470:
3432:
3431:
3404:
3403:
3353:
3352:
3301:
3300:
3247:
3246:
3221:
3220:
3200:
3199:
3150:
3139:
3138:
3109:
3108:
3079:
3078:
3036:
3035:
2985:
2984:
2937:
2936:
2922:
2902:
2870:
2844:
2843:
2810:
2797:
2784:
2779:
2778:
2730:
2717:
2712:
2711:
2701:
2670:
2647:
2531:
2530:
2515:
2482:
2481:
2444:
2440:
2428:
2427:
2407:
2406:
2387:
2386:
2365:
2352:
2339:
2316:
2315:
2296:
2295:
2271:
2257:
2243:
2230:
2217:
2186:
2181:
2180:
2177:product measure
2153:
2152:
2126:
2125:
2095:
2082:
2059:
2058:
2035:
2034:
2008:
1985:
1984:
1965:
1964:
1930:
1929:
1922:
1898:
1897:
1849:
1848:
1829:
1828:
1765:
1764:
1738:
1737:
1718:
1717:
1682:
1681:
1658:
1657:
1616:
1615:
1585:
1584:
1565:
1564:
1519:
1518:
1478:
1467:
1466:
1414:
1413:
1390:
1389:
1366:
1365:
1318:
1317:
1284:
1260:
1259:
1228:
1227:
1200:
1199:
1180:
1179:
1152:
1151:
1095:
1094:
1064:
1063:
1024:
998:
997:
964:
959:
958:
921:
920:
878:
877:
846:
845:
806:
805:
774:
773:
752:
751:
714:
713:
678:
645:
644:
613:
612:
557:
556:
505:
494:
493:
490:
433:
404:
403:
384:
383:
344:
343:
307:
306:
272:
271:
220:
219:
198:
197:
170:
169:
148:
147:
94:
93:
79:
67:non-dissipative
35:
28:
23:
22:
15:
12:
11:
5:
7575:
7573:
7565:
7564:
7562:Measure theory
7559:
7554:
7549:
7544:
7534:
7533:
7527:
7526:
7524:
7523:
7518:
7513:
7508:
7503:
7497:
7495:
7491:
7490:
7487:
7486:
7484:
7483:
7478:
7473:
7468:
7467:
7466:
7456:
7450:
7448:
7439:
7438:
7436:
7435:
7430:
7428:Sard's theorem
7425:
7420:
7415:
7414:
7413:
7411:Lifting theory
7402:
7396:
7392:
7391:
7389:
7388:
7383:
7378:
7373:
7368:
7367:
7366:
7364:FubiniâTonelli
7356:
7351:
7346:
7345:
7344:
7339:
7334:
7326:
7325:
7324:
7319:
7314:
7306:
7300:
7298:
7292:
7291:
7289:
7288:
7283:
7278:
7273:
7268:
7263:
7258:
7252:
7247:
7246:
7245:
7243:in probability
7240:
7230:
7225:
7220:
7214:
7213:
7212:
7207:
7202:
7191:
7189:
7185:
7184:
7182:
7181:
7176:
7171:
7166:
7161:
7156:
7155:
7154:
7144:
7139:
7138:
7137:
7127:
7122:
7117:
7112:
7107:
7102:
7097:
7092:
7086:
7084:
7078:
7077:
7075:
7074:
7069:
7064:
7059:
7054:
7049:
7044:
7039:
7034:
7029:
7024:
7023:
7022:
7017:
7012:
7002:
6997:
6992:
6987:
6977:
6972:
6967:
6962:
6957:
6952:
6950:Locally finite
6947:
6937:
6932:
6927:
6922:
6917:
6912:
6902:
6897:
6892:
6887:
6882:
6877:
6872:
6867:
6862:
6856:
6854:
6847:
6846:
6844:
6843:
6838:
6833:
6828:
6823:
6822:
6821:
6811:
6806:
6798:
6793:
6792:
6791:
6781:
6776:
6775:
6774:
6764:
6759:
6754:
6753:
6752:
6742:
6737:
6732:
6726:
6724:
6720:
6719:
6717:
6716:
6707:
6706:
6705:
6695:
6690:
6682:
6677:
6667:
6665:
6664:Basic concepts
6661:
6660:
6657:Measure theory
6655:
6653:
6652:
6645:
6638:
6630:
6624:
6623:
6610:
6588:Lai-Sang Young
6585:
6566:
6563:
6560:
6559:
6552:
6534:
6519:
6506:(3): 337â352.
6481:
6460:
6441:(2): 332â350.
6425:
6418:
6397:
6356:
6311:
6266:
6259:
6239:
6226:
6208:
6187:
6180:
6158:
6157:
6155:
6152:
6151:
6150:
6144:
6136:
6133:
6119:
6099:
6096:
6093:
6070:
6050:
6030:
6027:
6024:
6021:
6016:
6012:
5992:{\textstyle T}
5988:
5974:(Krieger 1970)
5967:
5966:
5965:
5955:
5926:
5900:
5875:
5853:
5850:
5847:
5844:
5841:
5836:
5811:
5808:
5805:
5785:
5782:
5779:
5759:
5756:
5753:
5750:
5747:
5742:
5737:
5734:
5731:
5711:
5691:
5688:
5685:
5680:
5675:
5672:
5669:
5657:
5654:
5639:
5635:
5631:
5626:
5622:
5618:
5615:
5610:
5606:
5602:
5599:
5579:
5576:
5571:
5567:
5563:
5560:
5555:
5551:
5547:
5544:
5522:
5518:
5497:
5494:
5491:
5488:
5484:
5479:
5476:
5471:
5450:
5447:
5444:
5424:
5404:
5384:
5381:
5378:
5375:
5372:
5367:
5364:
5360:
5332:
5329:
5326:
5323:
5320:
5316:
5312:
5309:
5305:
5301:
5298:
5294:
5290:
5287:
5284:
5281:
5278:
5275:
5272:
5267:
5263:
5242:
5221:
5217:
5214:
5194:
5134:
5133:
5122:
5119:
5114:
5109:
5106:
5103:
5098:
5094:
5087:
5082:
5078:
5075:
5072:
5069:
5064:
5060:
5047:is defined as
5036:
5033:
5030:
5027:
5024:
5019:
5014:
5011:
5008:
4989:
4988:
4977:
4973:
4967:
4960:
4957:
4953:
4947:
4942:
4939:
4936:
4932:
4927:
4923:
4918:
4915:
4908:
4905:
4902:
4898:
4894:
4891:
4886:
4881:
4878:
4875:
4870:
4866:
4848:
4841:
4822:
4819:
4816:
4813:
4810:
4805:
4800:
4797:
4794:
4783:
4782:
4771:
4768:
4765:
4762:
4759:
4756:
4753:
4750:
4747:
4744:
4741:
4734:
4729:
4726:
4722:
4718:
4715:
4712:
4707:
4702:
4699:
4687:is defined as
4674:
4650:
4647:
4643:
4642:
4627:
4624:
4621:
4617:
4609:
4605:
4600:
4594:
4591:
4587:
4583:
4580:
4577:
4570:
4566:
4561:
4555:
4552:
4548:
4544:
4537:
4533:
4528:
4523:
4519:
4512:
4510:
4504:
4501:
4498:
4495:
4492:
4489:
4486:
4483:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4454:
4450:
4436:
4434:
4427:
4423:
4418:
4412:
4409:
4405:
4401:
4398:
4395:
4388:
4384:
4379:
4373:
4370:
4366:
4362:
4355:
4351:
4346:
4342:
4339:
4336:
4334:
4332:
4327:
4324:
4320:
4314:
4309:
4306:
4303:
4299:
4295:
4294:
4282:is defined as
4276:
4275:
4264:
4261:
4258:
4255:
4252:
4247:
4243:
4239:
4234:
4230:
4226:
4223:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4161:
4157:
4153:
4148:
4144:
4140:
4137:
4134:
4131:
4128:
4107:
4100:
4088:
4082:
4069:
4068:
4057:
4054:
4049:
4045:
4039:
4036:
4032:
4028:
4025:
4022:
4017:
4013:
4007:
4004:
4000:
3996:
3993:
3990:
3985:
3982:
3978:
3946:
3943:
3940:
3937:
3934:
3929:
3924:
3921:
3918:
3904:
3897:
3890:
3887:
3863:
3862:
3851:
3844:
3840:
3835:
3831:
3828:
3823:
3819:
3801:
3777:
3746:
3740:
3721:
3718:
3715:
3712:
3709:
3704:
3699:
3696:
3693:
3681:
3678:
3653:
3652:Generic points
3650:
3642:
3641:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3591:
3588:
3585:
3565:
3554:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3504:
3501:
3498:
3478:
3463:
3462:
3451:
3448:
3445:
3442:
3439:
3411:
3388:
3385:
3382:
3379:
3376:
3371:
3366:
3363:
3360:
3336:
3333:
3330:
3327:
3324:
3319:
3314:
3311:
3308:
3297:
3296:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3234:
3231:
3228:
3207:
3195:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3160:
3157:
3153:
3149:
3146:
3124:
3119:
3116:
3105:
3086:
3067:
3066:
3055:
3052:
3049:
3046:
3043:
3020:
3017:
3014:
3011:
3008:
3003:
2998:
2995:
2992:
2972:
2969:
2966:
2963:
2960:
2955:
2950:
2947:
2944:
2921:
2918:
2900:
2895:
2894:
2880:
2877:
2873:
2869:
2864:
2861:
2856:
2852:
2841:
2823:
2820:
2817:
2813:
2809:
2804:
2800:
2796:
2791:
2787:
2776:
2756:
2753:
2750:
2747:
2742:
2737:
2734:
2729:
2724:
2720:
2699:
2668:
2646:
2643:
2642:
2641:
2630:
2619:
2608:
2601:
2594:
2587:
2556:
2551:
2547:
2544:
2541:
2538:
2521:Example of a (
2514:
2511:
2489:
2466:
2462:
2457:
2454:
2451:
2447:
2443:
2437:
2414:
2394:
2372:
2368:
2364:
2359:
2355:
2351:
2346:
2342:
2338:
2335:
2332:
2329:
2326:
2323:
2303:
2283:
2278:
2274:
2269:
2264:
2260:
2255:
2250:
2246:
2242:
2237:
2233:
2229:
2224:
2220:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2193:
2189:
2160:
2151:, the measure
2133:
2112:
2107:
2104:
2099:
2094:
2089:
2085:
2081:
2078:
2075:
2072:
2069:
2066:
2042:
2022:
2017:
2012:
2007:
2004:
2001:
1998:
1995:
1992:
1972:
1963:consisting of
1952:
1949:
1946:
1943:
1940:
1937:
1921:
1918:
1905:
1884:
1881:
1878:
1875:
1872:
1867:
1862:
1859:
1856:
1836:
1825:thermalization
1800:
1797:
1794:
1791:
1788:
1783:
1778:
1775:
1772:
1745:
1725:
1705:
1700:
1695:
1692:
1689:
1665:
1645:
1642:
1639:
1634:
1629:
1626:
1623:
1592:
1583:; the measure
1572:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1502:
1499:
1496:
1493:
1488:
1485:
1481:
1477:
1474:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1423:
1399:
1375:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1327:
1305:
1302:
1299:
1294:
1291:
1287:
1283:
1280:
1277:
1274:
1269:
1247:
1244:
1241:
1238:
1235:
1207:
1187:
1161:
1148:
1147:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1104:
1073:
1048:
1045:
1042:
1039:
1034:
1031:
1027:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
985:
982:
979:
974:
971:
967:
946:
943:
940:
937:
934:
931:
928:
905:
902:
899:
896:
892:
888:
885:
865:
862:
859:
856:
853:
833:
830:
826:
822:
819:
816:
813:
793:
790:
787:
784:
781:
768:. This is the
755:
750:
746:
742:
739:
736:
731: if
728:
725:
722:
719:
716:
715:
712:
708:
704:
701:
698:
693: if
690:
687:
684:
683:
681:
676:
673:
668:
664:
661:
658:
655:
652:
632:
629:
626:
623:
620:
597:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
544:
541:
538:
535:
532:
529:
526:
523:
520:
515:
512:
508:
504:
501:
489:
486:
485:
484:
472:
469:
466:
463:
460:
457:
454:
451:
448:
443:
440:
436:
432:
429:
422:
417:
414:
411:
391:
363:
360:
357:
354:
351:
341:
329:
326:
323:
320:
317:
314:
294:
291:
288:
285:
282:
279:
255:
252:
249:
246:
243:
240:
235:
230:
227:
217:
205:
194:σ-algebra
179:
167:
155:
141:
140:
129:
126:
123:
120:
117:
112:
107:
104:
101:
78:
75:
51:ergodic theory
32:Equal-area map
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7574:
7563:
7560:
7558:
7555:
7553:
7550:
7548:
7545:
7543:
7540:
7539:
7537:
7522:
7519:
7517:
7516:Real analysis
7514:
7512:
7509:
7507:
7504:
7502:
7499:
7498:
7496:
7492:
7482:
7479:
7477:
7474:
7472:
7469:
7465:
7462:
7461:
7460:
7457:
7455:
7452:
7451:
7449:
7446:
7440:
7434:
7431:
7429:
7426:
7424:
7421:
7419:
7416:
7412:
7409:
7408:
7407:
7404:
7403:
7400:
7397:
7395:Other results
7393:
7387:
7384:
7382:
7381:RadonâNikodym
7379:
7377:
7374:
7372:
7369:
7365:
7362:
7361:
7360:
7357:
7355:
7354:Fatou's lemma
7352:
7350:
7347:
7343:
7340:
7338:
7335:
7333:
7330:
7329:
7327:
7323:
7320:
7318:
7315:
7313:
7310:
7309:
7307:
7305:
7302:
7301:
7299:
7297:
7293:
7287:
7284:
7282:
7279:
7277:
7274:
7272:
7269:
7267:
7264:
7262:
7259:
7257:
7253:
7251:
7248:
7244:
7241:
7239:
7236:
7235:
7234:
7231:
7229:
7226:
7224:
7221:
7219:
7216:Convergence:
7215:
7211:
7208:
7206:
7203:
7201:
7198:
7197:
7196:
7193:
7192:
7190:
7186:
7180:
7177:
7175:
7172:
7170:
7167:
7165:
7162:
7160:
7157:
7153:
7150:
7149:
7148:
7145:
7143:
7140:
7136:
7133:
7132:
7131:
7128:
7126:
7123:
7121:
7118:
7116:
7113:
7111:
7108:
7106:
7103:
7101:
7098:
7096:
7093:
7091:
7088:
7087:
7085:
7083:
7079:
7073:
7070:
7068:
7065:
7063:
7060:
7058:
7055:
7053:
7050:
7048:
7045:
7043:
7040:
7038:
7035:
7033:
7030:
7028:
7025:
7021:
7020:Outer regular
7018:
7016:
7015:Inner regular
7013:
7011:
7010:Borel regular
7008:
7007:
7006:
7003:
7001:
6998:
6996:
6993:
6991:
6988:
6986:
6982:
6978:
6976:
6973:
6971:
6968:
6966:
6963:
6961:
6958:
6956:
6953:
6951:
6948:
6946:
6942:
6938:
6936:
6933:
6931:
6928:
6926:
6923:
6921:
6918:
6916:
6913:
6911:
6907:
6903:
6901:
6898:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6876:
6873:
6871:
6868:
6866:
6863:
6861:
6858:
6857:
6855:
6853:
6848:
6842:
6839:
6837:
6834:
6832:
6829:
6827:
6824:
6820:
6817:
6816:
6815:
6812:
6810:
6807:
6805:
6799:
6797:
6794:
6790:
6787:
6786:
6785:
6782:
6780:
6777:
6773:
6770:
6769:
6768:
6765:
6763:
6760:
6758:
6755:
6751:
6748:
6747:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6727:
6725:
6721:
6715:
6711:
6708:
6704:
6701:
6700:
6699:
6698:Measure space
6696:
6694:
6691:
6689:
6687:
6683:
6681:
6678:
6676:
6672:
6669:
6668:
6666:
6662:
6658:
6651:
6646:
6644:
6639:
6637:
6632:
6631:
6628:
6622:
6619:
6615:
6611:
6609:
6608:0-691-11338-6
6605:
6601:
6597:
6593:
6589:
6586:
6584:
6581:
6580:0-19-853390-X
6577:
6573:
6569:
6568:
6564:
6555:
6549:
6545:
6538:
6535:
6530:
6523:
6520:
6514:
6509:
6505:
6501:
6500:
6495:
6491:
6485:
6482:
6477:
6473:
6472:
6464:
6461:
6456:
6452:
6448:
6444:
6440:
6436:
6429:
6426:
6421:
6419:0-8218-2262-4
6415:
6411:
6407:
6401:
6398:
6393:
6389:
6384:
6379:
6375:
6371:
6367:
6360:
6357:
6351:
6346:
6341:
6336:
6332:
6328:
6327:
6322:
6315:
6312:
6307:
6303:
6299:
6295:
6290:
6285:
6281:
6277:
6270:
6267:
6262:
6256:
6252:
6251:
6243:
6240:
6237:
6236:
6230:
6227:
6219:
6212:
6209:
6204:
6200:
6199:
6191:
6188:
6183:
6181:0-387-95152-0
6177:
6173:
6166:
6164:
6160:
6153:
6148:
6145:
6142:
6139:
6138:
6134:
6131:
6117:
6097:
6094:
6091:
6082:
6068:
6048:
6028:
6025:
6022:
6019:
6014:
6010:
6000:
5986:
5963:
5959:
5956:
5953:
5952:
5951:
5948:
5946:
5942:
5914:
5913:weak topology
5898:
5889:
5851:
5848:
5845:
5842:
5839:
5825:
5809:
5806:
5803:
5783:
5780:
5777:
5754:
5751:
5748:
5745:
5735:
5732:
5709:
5686:
5683:
5673:
5670:
5655:
5653:
5637:
5633:
5629:
5624:
5620:
5616:
5608:
5604:
5597:
5574:
5569:
5565:
5561:
5553:
5549:
5542:
5520:
5516:
5495:
5492:
5489:
5486:
5477:
5474:
5448:
5445:
5442:
5422:
5402:
5382:
5379:
5373:
5365:
5362:
5358:
5348:
5346:
5327:
5324:
5318:
5310:
5307:
5303:
5299:
5296:
5288:
5285:
5282:
5279:
5273:
5265:
5261:
5240:
5215:
5212:
5192:
5183:
5181:
5177:
5174:If the space
5172:
5170:
5166:
5162:
5161:unit interval
5158:
5155:has a unique
5154:
5151:
5147:
5143:
5139:
5120:
5107:
5104:
5096:
5092:
5076:
5070:
5062:
5058:
5050:
5049:
5048:
5031:
5028:
5025:
5022:
5012:
5009:
4998:
4994:
4991:Finally, the
4975:
4971:
4958:
4955:
4951:
4945:
4940:
4937:
4934:
4930:
4925:
4921:
4916:
4913:
4900:
4892:
4879:
4876:
4868:
4864:
4856:
4855:
4854:
4851:
4847:
4840:
4836:
4817:
4814:
4811:
4808:
4798:
4795:
4769:
4763:
4757:
4754:
4751:
4745:
4739:
4727:
4724:
4720:
4716:
4713:
4697:
4690:
4689:
4688:
4662:
4656:
4648:
4646:
4622:
4619:
4615:
4607:
4603:
4598:
4592:
4589:
4585:
4581:
4578:
4575:
4568:
4564:
4559:
4553:
4550:
4546:
4542:
4535:
4531:
4526:
4521:
4517:
4502:
4499:
4496:
4493:
4490:
4487:
4484:
4481:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4452:
4448:
4425:
4421:
4416:
4410:
4407:
4403:
4399:
4396:
4393:
4386:
4382:
4377:
4371:
4368:
4364:
4360:
4353:
4349:
4344:
4337:
4335:
4330:
4325:
4322:
4318:
4312:
4307:
4304:
4301:
4297:
4285:
4284:
4283:
4281:
4262:
4256:
4253:
4245:
4241:
4237:
4232:
4228:
4221:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4159:
4155:
4151:
4146:
4142:
4135:
4132:
4129:
4126:
4119:
4118:
4117:
4115:
4110:
4106:
4099:
4095:
4091:
4081:
4077:
4074:
4055:
4047:
4043:
4037:
4034:
4030:
4026:
4023:
4020:
4015:
4011:
4005:
4002:
3998:
3991:
3988:
3983:
3980:
3976:
3968:
3967:
3966:
3964:
3961:-pullback of
3960:
3957:, define the
3941:
3938:
3935:
3932:
3922:
3919:
3907:
3903:
3896:
3888:
3886:
3884:
3880:
3876:
3872:
3868:
3849:
3842:
3838:
3833:
3829:
3826:
3821:
3817:
3809:
3808:
3807:
3804:
3800:
3796:
3792:
3788:
3787:symbolic name
3784:
3780:
3773:
3769:
3765:
3761:
3757:
3753:
3749:
3739:
3735:
3716:
3713:
3710:
3707:
3697:
3694:
3679:
3677:
3675:
3671:
3667:
3666:generic point
3663:
3659:
3651:
3649:
3647:
3624:
3621:
3615:
3612:
3609:
3589:
3586:
3583:
3563:
3555:
3537:
3534:
3528:
3525:
3522:
3502:
3499:
3496:
3476:
3468:
3467:
3466:
3449:
3443:
3440:
3437:
3430:
3429:
3428:
3426:
3409:
3400:
3383:
3380:
3377:
3374:
3364:
3361:
3350:
3331:
3328:
3325:
3322:
3312:
3309:
3279:
3276:
3270:
3267:
3261:
3258:
3252:
3232:
3229:
3226:
3219:
3205:
3196:
3178:
3172:
3169:
3163:
3158:
3155:
3151:
3144:
3117:
3114:
3106:
3103:
3084:
3076:
3075:
3074:
3072:
3053:
3047:
3044:
3041:
3034:
3033:
3032:
3015:
3012:
3009:
3006:
2996:
2993:
2967:
2964:
2961:
2958:
2948:
2945:
2933:
2931:
2927:
2920:Homomorphisms
2919:
2917:
2915:
2911:
2907:
2903:
2878:
2875:
2871:
2867:
2862:
2859:
2854:
2850:
2842:
2839:
2821:
2818:
2815:
2811:
2807:
2802:
2798:
2794:
2789:
2785:
2777:
2774:
2770:
2754:
2748:
2745:
2740:
2727:
2722:
2718:
2710:
2709:
2708:
2706:
2702:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2664:
2660:
2656:
2652:
2644:
2639:
2635:
2631:
2628:
2624:
2620:
2617:
2613:
2609:
2606:
2602:
2599:
2595:
2592:
2588:
2585:
2581:
2577:
2573:
2572:
2571:
2554:
2549:
2545:
2542:
2536:
2528:
2524:
2519:
2512:
2510:
2506:
2503:
2487:
2478:
2464:
2460:
2455:
2452:
2449:
2445:
2441:
2412:
2392:
2370:
2366:
2362:
2357:
2353:
2349:
2344:
2340:
2336:
2333:
2330:
2327:
2324:
2321:
2301:
2281:
2276:
2272:
2267:
2262:
2258:
2248:
2244:
2240:
2235:
2231:
2227:
2222:
2218:
2214:
2211:
2208:
2205:
2202:
2199:
2191:
2187:
2179:, in that if
2178:
2174:
2158:
2150:
2145:
2131:
2110:
2105:
2102:
2092:
2087:
2079:
2076:
2073:
2070:
2067:
2056:
2040:
2020:
2015:
2005:
2002:
1999:
1996:
1993:
1990:
1970:
1950:
1947:
1944:
1941:
1938:
1935:
1927:
1919:
1917:
1903:
1879:
1876:
1873:
1870:
1860:
1857:
1834:
1826:
1822:
1818:
1814:
1795:
1792:
1789:
1786:
1776:
1773:
1761:
1759:
1743:
1723:
1693:
1690:
1679:
1663:
1640:
1637:
1627:
1624:
1612:
1610:
1606:
1590:
1570:
1562:
1558:
1536:
1530:
1524:
1516:
1494:
1486:
1483:
1479:
1472:
1464:
1450:
1444:
1438:
1435:
1429:
1348:
1342:
1339:
1333:
1300:
1292:
1289:
1285:
1281:
1275:
1245:
1239:
1236:
1233:
1225:
1221:
1205:
1185:
1177:
1131:
1125:
1116:
1110:
1107:
1093:
1092:
1091:
1089:
1060:
1040:
1032:
1029:
1025:
1018:
1015:
1009:
1003:
980:
972:
969:
965:
941:
938:
935:
929:
926:
917:
900:
897:
894:
890:
886:
860:
857:
854:
828:
824:
820:
817:
814:
788:
785:
782:
771:
770:Bernoulli map
748:
744:
740:
737:
734:
726:
723:
720:
717:
710:
706:
702:
699:
696:
688:
685:
679:
674:
671:
666:
662:
659:
656:
653:
650:
627:
624:
621:
609:
592:
586:
583:
574:
568:
562:
539:
533:
530:
521:
513:
510:
506:
499:
487:
467:
461:
458:
449:
441:
438:
434:
427:
415:
412:
389:
381:
377:
361:
355:
352:
349:
342:
327:
324:
312:
292:
289:
283:
277:
269:
250:
247:
244:
228:
225:
218:
203:
195:
168:
153:
146:
145:
144:
124:
121:
118:
115:
105:
102:
92:
91:
90:
88:
84:
76:
74:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
7296:Main results
7032:Set function
6960:Metric outer
6915:Decomposable
6772:Cylinder set
6685:
6620:
6618:PDF-Document
6613:
6599:
6582:
6571:
6543:
6537:
6528:
6522:
6503:
6497:
6490:Ornstein, D.
6484:
6475:
6469:
6463:
6438:
6434:
6428:
6409:
6406:Ornstein, D.
6400:
6376:(1): 51â84.
6373:
6369:
6359:
6330:
6324:
6314:
6279:
6275:
6269:
6249:
6242:
6233:
6229:
6211:
6202:
6196:
6190:
6174:. Springer.
6171:
6083:
6001:
5968:
5949:
5890:
5659:
5349:
5345:logistic map
5184:
5175:
5173:
5168:
5164:
5150:almost every
5135:
4996:
4992:
4990:
4849:
4845:
4838:
4834:
4784:
4658:
4644:
4279:
4277:
4108:
4104:
4097:
4093:
4086:
4079:
4075:
4070:
3962:
3958:
3905:
3901:
3894:
3892:
3882:
3878:
3874:
3873:is called a
3870:
3864:
3806:} such that
3802:
3798:
3794:
3790:
3786:
3782:
3775:
3771:
3767:
3763:
3759:
3755:
3744:
3737:
3733:
3683:
3665:
3664:is called a
3661:
3657:
3655:
3643:
3576:-almost all
3489:-almost all
3464:
3424:
3401:
3348:
3298:
3070:
3068:
2934:
2926:homomorphism
2923:
2913:
2909:
2905:
2898:
2896:
2838:well-defined
2772:
2704:
2697:
2693:
2689:
2685:
2681:
2677:
2673:
2666:
2650:
2648:
2579:
2569:
2526:
2507:
2479:
2385:, then, for
2146:
2055:single point
2054:
1923:
1762:
1613:
1465:
1220:conservative
1149:
1061:
918:
643:, and a map
610:
491:
382:the measure
142:
80:
42:
36:
7256:compact set
7223:of measures
7159:Pushforward
7152:Projections
7142:Logarithmic
6985:Probability
6975:Pre-measure
6757:Borel space
6675:of measures
6282:: 345â423.
6081:generator.
5822:defines an
5153:real number
5142:Yakov Sinai
3299:The system
3218:-almost all
2930:isomorphism
2657:(or even a
2576:unit circle
1515:pushforward
1258:by writing
39:mathematics
7536:Categories
7228:in measure
6955:Maximising
6925:Equivalent
6819:Vitali set
6478:: 797â800.
6370:Fund. Math
6340:1705.04414
6289:1703.07093
6205:: 768â771.
6154:References
5136:where the
4653:See also:
4114:refinement
4073:partitions
3770:, clearly
3732:, and let
3602:, one has
3515:, one has
3245:, one has
3137:, one has
3102:measurable
2175:. It is a
1821:turbulence
1517:, whereas
1224:surjective
1176:Borel sets
1088:power sets
488:Discussion
376:measurable
270:, so that
77:Definition
7342:Maharam's
7312:Dominated
7125:Intensity
7120:Hausdorff
7027:Saturated
6945:Invariant
6850:Types of
6809:Ï-algebra
6779:đ-system
6745:Borel set
6740:Baire set
6306:119128525
6095:
6026:
6020:≤
5964:for more.
5941:Borel set
5939:is not a
5846:×
5840:⊂
5781:∼
5749:μ
5687:μ
5617:∩
5578:∅
5562:∩
5496:ϵ
5487:≥
5443:ϵ
5363:−
5319:μ
5289:
5283:∫
5266:μ
5241:μ
5216:⊂
5097:μ
5063:μ
5032:μ
4956:−
4931:⋁
4907:∞
4904:→
4869:μ
4818:μ
4758:μ
4755:
4740:μ
4728:∈
4721:∑
4717:−
4590:−
4582:∩
4579:⋯
4576:∩
4551:−
4543:∩
4518:μ
4494:…
4482:ℓ
4467:…
4453:ℓ
4408:−
4400:∩
4397:⋯
4394:∩
4369:−
4361:∩
4323:−
4298:⋁
4238:∩
4222:μ
4207:…
4180:…
4165:∣
4152:∩
4130:∨
4035:−
4024:…
4003:−
3981:−
3942:μ
3875:generator
3830:∈
3752:partition
3717:μ
3622:ψ
3616:φ
3587:∈
3564:ν
3535:φ
3529:ψ
3500:∈
3477:μ
3447:→
3438:ψ
3410:φ
3378:μ
3326:ν
3277:φ
3253:φ
3230:∈
3206:μ
3173:ν
3156:−
3152:φ
3145:μ
3118:∈
3107:For each
3085:φ
3051:→
3042:φ
3010:ν
2962:μ
2876:−
2860:−
2795:∘
2752:→
2612:base flow
2540:↦
2450:−
2159:μ
2149:ideal gas
2093:×
2077:×
2071:×
2006:×
2000:×
1994:×
1945:×
1939:×
1904:μ
1874:μ
1790:μ
1744:μ
1641:μ
1591:μ
1525:μ
1484:−
1473:μ
1290:−
1243:→
1123:→
1030:−
1019:μ
1004:μ
970:−
930:⊂
724:−
587:μ
563:μ
534:μ
511:−
500:μ
462:μ
439:−
428:μ
416:∈
410:∀
390:μ
380:preserves
359:→
319:∅
313:μ
278:μ
239:→
226:μ
166:is a set,
119:μ
7359:Fubini's
7349:Egorov's
7317:Monotone
7276:variable
7254:Random:
7205:Strongly
7130:Lebesgue
7115:Harmonic
7105:Gaussian
7090:Counting
7057:Spectral
7052:Singular
7042:s-finite
7037:Ï-finite
6920:Discrete
6895:Complete
6852:Measures
6826:Null set
6714:function
6492:(1970).
6392:55619325
6135:See also
5478:′
5395:implies
5138:supremum
3656:A point
3646:category
3402:The map
3077:The map
2672: :
2513:Examples
1680:, fixes
1557:pullback
402:, i.e.,
7547:Entropy
7271:process
7266:measure
7261:element
7200:Bochner
7174:Trivial
7169:Tangent
7147:Product
7005:Regular
6983:)
6970:Perfect
6943:)
6908:)
6900:Content
6890:Complex
6831:Support
6804:-system
6693:Measure
6600:Entropy
6455:1968872
6437:. (2).
4844:, ...,
4661:entropy
4103:, ...,
4085:, ...,
3900:, ...,
3750:} be a
3743:, ...,
3668:if the
2928:and an
2144:above.
7337:Jordan
7322:Vitali
7281:vector
7210:Weakly
7072:Vector
7047:Signed
7000:Random
6941:Quasi-
6930:Finite
6910:Convex
6870:Banach
6860:Atomic
6688:spaces
6673:
6606:
6578:
6550:
6453:
6416:
6390:
6304:
6257:
6178:
5972:
5233:, and
4506:
4479:
4219:
4192:
4092:} and
3423:is an
3349:factor
3088:
2767:, the
2655:monoid
2636:, the
2578:, and
1817:mixing
305:, and
85:and a
49:, and
7179:Young
7100:Euler
7095:Dirac
7067:Tight
6995:Radon
6965:Outer
6935:Inner
6885:Brown
6880:Borel
6875:Besov
6865:Baire
6451:JSTOR
6388:S2CID
6335:arXiv
6302:S2CID
6284:arXiv
6221:(PDF)
3758:into
3670:orbit
3069:is a
2692:, or
2659:group
2625:) by
2614:of a
374:is a
266:is a
196:over
192:is a
7443:For
7332:Hahn
7188:Maps
7110:Haar
6981:Sub-
6735:Atom
6723:Sets
6604:ISBN
6576:ISBN
6548:ISBN
6414:ISBN
6255:ISBN
6176:ISBN
5446:>
4659:The
4620:>
4254:>
3556:for
3469:for
3197:For
2983:and
2908:for
2688:(or
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2596:the
2589:the
1924:The
1716:and
1226:map
738:>
700:<
41:, a
6592:pdf
6508:doi
6476:147
6443:doi
6378:doi
6374:169
6345:doi
6294:doi
6203:124
6002:If
5590:or
5185:If
5081:sup
4995:or
4897:lim
4837:= {
4752:log
4116:as
4096:= {
4078:= {
3965:as
3877:or
3789:of
3754:of
3736:= {
3351:of
3100:is
2771:on
2550:mod
1198:to
1086:of
667:mod
37:In
7538::
6596:ps
6594:;
6502:.
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6439:43
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6372:.
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6343:.
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6329:.
6323:.
6300:.
6292:.
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6278:.
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6162:^
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5852:.
5849:U
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5835:R
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3179:B
3176:(
3170:=
3167:)
3164:B
3159:1
3148:(
3123:B
3115:B
3104:.
3054:Y
3048:X
3045::
3019:)
3016:S
3013:,
3007:,
3002:B
2997:,
2994:Y
2991:(
2971:)
2968:T
2965:,
2959:,
2954:A
2949:,
2946:X
2943:(
2914:N
2910:s
2906:T
2901:s
2899:T
2879:s
2872:T
2868:=
2863:1
2855:s
2851:T
2840:;
2822:s
2819:+
2816:t
2812:T
2808:=
2803:t
2799:T
2790:s
2786:T
2775:;
2773:X
2755:X
2749:X
2746::
2741:X
2736:d
2733:i
2728:=
2723:0
2719:T
2705:T
2700:s
2698:T
2694:N
2690:R
2686:Z
2682:s
2678:X
2674:X
2669:s
2667:T
2651:T
2629:;
2618:;
2607:;
2600:;
2593:;
2586:;
2580:T
2546:x
2543:2
2537:x
2527:T
2488:T
2465:.
2461:)
2456:N
2453:3
2446:2
2442:(
2436:O
2413:N
2393:N
2371:z
2367:v
2363:,
2358:y
2354:v
2350:,
2345:x
2341:v
2337:,
2334:z
2331:,
2328:y
2325:,
2322:x
2302:i
2282:p
2277:3
2273:d
2268:x
2263:3
2259:d
2254:)
2249:z
2245:v
2241:,
2236:y
2232:v
2228:,
2223:x
2219:v
2215:,
2212:z
2209:,
2206:y
2203:,
2200:x
2197:(
2192:i
2188:p
2132:X
2111:.
2106:N
2103:3
2098:R
2088:N
2084:)
2080:h
2074:l
2068:w
2065:(
2041:N
2021:.
2016:3
2011:R
2003:h
1997:l
1991:w
1971:N
1951:,
1948:h
1942:l
1936:w
1883:)
1880:T
1877:,
1871:,
1866:B
1861:,
1858:X
1855:(
1835:T
1799:)
1796:T
1793:,
1787:,
1782:B
1777:,
1774:X
1771:(
1724:T
1704:)
1699:B
1694:,
1691:X
1688:(
1664:T
1644:)
1638:,
1633:B
1628:,
1625:X
1622:(
1571:T
1543:)
1540:)
1537:A
1534:(
1531:T
1528:(
1501:)
1498:)
1495:A
1492:(
1487:1
1480:T
1476:(
1451:;
1448:)
1445:A
1442:(
1439:T
1436:=
1433:)
1430:A
1427:(
1422:T
1398:T
1374:T
1352:)
1349:A
1346:(
1343:T
1340:=
1337:)
1334:A
1331:(
1326:T
1304:)
1301:A
1298:(
1293:1
1286:T
1282:=
1279:)
1276:A
1273:(
1268:T
1246:X
1240:X
1237::
1234:T
1206:X
1186:X
1160:T
1135:)
1132:X
1129:(
1126:P
1120:)
1117:X
1114:(
1111:P
1108::
1103:T
1072:T
1047:)
1044:)
1041:A
1038:(
1033:1
1026:T
1022:(
1016:=
1013:)
1010:A
1007:(
984:)
981:A
978:(
973:1
966:T
945:]
942:1
939:,
936:0
933:[
927:A
904:]
901:1
898:,
895:2
891:/
887:1
884:[
864:]
861:1
858:,
855:0
852:[
832:]
829:2
825:/
821:1
818:,
815:0
812:[
792:]
789:1
786:,
783:0
780:[
749:2
745:/
741:1
735:x
727:1
721:x
718:2
711:2
707:/
703:1
697:x
689:x
686:2
680:{
675:=
672:1
663:x
660:2
657:=
654:x
651:T
631:]
628:1
625:,
622:0
619:[
596:)
593:A
590:(
584:=
581:)
578:)
575:A
572:(
569:T
566:(
543:)
540:A
537:(
531:=
528:)
525:)
522:A
519:(
514:1
507:T
503:(
483:.
471:)
468:A
465:(
459:=
456:)
453:)
450:A
447:(
442:1
435:T
431:(
421:B
413:A
362:X
356:X
353::
350:T
340:,
328:0
325:=
322:)
316:(
293:1
290:=
287:)
284:X
281:(
254:]
251:1
248:,
245:0
242:[
234:B
229::
216:,
204:X
178:B
154:X
128:)
125:T
122:,
116:,
111:B
106:,
103:X
100:(
34:.
20:)
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