Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ledrappier-Young entropy formula for $C^1$ diffeomorphisms with dominated splitting Part 1: Unstable entropy formula and invariance principle

Shaobo Gan, Yao Tong, Jiagang Yang

TL;DR

The paper extends the Ledrappier–Young entropy framework to $C^1$ diffeomorphisms with dominated splitting, proving that when the center bundle is one-dimensional, the unstable entropy equals the metric entropy for ergodic measures and the center contributes no entropy. It overcomes lack of Lipschitz unstable holonomies by introducing fake foliations and establishing uniform Hölder holonomy, along with uncentered maximal-function techniques to obtain convergence arguments. A precise unstable entropy theory is developed with subordinate and quotient partitions, transverse metrics, and adapted partitions, enabling the main equality $h^u_m(f)=h_m(f)$ under dominated splitting and $\,\dim E^c\leq 1$. The results yield a $C^1$ invariance principle and various applications, including extensions of Avila–Viana type invariance results and entropy-measure classifications in the $C^1$ partially hyperbolic setting, notably for SPH$(M)$. Overall, the work paves the way for a $C^1$ Ledrappier–Young theory in the one-dimensional center case and lays groundwork for further Hölder-regularity relaxations in subsequent parts.

Abstract

We study the unstable entropy of $C^1$ diffeomorphisms with dominated splittings. Our main result shows that when the zero Lyapunov exponent has multiplicity one, the center direction contributes no entropy, and the unstable entropy coincides with the metric entropy. This extends the celebrated work of Ledrappier-Young [18] for $C^2$ diffeomorphisms to the $C^1$ setting under these assumptions. In particular, our results apply to $C^1$ diffeomorphisms away from homoclinic tangencies due to [20]. As consequences, we obtain several applications at $C^1$ regularity. The Avila-Viana invariance principle [7, 33] holds when the center is one-dimensional. Results on measures of maximal entropy due to Hertz-Hertz-Tahzibi-Ures [25], Tahzibi-Yang [33], and Ures-Viana-Yang-Yang [34, 35] also remain valid for $C^1$ diffeomorphisms.

Ledrappier-Young entropy formula for $C^1$ diffeomorphisms with dominated splitting Part 1: Unstable entropy formula and invariance principle

TL;DR

The paper extends the Ledrappier–Young entropy framework to diffeomorphisms with dominated splitting, proving that when the center bundle is one-dimensional, the unstable entropy equals the metric entropy for ergodic measures and the center contributes no entropy. It overcomes lack of Lipschitz unstable holonomies by introducing fake foliations and establishing uniform Hölder holonomy, along with uncentered maximal-function techniques to obtain convergence arguments. A precise unstable entropy theory is developed with subordinate and quotient partitions, transverse metrics, and adapted partitions, enabling the main equality under dominated splitting and . The results yield a invariance principle and various applications, including extensions of Avila–Viana type invariance results and entropy-measure classifications in the partially hyperbolic setting, notably for SPH. Overall, the work paves the way for a Ledrappier–Young theory in the one-dimensional center case and lays groundwork for further Hölder-regularity relaxations in subsequent parts.

Abstract

We study the unstable entropy of diffeomorphisms with dominated splittings. Our main result shows that when the zero Lyapunov exponent has multiplicity one, the center direction contributes no entropy, and the unstable entropy coincides with the metric entropy. This extends the celebrated work of Ledrappier-Young [18] for diffeomorphisms to the setting under these assumptions. In particular, our results apply to diffeomorphisms away from homoclinic tangencies due to [20]. As consequences, we obtain several applications at regularity. The Avila-Viana invariance principle [7, 33] holds when the center is one-dimensional. Results on measures of maximal entropy due to Hertz-Hertz-Tahzibi-Ures [25], Tahzibi-Yang [33], and Ures-Viana-Yang-Yang [34, 35] also remain valid for diffeomorphisms.

Paper Structure

This paper contains 23 sections, 33 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Diffeomorphisms away from tangencies
  5. The non-ergodic case
  6. The geometric structure
  7. Fake foliation
  8. Uniformly Hölder holonomy
  9. Unstable manifolds and center unstable sets
  10. Adapted partition
  11. Special partitions and unstable entropy
  12. The unstable entropy
  13. Special partitions and the quotient structure
  14. Transverse metrics
  15. Uncentered maximal functions
  16. ...and 8 more sections

Key Result

theorem 1

Let $f\in \mathrm{Diff}^1(M)$ preserve an ergodic Borel probability $m$. If the splitting $E^u \oplus E^c \oplus E^s$ is dominated on the support of $m$ and $\dim E^c \leq1$, then $h^u_m(f)=h_m(f)$, where $h_m(f)$ denotes the metric entropy of $f$.

Figures (1)

  • Figure 1: The construction of good balls $b(x,n,k)$.

Theorems & Definitions (61)

  • Definition 1: Subordinate partition
  • theorem 1: Main Theorem
  • Remark 1
  • proposition 1
  • theorem 2
  • theorem 3
  • theorem 4
  • lemma 1: ABC
  • lemma 2
  • lemma 3: Fake foliations
  • ...and 51 more