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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\... 6 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 ... 5 • 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 ... 5 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 ... 4 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], ... 4 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 ... 3 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 ... 2 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 ... 1 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. 1 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-... 1 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 ... 1 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 ... 1 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 ...