[Paper Review] The maximal entropy measure detects non-uniform hyperbolicity

Juan Rivera‐Letelier|arXiv (Cornell University)|May 14, 2010

Mathematical Dynamics and Fractals20 references3 citations

TL;DR

This paper establishes that semi-hyperbolicity and the Topological Collet-Eckmann (TCE) condition for rational maps are characterized by properties of the maximal entropy measure: semi-hyperbolicity is equivalent to the measure being doubling on the Julia set, and the TCE condition is equivalent to the measure having polynomial lower bounds on small balls. The results extend classical theorems of Carleson, Jones, and Yoccoz to rational maps with completely invariant attracting basins, showing such basins are John domains if and only if the map is semi-hyperbolic.

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. Using the same tools in the proof of these results we give an extension of a 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.

Motivation & Objective

• To characterize semi-hyperbolicity and the TCE condition for rational maps using the maximal entropy measure.
• To extend the Carleson-Jones-Yoccoz theorem on polynomial basins being John domains to rational maps with completely invariant attracting basins.
• To show that the maximal entropy measure is doubling on the Julia set if and only if the map is semi-hyperbolic.
• To establish that the TCE condition is equivalent to polynomial lower bounds on the measure of small balls.

Proposed method

• Define the semi-local degree of iterates at points in the Julia set using local preimage components under iterated maps.
• Use the maximal entropy measure's support on the Julia set and analyze its doubling property via measure-theoretic estimates.
• Prove that semi-hyperbolicity implies doubling of the measure by bounding the growth of preimage components.
• Establish the TCE condition via lower polynomial bounds on measure of balls, using distortion estimates and covering arguments.
• Apply the theory of John domains and porosity to link geometric properties of attracting basins to dynamical conditions.
• Use results from Haïssinsky and Pilgrim on doubling measures and extend them to rational maps via covering and distortion control.

Experimental results

Research questions

• RQ1 Under what conditions on the maximal entropy measure is a rational map semi-hyperbolic?
• RQ2 Is the TCE condition equivalent to a polynomial lower bound on the measure of small balls in the Julia set?
• RQ3 Can the characterization of polynomial basins as John domains via semi-hyperbolicity be extended to rational maps with completely invariant attracting basins?
• RQ4 Does the absence of recurrent critical points in the Julia set follow from the porosity of the Julia set in a rational map with a John domain basin?
• RQ5 What is the optimal exponent α in the polynomial lower bound for the maximal entropy measure under the TCE condition?

Key findings

• A rational map is semi-hyperbolic if and only if its maximal entropy measure is doubling on the Julia set.
• The TCE condition holds if and only if there exist constants r₀ > 0, α > 0, and C > 0 such that ρ_f(B(x,r)) ≥ Cr^α for all x ∈ J(f) and r ∈ (0,r₀).
• The optimal exponent α in the lower bound is determined and explicitly characterized in Remark 4.
• If a rational map has a completely invariant attracting basin that is a John domain, then it is semi-hyperbolic.
• The absence of recurrent critical points in the Julia set follows from the porosity of the Julia set when the basin is a John domain.
• The result extends the Carleson-Jones-Yoccoz theorem to rational maps: a rational map with a completely invariant attracting basin is semi-hyperbolic if and only if the basin is a John domain.