[Paper Review] Entropy along expanding foliations

Jiagang Yang|arXiv (Cornell University)|Jan 21, 2016

Mathematical Dynamics and Fractals25 references32 citations

TL;DR

This paper establishes the upper semi-continuity of partial entropy along expanding foliations under C¹ perturbations of diffeomorphisms, invariant measures (weak* topology), and foliations. It proves that the metric entropy along unstable foliations varies upper semi-continuously, leading to key results: the set of Gibbs u-states is upper semi-continuous in the C¹ topology, partially hyperbolic diffeomorphisms with mostly expanding or contracting center are C¹ open, and new robustly transitive C² volume-preserving diffeomorphisms are constructed with non-vanishing center exponent.

ABSTRACT

The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper semi-continuously with the diffeomorphism ($\C^1$ topology), the invariant measure (weak* topology) and the foliation itself in a suitable sense. This has several important consequences. For one thing, it implies that the set of Gibbs $u$-states of $\C^{1+}$ partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the $\C^1$ topology. Another consequence is that the sets of partially hyperbolic diffeomorphisms with mostly contracting or mostly expanding center are $\C^1$ open. New examples of partially hyperbolic diffeomorphisms with mostly expanding center are provided, and the existence of physical measures for $C^1$ residual subset of diffeomorphisms are discussed. We also provide a new class of robustly transitive diffeomorphisms: every $C^2$ volume preserving, accessible partially hyperbolic diffeomorphism with one dimensional center and non-vanishing center exponent is $C^1$ robustly transitive (among neighborhood of diffeomorphisms which are not necessarily volume preserving).

Motivation & Objective

• To establish the upper semi-continuity of metric entropy along expanding foliations under C¹ perturbations of diffeomorphisms.
• To analyze the regularity of partial entropy in the context of partially hyperbolic systems with invariant expanding foliations.
• To apply the regularity result to prove openness of sets of diffeomorphisms with mostly expanding or contracting center directions.
• To establish robust transitivity for C² volume-preserving accessible partially hyperbolic diffeomorphisms with non-vanishing center exponent.
• To provide new examples of partially hyperbolic diffeomorphisms with mostly expanding center and discuss physical measures for C¹ residual diffeomorphisms.

Proposed method

• Uses the theory of subordinate measurable partitions to define and analyze partial entropy along expanding foliations.
• Applies dimension theory and the Pesin entropy formula to control the growth of measure-theoretic complexity on leaves.
• Employs weak* convergence of invariant measures and C¹ convergence of diffeomorphisms to establish upper semi-continuity of entropy.
• Utilizes the notion of Gibbs u-states—measures with absolutely continuous disintegration along unstable leaves—to link entropy regularity to statistical properties.
• Applies a modified version of a lemma from Pesin theory (Lemma A.1) to control measure decay on small balls, enabling the construction of good partitions.
• Combines the entropy regularity with ergodic decomposition and basin arguments to derive contradictions in the case of non-expanding center behavior.

Experimental results

Research questions

• RQ1 Does the partial entropy along an expanding foliation vary upper semi-continuously under C¹ perturbations of the diffeomorphism?
• RQ2 Is the set of Gibbs u-states upper semi-continuous in the C¹ topology for C¹+ partially hyperbolic diffeomorphisms?
• RQ3 Are the sets of C¹ diffeomorphisms with mostly expanding or mostly contracting center directions open?
• RQ4 Can new examples of partially hyperbolic diffeomorphisms with mostly expanding center be constructed?
• RQ5 Do C¹ residual diffeomorphisms admit physical measures, and what is the role of partial entropy in this context?

Key findings

• The partial entropy along expanding foliations is upper semi-continuous with respect to C¹ convergence of diffeomorphisms, weak* convergence of invariant measures, and foliation convergence in the sense of Definition 2.2.
• The set of Gibbs u-states of C¹+ partially hyperbolic diffeomorphisms is upper semi-continuous in the C¹ topology.
• The sets of C¹ diffeomorphisms with mostly contracting or mostly expanding center directions are open in the C¹ topology.
• New examples of partially hyperbolic diffeomorphisms with mostly expanding center are constructed, extending known classes.
• Every C² volume-preserving, accessible, partially hyperbolic diffeomorphism with one-dimensional center and non-vanishing center exponent is C¹ robustly transitive, even without assuming volume preservation in the neighborhood.
• Physical measures exist for a C¹ residual subset of diffeomorphisms, as a consequence of the entropy regularity and ergodic decomposition arguments.