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Critical set for surface diffeomorphisms revisited

Sylvain Crovisier, Enrique Pujals

TL;DR

This work defines a two-dimensional analogue of critical points for surface diffeomorphisms through the projective-cocycle dynamics, encapsulated in the critical set Crit(f,Λ). It proves that absence of critical points is equivalent to the existence of a one-dimensional dominated splitting on Λ, and identifies precise conditions under which a system is far from homotheties. The paper develops a comprehensive framework—covering dominated splitting preliminaries, dissipative disk dynamics, and Misiurewicz-type maps—to show that critical points organize non-hyperbolic behavior and yield finite decompositions into transitive components or normally hyperbolic arcs in many cases. It also connects tangencies, non-recurrence, and non-uniform hyperbolicity (à la Benedicks–Carleson and Wang–Young) to the critical-set structure, demonstrating how critical points underpin the global dynamics and possible SRB-type behavior in dissipative settings. Overall, the results provide intrinsic criteria linking criticality to domination, enabling a structured classification of non-hyperbolic surface diffeomorphisms and clarifying when hyperbolicity-like conclusions hold.

Abstract

We propose a notion of critical set for two-dimensional surface diffeomorphisms as an intrinsically defined object designed to play a role analogous to that of critical points in one-dimensional dynamics.

Critical set for surface diffeomorphisms revisited

TL;DR

This work defines a two-dimensional analogue of critical points for surface diffeomorphisms through the projective-cocycle dynamics, encapsulated in the critical set Crit(f,Λ). It proves that absence of critical points is equivalent to the existence of a one-dimensional dominated splitting on Λ, and identifies precise conditions under which a system is far from homotheties. The paper develops a comprehensive framework—covering dominated splitting preliminaries, dissipative disk dynamics, and Misiurewicz-type maps—to show that critical points organize non-hyperbolic behavior and yield finite decompositions into transitive components or normally hyperbolic arcs in many cases. It also connects tangencies, non-recurrence, and non-uniform hyperbolicity (à la Benedicks–Carleson and Wang–Young) to the critical-set structure, demonstrating how critical points underpin the global dynamics and possible SRB-type behavior in dissipative settings. Overall, the results provide intrinsic criteria linking criticality to domination, enabling a structured classification of non-hyperbolic surface diffeomorphisms and clarifying when hyperbolicity-like conclusions hold.

Abstract

We propose a notion of critical set for two-dimensional surface diffeomorphisms as an intrinsically defined object designed to play a role analogous to that of critical points in one-dimensional dynamics.
Paper Structure (26 sections, 35 theorems, 40 equations)

This paper contains 26 sections, 35 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. a -- The critical set.
  3. b-- Hypothesis that imply that $f_{|\Lambda}$ is far from homotheties.
  4. c-- Properties of the critical set.
  5. d-- Critical points for dissipative dynamics of the disk.
  6. f-- Misiurewicz diffeomorphisms.
  7. Examples
  8. i- Tangencies and critical points.
  9. ii- Misiurewicz diffeomorphisms.
  10. iii- Recurrent critical points and no exponential growth of the projective cocycle.
  11. iv- Critical points for non-uniform hyperbolic diffeomorphisms in the works of Benedicks-Carlesson and Wang-Young.
  12. v- Critical points in Odometers and unicritical sets.
  13. vi- Critical points outside the Limit set.
  14. Preliminaries about dominated splitting
  15. Proof of theorem \ref{['p.existence']}
  16. ...and 11 more sections

Key Result

Theorem A

Let $f\in Diff^1(M)$ and $\Lambda$ be a compact invariant set of $f$. If $f_{|\Lambda}$ is far from homotheties it follows that $\Lambda$ admits a dominated splitting $T_\Lambda=E\oplus F$, with $\dim(E)=\dim(F)=1$ if and only if ${\rm Crit}(f,\Lambda)$ is empty.

Theorems & Definitions (76)

  • Definition 1
  • Definition 2
  • Theorem A
  • Remark 3
  • Theorem B
  • Theorem C
  • Proposition 4
  • Proposition 5
  • Definition 6
  • Theorem D
  • ...and 66 more