Visible measures of maximal entropy in dimension one.

*(English)*Zbl 1132.37017A map \(f:[0,1]\to [0,1]\) is called a C-map, if it is \(C^{1}\), \(f^{\prime}\) is HĂ¶lder-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.

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)

##### MSC:

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 |