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Visible measures of maximal entropy in dimension one. (English) Zbl 1132.37017
A map \(f:[0,1]\to [0,1]\) is called a C-map, if it is \(C^{1}\), \(f^{\prime}\) is Hölder-continuous, there are only finitely many critical points, i.e. points \(x\) with \(f^{\prime}(x)=0\), and all critical points are nonflat. The post-critical set \(P\) is defined as \(P:=\bigcup_{k=0}^{\infty}f^{k}(C)\), where \(C\) is the set of all critical points of \(f\). For a Borel probability measure \(\mu\) the Hausdorff dimension \(\text{HD}(\mu )\) of \(\mu\) is defined by \(\text{HD}(\mu ):= \inf\{\text{HD}(Y):\mu (Y)=1\}\). An \(f\)-invariant Borel probability measure \(\mu\) is said to be a measure of maximal entropy, if its measure-theoretic entropy coincides with the topological entropy of \(f\). If \(J\subset [0,1]\) is a nontrivial interval, \(J\), \(f(J)\), …, \(f^{n-1}(J)\) are pairwise disjoint, \(f^{n}(J)\subset J\) and \(f^{n}\) maps the boundary of \(J\) to itself, then \(J\) is called a restrictive interval of period \(n\). The map \(f\) is called renormalisable of type \(n\geq 2\), if it has a restrictive interval of period \(n\) containing at least one critical point.
Assume that \(f\) is a C-map of type \(C^{r}\) for some \(r\geq 2\), and suppose that \(f\) has positive topological entropy. Then the main result of this paper states that either \(\text{HD}(\mu )<1\) for every measure \(\mu\) of maximal entropy or there exists a restrictive interval \(J\) of period \(k\geq 1\), the restriction of \(f^{k}\) to \(J\) has finite post-critical set \(P\), \(f^{k}\) restricted to \(J\setminus P\) is \(C^{r}\)-conjugate to a continuous piecewise linear map with constant slope, and there is a measure \(\mu\) of maximal entropy with \(\mu (J)>0\). If \(f\) is not renormalisable, then \(\text{HD}(\mu )=1\) for a measure \(\mu\) of maximal entropy implies that \(f\) has finite post-critical set \(P\) and \(f\) is \(C^{r}\)-conjugate to a continuous piecewise linear map with constant slope on \([0,1]\setminus P\). A C-map is called unimodal, if it has exactly one critical point which is a turning point, exactly two fixed points, and \(\{0,1\}\) is mapped to itself. Unimodal maps with positive entropy have a unique measure \(\mu\) of maximal entropy. For unimodal C-maps of type \(C^{r}\) for some \(r\geq 2\) having positive topological entropy the property \(\text{HD}(\mu )=1\) implies that there is \(0\leq k<\infty\) such that \(f\) is \(k\)-times renormalisable, each renormalisation is of type \(2\) and the resultant non-renormalisable map \(f^{2^{k}}\) restricted to \(J\setminus P\) is \(C^{r}\)-conjugate to the full tent map.
Reviewer: Peter Raith (Wien)

37E05 Dynamical systems involving maps of the interval
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37B40 Topological entropy
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37C45 Dimension theory of smooth dynamical systems
37E20 Universality and renormalization of dynamical systems
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