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Wild dynamics on manifolds

Pierre Berger

TL;DR

The paper surveys the landscape of wild dynamics on manifolds, bridging differentiable, symplectic, and analytic contexts. It highlights how renormalization and blender-type constructions drive universal behavior near homoclinic tangencies and generate prolific attractors, including the Newhouse phenomena and coexisting sinks. Emergence theory is developed to quantify how nontrivial ergodic decompositions can be, in some settings, as rich as the space of invariant measures, with high emergence being shown to be typical in several symplectic settings via localizable properties and AbC-type constructions. Collectively, the work connects robust dynamical mechanisms (renormalization, universality, AbC) with quantitative descriptions of complexity (emergence, maximal order) and outlines open problems for analytic and higher-dimensional regimes.

Abstract

We survey a few results on differentiable, symplectic, or analytic wild dynamics.

Wild dynamics on manifolds

TL;DR

The paper surveys the landscape of wild dynamics on manifolds, bridging differentiable, symplectic, and analytic contexts. It highlights how renormalization and blender-type constructions drive universal behavior near homoclinic tangencies and generate prolific attractors, including the Newhouse phenomena and coexisting sinks. Emergence theory is developed to quantify how nontrivial ergodic decompositions can be, in some settings, as rich as the space of invariant measures, with high emergence being shown to be typical in several symplectic settings via localizable properties and AbC-type constructions. Collectively, the work connects robust dynamical mechanisms (renormalization, universality, AbC) with quantitative descriptions of complexity (emergence, maximal order) and outlines open problems for analytic and higher-dimensional regimes.

Abstract

We survey a few results on differentiable, symplectic, or analytic wild dynamics.
Paper Structure (15 sections, 20 theorems, 22 equations, 4 figures)

This paper contains 15 sections, 20 theorems, 22 equations, 4 figures.

Table of Contents

  1. Wild hyperbolicity
  2. Definition and examples
  3. Infinitude of the number of attractors
  4. A symplectic counterpart of the problem of the finitude of attractors
  5. Strange attractors, stochastic seas and strong regularity
  6. Wild dynamics from renormalization
  7. Renormalization near homoclinic saddle point
  8. Universality
  9. Density of positive metric entropy
  10. Generic local property and Kolmogorov typicality
  11. Emergence
  12. Genericity and typicality of emergence of maximal order
  13. Maximal oscillation
  14. Wandering stable component
  15. Pseudo-rotation

Key Result

Theorem 1.7

For every $1\le r< \infty$, for every $d\ge 0$, there exists a nonempty open subset $\mathcal{U}_d$ of $d$-dimensional $C^r$-families of $C^r$-local diffeomorphisms of the annulus, such that a generic familyThis common way of writing means that there exists a topologically generic subset $\cal G$ of

Figures (4)

  • Figure 1.1: The bicycle configuration: a dissipative saddle point $P$ displaying both a homoclinic tangency and a heterodimensional cycle with a projectively hyperbolic source $S$, such that $W^{uu}(S)$ intersects $W^s(P)$.
  • Figure 1.2: Each vertical box is sent by an iterate to a horizontal box and then folded at another vertical box. First and second steps of the induction.
  • Figure 2.1: Chain of heteroclinic tangencies between saddle points of a same horseshoe.
  • Figure 2.2: A stochastic sea given by the surgery of a linear hyperbolic map.

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.5
  • Conjecture 1.6: PS96
  • Theorem 1.7: Be16Be17
  • Proof 1: Sketch of proof
  • Conjecture 1.8
  • Theorem 1.9: BC2BV01BY00MV93WY08Ta11berhen
  • Definition 1.12
  • ...and 38 more