[Paper Review] On measures of maximal and full dimension for polynomial automorphisms of $\C^2$

Christian Wolf|ArXiv.org|Apr 2, 2001

Advanced Differential Equations and Dynamical Systems3 references4 citations

TL;DR

This paper establishes the existence of a measure of maximal dimension for hyperbolic polynomial automorphisms of ℂ² using thermodynamic formalism, showing such measures are equilibrium states for a one-parameter family of potentials. It further proves that for non-volume preserving maps, a measure of full dimension must be the measure of maximal entropy, and that the dynamical dimension d(g) is continuous and plurisubharmonic in parameter families.

ABSTRACT

For a hyperbolic polynomial automorphism of $\C^2$, we show the existence of a measure of maximal dimension, and identify the conditions under which a measure of full dimension exists.

Motivation & Objective

• To establish the existence of a measure of maximal dimension for hyperbolic polynomial automorphisms of ℂ².
• To determine conditions under which a measure of full dimension exists.
• To analyze the dependence of the dynamical dimension d(g) on the parameters of the mapping.
• To clarify the relationship between measures of full dimension and measures of maximal entropy in non-volume preserving systems.

Proposed method

• Utilizes thermodynamic formalism by constructing a one-parameter family of potentials associated with the dynamics.
• Identifies measures of maximal dimension as equilibrium measures for these potentials.
• Applies Young’s formula relating metric entropy and Hausdorff dimension to link ergodic properties with dimension theory.
• Employs real analyticity and uniqueness theorems to show finitely many maxima in the pressure function, ensuring existence of maximal dimension measures.
• Uses conjugacy families and harmonic functions to prove continuity and plurisubharmonicity of d(g) in holomorphic parameter families.
• Applies results from [VW] and [Wo1] on real analytic pressure functions and conjugacies to establish regularity of d(g).

Experimental results

Research questions

• RQ1 Does a hyperbolic polynomial automorphism of ℂ² always admit a measure of maximal dimension?
• RQ2 Under what conditions does a measure of full dimension exist for such maps?
• RQ3 Is the measure of full dimension unique when it exists?
• RQ4 Can a measure of full dimension exist for non-volume preserving maps, and if so, what characterizes it?
• RQ5 How does the dynamical dimension d(g) vary with parameters in a holomorphic family of maps?

Key findings

• A measure of maximal dimension exists for every hyperbolic polynomial automorphism of ℂ², and the set of such measures is finite.
• For non-volume preserving maps, a measure of full dimension must be the measure of maximal entropy, indicating rarity of such measures.
• There exists at most one measure of full dimension, as shown in Corollary 3.6.
• The dynamical dimension d(g) is strictly less than dim_H J for a dense open subset of the hyperbolic parameter space.
• The function λ ↦ d(g_λ) is continuous and plurisubharmonic in holomorphic families of hyperbolic polynomial automorphisms.
• Every measure of maximal dimension is Bernoulli, due to topological mixing and equilibrium state properties.