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Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms

Lorenzo J. Díaz, Katrin Gelfert, Michal Rams, Jinhua Zhang

TL;DR

The paper develops a robust framework to realize Katok-style flexibility between entropy and center Lyapunov exponent for a broad class of nonhyperbolic partially hyperbolic diffeomorphisms with a one-dimensional center. By introducing ${\rm scu}$-cubes and ${\rm scu}$-horseshoes, and coupling them with blender-horseshoes under minimal strong foliations, it constructs cascades that approximate ergodic measures with prescribed center exponents and controllable entropies. The method uses discrete-time suspensions and FK/d-bar distance techniques to ensure weak$^*$ convergence and entropy continuity to ergodic limits, yielding continuous entropy paths within each exponent level and a restricted variational principle for $\chi=0$. These results illuminate the structure of the ergodic landscape in partially hyperbolic systems, demonstrate Kadok-like intermediate entropy phenomena in non-specification settings, and apply to a wide range of systems including circle-fibered, flow-type, and some anomalous diffeomorphisms, with implications for the interaction between coexisting hyperbolicity types.

Abstract

We study nonhyperbolic and transitive partially hyperbolic diffeomorphisms having a one-dimensional center. We prove joint flexibility with respect to entropy and center Lyapunov exponent for a broad class of these systems. Flexibility means that for any given value of the center Lyapunov exponent and any value of entropy less than the supremum of entropies of ergodic measures with that exponent, there is an ergodic measure with exactly this entropy and exponent. Our hypotheses involve minimal foliations and blender-horseshoes, they formalize the interplay between two regions of the ambient space, one of center expanding and the other of center contracting type. The list of examples our results apply is rather long, a non-exhaustive list includes fibered by circles, flow-type, some Derived from Anosov diffeomorphisms, and some anomalous (non-dynamically coherent) diffeomorphisms.

Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms

TL;DR

The paper develops a robust framework to realize Katok-style flexibility between entropy and center Lyapunov exponent for a broad class of nonhyperbolic partially hyperbolic diffeomorphisms with a one-dimensional center. By introducing -cubes and -horseshoes, and coupling them with blender-horseshoes under minimal strong foliations, it constructs cascades that approximate ergodic measures with prescribed center exponents and controllable entropies. The method uses discrete-time suspensions and FK/d-bar distance techniques to ensure weak convergence and entropy continuity to ergodic limits, yielding continuous entropy paths within each exponent level and a restricted variational principle for . These results illuminate the structure of the ergodic landscape in partially hyperbolic systems, demonstrate Kadok-like intermediate entropy phenomena in non-specification settings, and apply to a wide range of systems including circle-fibered, flow-type, and some anomalous diffeomorphisms, with implications for the interaction between coexisting hyperbolicity types.

Abstract

We study nonhyperbolic and transitive partially hyperbolic diffeomorphisms having a one-dimensional center. We prove joint flexibility with respect to entropy and center Lyapunov exponent for a broad class of these systems. Flexibility means that for any given value of the center Lyapunov exponent and any value of entropy less than the supremum of entropies of ergodic measures with that exponent, there is an ergodic measure with exactly this entropy and exponent. Our hypotheses involve minimal foliations and blender-horseshoes, they formalize the interplay between two regions of the ambient space, one of center expanding and the other of center contracting type. The list of examples our results apply is rather long, a non-exhaustive list includes fibered by circles, flow-type, some Derived from Anosov diffeomorphisms, and some anomalous (non-dynamically coherent) diffeomorphisms.
Paper Structure (50 sections, 35 theorems, 288 equations, 3 figures, 1 table)

This paper contains 50 sections, 35 theorems, 288 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Flexibility program
  3. Setting
  4. Main results
  5. The set $\mathrm{BM}^1(M)$
  6. Further results
  7. The space of ergodic measures
  8. Techniques
  9. Organization
  10. ${\rm scu}$-cubes and their properties
  11. ${\rm scu}$-cubes: covering relations and iterations
  12. Inner sizes and distances
  13. Cubes and distortion
  14. ${\rm scu}$-horseshoes
  15. Approximating and generating basic sets
  16. ...and 35 more sections

Key Result

Theorem 1

Let $f\in \mathrm{PH}^1_{{\rm c}=1}(M)$ such that Then for every $\chi\in(\chi_{\rm min},\chi_{\rm max})$ and every $\varepsilon>0$ small, there exists a continuous path such that

Figures (3)

  • Figure 1: A ${\rm scu}$-cube centered at some center curve $\gamma$ of size $r$
  • Figure 2: Choice of $T_{\rm b}(\mathbf{a})$
  • Figure 3: The internal structure of $\EuScript{S}_{k+1}$ (bottom), $\EuScript{S}_{k+2}$ (middle), and $\EuScript{S}_{k+3}$ (top) and the $k$-strips inside them. Light green is $t_{k+1}$, medium green $t_{k+2}$, dark green $t_{k+3}$.

Theorems & Definitions (138)

  • Theorem 1
  • Definition 1.1: The class $\mathrm{BM}^1(M)$
  • Remark 1.2
  • Remark 1.3
  • Corollary 2: Restricted variational principle for $\chi=0$
  • Remark 1.4: Occurrence of blender-horseshoes is open and dense in $\mathrm{RNT}^1_{{\rm c}=1}(M)$
  • Remark 1.5: Robust minimality of the strong foliations
  • Remark 1.6
  • Theorem 3
  • Definition 1.7: The classes $\mathrm{BM}^1_{\rm u}(M)$ and $\mathrm{BM}^1_{\rm s}(M)$
  • ...and 128 more