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$\newcommand\al{\alpha}$Let us drop the factor $1-\al$, by considering $$Y:=X_\infty/(1-\al)=\sum_{k=0}^\infty\al^k X_k.$$ By Kolmogorov's three-series theorem, this series will converge almost surely (a.s.) unless at least one of the tails of the distribution of $X_0$ is too heavy. Assume that the series indeed converges a.s. Then, obviously, $$Y\...


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It seems to me that the structure of ${\rm Aut}^{m}(C_{n})$ also depends heavily on the prime factorization of $n$, and I don't really see any reason to expect the answer to Q1 to be any more tractable than determining the structure of ${\rm Aut}(C_{n})$. For example (just to illustrate) , if we choose a prime $p$, greater than $3$, and then we take a pair ...


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• Q1 (the first yellow boxed question in the OP): The current status of the "persistence problem" has been discussed by Ilyashenko in Complex length and persistence of limit cycles in a neighborhood of a hyperbolic polycycle (2014), see also Persistence Theorems and Simultaneous Uniformization (2006). As discussed on page 285 of the 2014 paper, the ...


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The simplest and earliest example I know regarding the renormalization group idea is the following. Suppose we want to study some feature $\mathcal{Z}(\vec{V})$ of some object $\vec{V}$ which is in a set $\mathcal{E}$ of similar objects. Suppose that unfortunately this question is too hard. What can one do? The renormalization group philosophy is try to ...


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I just figured I added some Mathematica code and a picture for the attractor in the question. With[{dt = 0.001}, iter[{x_, y_, z_}] := {x, y, z} + dt {(z - y), x/2 - 1, -x y/2 - z} ]; pts = NestList[iter, {0.1, 0.1, 1/2}, 500000]; ListPlot[{#1, #2} & @@@ pts[[1 ;; ;; 5]], PlotRange -> All, Axes -> False, PlotStyle -> {Opacity[0.9], ...


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I have implemented some algorithms based on Kawahira's paper, which as presented goes $\theta \to c \not\in M$, but can be adapted to go $c \to \theta$. $\theta$ is conveniently expressed in turns as a binary expansion. When tracing inwards, one peels off the most-significant bit (aka angle doubling) each time the ray crosses a dwell band (integer part of ...


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Q: Might there be a reason why stable physical processes would tend to have low-dimensional phase spaces. Yes. One reason is physical processes have dissipation. E.g., turbulence is "known" to be chaotic dynamics on a low dimensional manifold (i.e., strange attractor) in the infinite dimensional phase space (of $L^2$ velocity fields). Even its dimension can ...


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This is not an answer, but too long for a comment. This promotion operator is (apparently) governed by local rules, similar to https://arxiv.org/abs/1804.06736, as follows: regard each path as a sequence of coordinates, that is, $A$ adds $(1,1)$, $B$ adds $(-1,0)$ and $C$ adds $(0,-1)$ to the current coordinate create a cylindrical array from each ...


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Let $f(z)=az^2(z-1)$. Zero is superattracting. Now choose $a$ so small that $|f(z)|<|z|/2$ for $|z|<2$. Then the root $z_0=1$ is in the immediate domain of attraction.


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In general, such a homeomorphism is not necessary equicontinuous. The existence of such examples on $X=\mathbb{T}^2$, i.e. the $2$-torus, can be shown as follows: let $\mathcal{O}$ be the $C^\infty$ closure of the set $\{h\circ R_\alpha\circ h^{-1} : h\in\mathrm{Diff}^\infty(\mathbb{T^2}),\ \alpha\in\mathbb{T}^2\}$, where $\mathbb{T}^2$ denotes the $2$-...


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Let $\alpha$ be irrational, let $Y$ be the two-dimensional torus equipped with the map $S(u,v)=(u+\alpha,v+u)$. Then the action of $S$ on $Y$ is minimal. Now partition the torus into two pieces, say $A_0=S^1\times[0,\frac 12)$ and $A_1=S^1\times[\frac 12,1)$ and let $j(y)=0$ if $y\in A_0$ and $j(y)=1$ if $y\in A_1$. Let $X$ be the set of bi-infinite ...


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The only values of $\alpha$ for which $F(\alpha)$ is non-empty are $-\infty$ and $\log 2$. This is because the map is conjugate (by the conjugacy $H\colon x\mapsto \sin^2(\pi x/2)$ to the full tent map $S$ (that is, $T\circ H=H\circ S$). In particular, $T^n\circ H=H\circ S^n$, so that for any $x$, $|(T^n)'(H(x))|H'(x)=H'(S^nx)\cdot 2^n$, so that for any $x$ ...


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If $f$ has order $<1/2$ then there is a sequence $r_k\to\infty$ with the property that $$\min_{|z|=r_k}|f(z)|>r_k.$$ Restricting $f$ on $\{ z:|z|<r_k\}$ we obtain a polynomial-like map in the sense of Douady and Hubbard. If $J_k$ is the Julia set of this map, then evidently $J_k\subset J(f)$, and the points of $J_k$ are not escaping. Now, if $f$ has ...


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There are examples of minimal real analytics torus diffeomorphisms which are not just weak mixing, but strong mixing. In fact, Bassam Fayad has constructed some mixing reparametrizations of minimal linear flows on tori in this paper. To get a diffeomorphism with those properties from such a flow, one can show that given any minimal flow, there is a ...


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