The generalized Milnor-Thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters
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 two such maps \(f\) and \(g\) have the same kneading data \((K^1, K^2)\) then their topological entropies \(h(f)\) and \(h(g)\) are the same. Then \(h(K^1, K^2):=h(f)\) is well defined. The main aim of the paper under review is to show that the kneading plane \(\Sigma_3\) includes infinitely many regions of (pairwise different) constant topological entropy (\textit{``plateaus''}), each of them containing uncountably many pairs \((K^1,K^2)\). Moreover, the union set of these regions is dense in \(\Sigma_3\) (when endowed with the natural product topology). A Sharkovskii-like order embedding theorem in these plateaus is also proved, further emphasizing the dynamical complexity of such regions. For these purposes, a generalization of the well known Derrida-Gervois-Pomeau \(*\)-product for unimodal maps (so-called \textit{dual star products}) is used. While the paper is interesting, it has some expository drawbacks which may render its reading rather difficult to the non-specialist. In particular, the connexions with the so-called ``generalized Milnor-Thurston conjecture'' should be more clearly stated, as in its usual formulation it concerns with the space of parameters of a family of cubic polynomials as for instance those in \textit{J. Milnor} and \textit{C. Tresser} [Commun. Math. Phys. 209, 123-178 (2000; Zbl 0971.37007)], and not \(\Sigma_3\). It must be said that the existence of infinitely many plateaus in \(\Sigma_3\) also follows from \textit{J. Llibre} and \textit{P. Mumbrú} [Extending the \({}^*\)-product operator, in European Conference on Iteration Theory (Batschuns, 1989), World Sci. Publishing, 199-214 (1991), see also Renormalisation and periodic structure for bimodal maps, in European Conference on Iteration Theory (Caldes de Malavella, 1987), World Sci. Publishing, 253-262 (1989)]. They use an alternative approach to that of the paper under review.
scientific article; zbMATH DE number 1529107