Characterization of Ergodic Measures of Maximal Entropy for Topologically Transitive Partially Hyperbolic Diffeomorphisms with compact center leaves

TL;DR

This work studies the number and supports of ergodic maximal entropy measures with zero center Lyapunov exponent for topologically transitive, dynamically coherent partially hyperbolic diffeomorphisms on $\mathbb{T}^3$ with compact center leaves. It develops a framework of disintegration along increasing partitions induced by the center-quotient dynamics, uses holonomy invariance, and introduces AB- and AI-systems to model non-accessible behavior, enabling a reduction to canonical prototypes on finite covers. The authors prove an upper bound of two on the number of ergodic maximal entropy measures with $\lambda_c(\mu)=0$, and show their supports are either the entire $\mathbb{T}^3$ or the orbit of a periodic $su$-torus; when a transversely hyperbolic $su$-torus exists, the total number of ergodic MMEs is $2$ or $3$. These results extend the understanding of maximal entropy phenomena for PHC diffeomorphisms on $\mathbb{T}^3$, connecting to prior work on AB/AI-systems and laying groundwork for further classification in low-dimensional dynamics.

Abstract

In this paper, we provide an upper bound on the number of maximal entropy ergodic measures with zero Lyapunov exponent for topologically transitive partially hyperbolic diffeomorphisms with compact one-dimensional center leaves on $\mathbb{T}^3$. Furthermore, we establish a comprehensive characterization of the support of these measures.

Figures (6)

• Figure 1: $cs$-holonomy $\mathcal{H}_{x_0, y_0}^{c s}$
• Figure 2: Mapping correspondence $\beta$
• Figure 3: The map $g_{\alpha}$
• Figure 4: Dynamic on $V_{\epsilon}(\mathbb{T}_{su})$
• Figure 5: support of $\widehat{\eta}_3$

Theorems & Definitions (52)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 3.1
• Lemma 3.2