[Paper Review] Teichmuller distance for some polynomial-like maps

Eduardo A. Prado|arXiv (Cornell University)|Jan 15, 1996

Mathematical Dynamics and Fractals10 references3 citations

TL;DR

This paper establishes that Teichmüller distance is a well-defined metric for a class of generalized polynomial-like maps—specifically, off-critically hyperbolic maps—by leveraging hyperbolic sets within their Julia sets and applying Sullivan’s rigidity theorem for non-linear analytic systems. The key result shows that for such maps, the measure of maximal entropy has strictly smaller Hausdorff dimension than the Julia set, except in the case of Chebyshev polynomials, extending Zdunik’s result beyond polynomials.

ABSTRACT

In this work we will show that the Teichmüller 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.

Motivation & Objective

• To establish that Teichmüller distance is a valid metric for a broad class of generalized polynomial-like maps.
• To extend Zdunik’s result on Hausdorff dimension of the measure of maximal entropy to non-polynomial generalized polynomial-like maps.
• To prove that strict inequality holds between the Hausdorff dimension of the measure of maximal entropy and the Julia set, except for Chebyshev polynomials.
• To avoid reliance on external classes or external classes by using internal dynamical structures such as hyperbolic sets.

Proposed method

• Utilizes hyperbolic sets within the Julia sets of the maps as a structural foundation for analysis.
• Applies Sullivan’s rigidity theorem for non-linear analytic hyperbolic systems to establish uniqueness of Teichmüller extremal maps.
• Employs Lyubich’s construction of the measure of maximal entropy on the Julia set of rational maps.
• Analyzes the non-existence of invariant affine structures to deduce rigidity properties.
• Uses the absence of invariant affine structures as a key analytic tool to rule out degeneracies in Teichmüller space.
• Relies on the intrinsic dynamics of the maps rather than external geometric or conformal classes.

Experimental results

Research questions

• RQ1 Is Teichmüller distance a well-defined metric for off-critically hyperbolic generalized polynomial-like maps?
• RQ2 Does the Hausdorff dimension of the measure of maximal entropy strictly less than that of the Julia set hold for such maps, excluding Chebyshev polynomials?
• RQ3 Can the rigidity of the Teichmüller metric be established without relying on external classes or external classes?
• RQ4 To what extent does Sullivan’s rigidity theorem apply to non-polynomial generalized polynomial-like maps?
• RQ5 What is the relationship between the existence of invariant affine structures and the Teichmüller distance in this class of maps?

Key findings

• Teichmüller distance is a valid metric for all off-critically hyperbolic generalized polynomial-like maps, including Yoccoz, Lyubich, Sullivan, and Fibonacci-type maps.
• The measure of maximal entropy has strictly smaller Hausdorff dimension than the Julia set for all such maps, except for Chebyshev polynomials.
• The strict inequality in Hausdorff dimension is established via the non-existence of invariant affine structures in the dynamical systems under study.
• The result extends Zdunik’s theorem beyond polynomial maps to the broader class of generalized polynomial-like maps.
• The proof relies on internal dynamical structures—hyperbolic sets—rather than external geometric or conformal invariants.
• Sullivan’s rigidity theorem is successfully applied to non-polynomial maps by using intrinsic hyperbolic dynamics and the absence of affine structures.