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Stable manifolds, Horseshoes and Lyapunov exponents for $C^1$ diffeomorphisms without domination

Yongluo Cao, Zeya Mi, Rui Zou

TL;DR

This work develops a full nonuniform hyperbolicity theory for C^1 diffeomorphisms lacking domination but possessing a continuous invariant splitting. A central device, resonance blocks, enables precise control of expansion/contraction and allows stable/unstable manifolds, shadowing, and horseshoe constructions to mirror classical Pesin theory in the C^1 setting. The authors prove stable manifold theorems with sub-exponential size variation, establish shadowing and closing on resonance blocks, and show horseshoes exist with entropy matching that of invariant measures; Lyapunov exponents are approximated by periodic orbits and horseshoes when the Oseledets splitting extends continuously. Overall, the paper extends nonuniform hyperbolicity to a broad C^1 regime, bridging domination-based theories and smooth Pesin theory with sharp, block-structured methods. The results broaden applicability to systems with minimal regularity while preserving key dynamical features such as hyperbolic behavior, orbit shadowing, and exponent approximation.

Abstract

We develop the nonuniformly hyperbolic theory for $C^1$ diffeomorphisms admitting continuous invariant splitting without domination. This framework includes stable manifold theorems, shadowing and closing lemmas, the existence of horseshoes and the approximation of Lyapunov exponents. The foundation is a new family of resonance blocks, each arising as the forward limit set of a typical point at carefully chosen resonance times where expansion, contraction and a weak scale-dependent domination coexist.

Stable manifolds, Horseshoes and Lyapunov exponents for $C^1$ diffeomorphisms without domination

TL;DR

This work develops a full nonuniform hyperbolicity theory for C^1 diffeomorphisms lacking domination but possessing a continuous invariant splitting. A central device, resonance blocks, enables precise control of expansion/contraction and allows stable/unstable manifolds, shadowing, and horseshoe constructions to mirror classical Pesin theory in the C^1 setting. The authors prove stable manifold theorems with sub-exponential size variation, establish shadowing and closing on resonance blocks, and show horseshoes exist with entropy matching that of invariant measures; Lyapunov exponents are approximated by periodic orbits and horseshoes when the Oseledets splitting extends continuously. Overall, the paper extends nonuniform hyperbolicity to a broad C^1 regime, bridging domination-based theories and smooth Pesin theory with sharp, block-structured methods. The results broaden applicability to systems with minimal regularity while preserving key dynamical features such as hyperbolic behavior, orbit shadowing, and exponent approximation.

Abstract

We develop the nonuniformly hyperbolic theory for diffeomorphisms admitting continuous invariant splitting without domination. This framework includes stable manifold theorems, shadowing and closing lemmas, the existence of horseshoes and the approximation of Lyapunov exponents. The foundation is a new family of resonance blocks, each arising as the forward limit set of a typical point at carefully chosen resonance times where expansion, contraction and a weak scale-dependent domination coexist.

Paper Structure

This paper contains 24 sections, 28 theorems, 272 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of Main results
  3. Stable and unstable manifolds
  4. Shadowing lemmas and horseshoe's approximation
  5. Extended continuity of Oseledets splitting
  6. Resonance blocks
  7. Basic properties on resonance blocks
  8. Large weight on resonance blocks
  9. Stable manifolds with sub-exponentially varying size
  10. Orbit segments with resonance times
  11. Stable manifolds on resonance blocks
  12. Proof of Theorem \ref{['main:stable']}
  13. Shadowing and closing lemmas
  14. Dynamics on admissible manifolds
  15. Auxiliary constants for $(g,\nu)$
  16. ...and 9 more sections

Key Result

Theorem A

Let $f \in \mathrm{Diff}^1(M)$ preserving an ergodic measure $\mu$. Suppose there exists an open neighborhood $\mathcal{U}$ of $\mathrm{supp}(\mu)$ admitting a continuous $Df$-invariant splitting: $T_{\mathcal{U}} M = E \oplus F.$ If the Lyapunov exponents satisfy then for every sufficiently small $\varepsilon > 0$, there exist Borel functions $\delta(x) > 0$ and $A(x) > 0$ such that for $\mu$-al

Figures (4)

  • Figure 1: Position of $z_1^0$
  • Figure 2: Positions of $w_m^k$ and $b_{-m}^k$
  • Figure 3: Position of $y_m^k$
  • Figure 4: Position of $u_m^k$

Theorems & Definitions (49)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary A
  • Theorem D
  • Theorem E
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 39 more