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and the unfolded version would take such a point near to (1, 1), whereas this image mapped the top right corner (1, 1) to the top left (0, 1). Also note that both bakers maps preserve chirality (clock-wise gets mapped to clockwise) in regions they don't split whereas this map turned the right 5 into it's (squashed) mirror image. Can someone please upload a correct demonstration as it was a good illustration aside from the fact it was wrong.
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I removed this image because it didn't demonstrate the folded bakers map (or indeed the unfolded bakers map) as it was defined in the article. As evidence to this fact, note how the folded baker's map (as defined) would map a point arbitrarily close to (1, 1) to a point arbitrarily close to (0, 1/2)
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The first uses \mathrm{baker-folded}; the second uses \text{baker-folded}. When \mathrm is used, the hyphen becomes a minus sign; when \text is used, it remains a hyphen. The first uses \mbox{for }; the second uses \text{for }. In some contexts, those look much more different from each other than
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My concern is that the tent map and the
Bernouilli map are not the same and since the Bernouilli map lops off digits, the tent map does not. I think you are saying that you could define some alternative representation of the unit interval by diadic expansions to make the tent map lop off digits and
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The tent map also lops off dyadic digits from the symbolic representation of a trajectory. Given a hyperbolic map (the tent map is hyperbolic), it always possible to set up symbolic representation for trajectories of a map (and flows too). The symbolic representation is usually set up by
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Yes, but the eigenfunctions of the tent map are ickier to understand; the symmetry about x=1/2 causes a degeneracy. The eigenfunctions of the non-flipped
Bernoulli map are the Bernoulli polynomials. Also, if you are interested in eigenfunctions of Baker's map on the torus/elliptic curve, the
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If you are working just with the baker's map, it's simpler not to flip. But if you are using the baker's map as some simplification of a smooth map, then you usually need to flip. Halmos and Arnold are the older references and as far as I can tell the origin of the citation chain (tree?).
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Well, yes, there is indeed an interesting interplay. If one has a string of dyadic digits, it can be represented by a real number, as binary; this is true both for
Bernoulli map orbits and tent map orbits. For the bernoulli map, one has a very direct correspondance between the value of
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non-flipped version makes more sense. (I think the flipped version gives the Klein bottle or something like that; I'm trying to understand this now). At any rate, the two maps are not equivalent, they behave differently; maybe this is what should be said.
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The baker's map, as defined, has the entire square as an attractor, so the
Hausdorff dimension is 2. The action of the baker's map is to squeeze the left half into the lower half and the right half into the upper half after a flip. The
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and the binary representation of the orbit -- indeed they are one and the same. For the tent map, they are not. Hmm. Come to think of it, I have never seen a graph of the binary values of the orbits of the tent map, as a function of
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The
Bernoulli map was changed to the tent map, but it's description was not changed. It is clear that the Bernoulli map lops off dyadic digits, so the tent map does something else. What is the section below supposed to indicate?
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The russian version of this article has a nice image of a baker, which should be copied to wiki commons and then included here. I tried to include the image directly from the russian site, but the inclusion did not work.
462:. I'll create a few graphs now, and post shortly. BTW, XaosBits, do you perchance have any contact with Andreas Wirzba (who I think is a coauthor of the chaos book?) I wanted to say hi to him, we went to school together.
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As with many deterministic dynamical systems, the baker's map is studied by its action on the space of functions defined on the unit square. The Baker's map defines an operator on the space of functions, known as the
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but its a stub. The eigenvalues of the tent map of unit height are linear combos of
Bernoulli. The interested reader is directed to the reference. I'd expand these articles, except for the fact that I am doing
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Agreed. The eigenvalues for the tent map are just the even powers of 1/2 (for analytic functions), but I have no idea about the eigenfunctions. I have no strong opinions about reverting the text the way
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854:{\displaystyle S_{\mbox{Bernoulli}}(x)=2x{\mbox{ mod }}1=\left\{{\begin{matrix}2x,&{\mbox{for }}0\leq x<{\frac {1}{2}}\\2x-1,&{\mbox{for }}{\frac {1}{2}}\leq x<1\end{matrix}}\right.}
680:{\displaystyle S_{\mbox{Baker}}(x,y)=\left\{{\begin{matrix}(2x,y/2),&{\mbox{for }}0\leq x<{\frac {1}{2}}\\(2x-1,1-y/2),&{\mbox{for }}{\frac {1}{2}}\leq x<1\end{matrix}}\right.}
1148:{\displaystyle S_{\mathrm {baker-folded} }(x,y)={\begin{cases}(2x,y/2)&{\mbox{for }}0\leq x<{\frac {1}{2}}\\(2-2x,1-y/2)&{\mbox{for }}{\frac {1}{2}}\leq x<1\end{cases}}}
1344:{\displaystyle S_{\text{baker-folded}}(x,y)={\begin{cases}(2x,y/2)&{\text{for }}0\leq x<{\frac {1}{2}}\\(2-2x,1-y/2)&{\text{for }}{\frac {1}{2}}\leq x<1\end{cases}}}
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had it or leaving it the way it is. I changed the article and only after that did I decided to poll the literature. If I had polled first, I would have left it as it was.
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The definition of the baker's map needs to include a flip of the right column to match the horseshoe map. Some authors do not include the flip and others do not:
229:= 1/4, which would be 1 1/2. Thanks. In fact, the way I learned it the constant y is multiplied by has to be less than 1/2, so I was like Hey, that can't be 2...
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is smaller than 1 (as it needs to be smaller than 1/2). So if a = 0.45, the fractal dimension of the attractor of the horseshoe would be approximately 1.736.
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When the formula changed to "human readable format" I think it also changed into the horseshoe map, because of the -y.--
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no more. This doesn't seem a very useful way of looking at it to me. If you think it is you should explain it better.--
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in very close proximity to this topic area, so whatever I wrote would be strongly colored. I'm happy to let it be.
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The
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The representation is not of the interval, but of the orbits. See sec 11.7 and chapter 12 of
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Oh yeah, thanks, I was thinking that the map of which we were speaking was the bakers map with
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The russian article also has a reatment of the symbolic dynamics which should be copied here.
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OK, see image, click on it to get detailed explanation.
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233:formula
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399:MarSch
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36:scale.
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1464::
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143:.
109:.
42::
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