Teichmuller distance for some polynomial-like maps

TL;DR

This paper demonstrates that the Teichmüller distance for a class of off-critically hyperbolic generalized polynomial-like maps is a true metric, extending known results for real polynomials and utilizing hyperbolic sets and Sullivan's rigidity theorem.

Contribution

It establishes the Teichmüller distance as a genuine metric for off-critically hyperbolic generalized polynomial-like maps, expanding on previous work and applying new analytic techniques.

Findings

Teichmüller distance is a true metric for the class studied.

The Hausdorff dimension of the maximal entropy measure is strictly less than that of the Julia set, except for Chebyshev polynomials.

The proof relies on the non-existence of invariant affine structures.

Abstract

In this work we will show that the Teichm\"{u}ller distance for all elements of a certain class of generalized polynomial-like maps (the class of off-critically hyperbolic generalized polynomial-like maps) is actually a distance, as in the case of real polynomials with connected Julia set, as studied by Sullivan. This class contains several important classes of generalized polynomial-like maps, namely: Yoccoz, Lyubich, Sullivan and Fibonacci. In our proof we can not use external arguments (like external classes). Instead we use hyperbolic sets inside the Julia sets of our maps. Those hyperbolic sets will allow us to use our main analytic tool, namely Sullivan's rigidity Theorem for non-linear analytic hyperbolic systems. Lyubich has constructed a measure of maximal entropy measure $m$ on the Julia set of any rational function $f$. Zdunik classified exactly when the Hausdorff dimension of $m$ equals the Hausdorff dimension of the Julia set. We show that the strict inequality holds if $f$ is off-crititcally hyperbolic, except for Chebyshev polynomials. This result is a particular case of Zdunik's result if we consider $f$ as a polynomial, but is an extension of Zdunik's result if $f$ is a generalized polynomial-like map. The proof follows from the non-existence of invariant affine structure.