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Observable Dynamics and the Generic Coincidence of Milnor, Statistical, and Physical Attractors

Magdalena Foryś-Krawiec, Jana Hantáková, Michał Kowalewski, Piotr Oprocha

TL;DR

The paper addresses how typical continuous interval dynamics exhibit observable long-term behavior by comparing Milnor, statistical, and physical attractors. It proves that for a residual subset of $C([0,1])$ these attractors coincide and equal the nonwandering set, which forms a Cantor set of zero Hausdorff dimension and depends continuously on the map in the Hausdorff metric. The attractor is robust to perturbations but not Lyapunov stable, and it contains no dense orbits, implying Palis attractors are generically absent and that observable statistical behavior can persist without SRB measures. Additionally, the authors construct an explicit interval map showing strict inclusions $A_{phys}\subsetneq A_{stat}\subsetneq \Lambda$, highlighting that even in one-dimensional continuous dynamics, multiple attractor notions can diverge in nontrivial ways.

Abstract

We study the observable long-term behavior of typical continuous dynamical systems on the interval $[0,1]$. For a residual subset of $C([0,1])$, the Milnor, statistical, and physical (in the sense of Ilyashenko) attractors coincide and are equal to the non-wandering set. This unified attractor governs the time-averaged dynamics of almost all initial conditions and depends continuously on the map with respect to the Hausdorff metric. From the physical viewpoint, it represents the ensemble of observable steady states describing the long-term statistical behavior of the system. Nevertheless, it is not Lyapunov stable and contains no dense orbits, implying the generic absence of Palis attractors. Thus, generic continuous dynamics admit a well-defined observable attractor even when all classical mechanisms of stability fail, showing how observable statistical behavior persists in the absence of SRB measures or hyperbolic structure.

Observable Dynamics and the Generic Coincidence of Milnor, Statistical, and Physical Attractors

TL;DR

The paper addresses how typical continuous interval dynamics exhibit observable long-term behavior by comparing Milnor, statistical, and physical attractors. It proves that for a residual subset of these attractors coincide and equal the nonwandering set, which forms a Cantor set of zero Hausdorff dimension and depends continuously on the map in the Hausdorff metric. The attractor is robust to perturbations but not Lyapunov stable, and it contains no dense orbits, implying Palis attractors are generically absent and that observable statistical behavior can persist without SRB measures. Additionally, the authors construct an explicit interval map showing strict inclusions , highlighting that even in one-dimensional continuous dynamics, multiple attractor notions can diverge in nontrivial ways.

Abstract

We study the observable long-term behavior of typical continuous dynamical systems on the interval . For a residual subset of , the Milnor, statistical, and physical (in the sense of Ilyashenko) attractors coincide and are equal to the non-wandering set. This unified attractor governs the time-averaged dynamics of almost all initial conditions and depends continuously on the map with respect to the Hausdorff metric. From the physical viewpoint, it represents the ensemble of observable steady states describing the long-term statistical behavior of the system. Nevertheless, it is not Lyapunov stable and contains no dense orbits, implying the generic absence of Palis attractors. Thus, generic continuous dynamics admit a well-defined observable attractor even when all classical mechanisms of stability fail, showing how observable statistical behavior persists in the absence of SRB measures or hyperbolic structure.

Paper Structure

This paper contains 9 sections, 15 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic definitions and properties
  4. Measures
  5. Limit sets
  6. Attractors
  7. Properties of typical attractor on interval
  8. Interval map satisfying $A_{\text{phys}}\subsetneq A_{\text{stat}}\subsetneq \Lambda$
  9. Symbolic dynamics

Key Result

Theorem A

There exists a residual subset $\mathcal{R} \subset C([0,1])$ such that for every $f \in \mathcal{R}$ the non-wandering set $\Omega(f)$ is a Cantor set of zero Hausdorff dimension, and the Milnor attractor, the statistical attractor, and the physical attractor of $f$ all coincide and equal $\Omega(f

Theorems & Definitions (38)

  • Conjecture 1.2: Palis2, Conjecture on the Finitude of Attractors (for interval maps)
  • Theorem A
  • Theorem B
  • Lemma 2.1
  • Lemma 2.1
  • Definition 2.2: Milnor local attractor
  • Definition 2.3: Milnor global attractor, likely limit set
  • Definition 2.4: Palis attractor
  • Definition 2.5: Statistical attractor
  • Theorem 2.6
  • ...and 28 more