The maximal entropy measure detects non-uniform hyperbolicity

Abstract

Abstract. We characterize two of the most studied non-uniform hyperbolicity conditions for rational maps, semi-hyperbolicity and the topological Collet-Eckmann condition, in terms of the maximal entropy measure.With the same tools we give an extension of the result of Carleson, Jones and Yoccoz that semi-hyperbolicity characterizes those polynomial maps whose basin of attraction of infinity is a John domain, to rational maps having a completely invariant attracting basin.

Analytic structures and harmonic measure at bifurcation locus

“…The proof of Theorem 2 is based on a global inductive estimate of conformal densities distributed over elements of Yoccoz partitions. The main technical difficulty to overcome is a tendency of conformal measures to excessively concentrate around recurrent critical points [18,45].…”

“…Even though not explicitely stated, the concept of hedgehog layers was introduced by J. Riviera-Letelier in his study of porosity at critical recurrent points for rational functions, see the proof of Theorem C' in ( [45]).…”

When do two rational functions have locally biholomorphic Julia sets?

“…Remark 3.3. The TCE property is detected by the maximal entropy measure: indeed it is equivalent to the property that the measure of small balls satisfies an estimate of the form µpBpx, rqq Á r θ for some θ ą 0 and for every x P J (see [36]). It is not difficult to see that if such an estimate holds for every x P U , where U is any open set interesting J f , then it holds everywhere (possibly with a different θ).…”

Interval maps quasi-symmetrically conjugate to a piecewise affine map

“…Lemma 9 (Lemma 1, [RL10]). Let (X, dist) be a compact metric space and let µ be a doubling measure on X.…”

“…Acknowledgments. The main idea of this paper came to the author after several discussions with Juan Rivera-Letelier on his work [RL10], and he read an earlier version very carefully. I am very grateful to him for those stimulating conversations and for his useful comments and corrections to an earlier version of this paper.…”