The generalized Milnor-Thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters
type: Item
instance of: scholarly article
title: The generalized Milnor-Thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters
zbMATH Open document ID: 0974.37010
published in: Communications in Mathematical Physics
publication date: 2001-09-23T00:00:00Z
review text: Write \(\mathcal U=\{L,C,M,D,R\}\), when \({\mathcal U}^\infty\) is the set of infinite sequences of the symbols \(L<C<M<D<R\), and define a partial ordering in \({\mathcal U}^\infty\) by writing \((A_0,A_1,\ldots) \prec (B_0,B_1,\ldots)\) whenever there is a \(k\geq 0\) such that \(A_i=B_i\in \{L,M,R\}\) for any \(i<k\), and either \(A_k<B_k\) [if \((A_0,A_1,\ldots,A_{k-1})\) has an even number of \(M\)'s] or \(A_k>B_k\) [if \((A_0,A_1,\ldots,A_{k-1})\) has an odd number of \(M\)'s]. Let \(\varphi: {\mathcal U}^\infty\rightarrow {\mathcal U}^\infty\) be the \textit{shift map}, that is, \(\varphi(A_0,A_1,\ldots)=(A_1,A_2,\ldots)\). We will say that two sequences \(K^1, K^2\in {\mathcal U}^\infty\) are \textit{admissible} if they satisfy the following conditions for any \(r\in \{1,2\}\): (a) if \(K^r=(A_0,A_1,\ldots ,A_k,\ldots)\) and \(A_k\) equals \(C\) (resp. \(D\)), then \(\varphi^k(K^r)=K^1\) [resp. \(\varphi^k(K^r)=K^2\)]; (b) \(K^1 \preceq \varphi^i(K^r)\preceq K^2\) for any positive integer \(i\). We call the set \(\Sigma_3\) of admissible pairs \((K^1, K^2)\) the \textit{kneading plane}. The reason is that if \(f\) is a \textit{boundary anchored bimodal map with the shape \((+,-,+)\)}, that is, a continuous map \(f:[a,b]\rightarrow [a,b]\) with \(f(a)=a\), \(f(b)=b\) such that, for some intermediate points \(a<c<d<b\), \(f\) is increasing in \(L=[a,c)\) and \(R=(c,b]\) and decreasing in \(M=(c,d)\) (monotonicity need not be strict in neither of these intervals), then its \textit{kneading data} \(K^1=(A(f(c)), A(f^2(c)),\ldots)\) and \(K^2=(A(f(d)), A(f^2(d)),\ldots)\) for the \textit{choice} \((c,d)\) are admissible in the above sense (here \(A(x)\) is the set \(L\), \(C=\{c\}\), \(M\), \(D=\{d\}\) or \(R\) containing the point \(x\in [a,b]\)). Conversely, if \((K^1, K^2)\in \Sigma_3\) then there is a boundary anchored bimodal map with the shape \((+,-,+)\) having \((K^1, K^2)\) as its kneading data for an appropriate choice \((c,d)\). It is known that if tw…
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 54C70
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37E05
zbMATH DE Number: 1529107
zbMATH Keywords: bimodal map
zbMATH Keywords: kneading plane
zbMATH Keywords: topological entropy
zbMATH Keywords: symbolic dynamics
zbMATH Keywords: \(*\)-product
Wikidata QID: Q123154257
author: Shou-Li Peng
author: Xu-Sheng Zhang
reviewed by: Víctor Jiménez López
MaRDI profile type: Publication
full work available at URL: s002200000245
OpenAlex ID: W2080050855
DOI: 10.1007/S002200000245
description: scientific article; zbMATH DE number 1529107