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

TL;DR

This paper demonstrates that for most continuous interval maps, the observable, statistical, and Milnor attractors coincide with the non-wandering set, providing a unified long-term behavior description that persists despite the absence of classical stability mechanisms.

Contribution

It proves that for a residual set of continuous maps on [0,1], the three types of attractors coincide and are stable under perturbations, even without hyperbolic or SRB structures.

Findings

Observable attractors are equal to the non-wandering set for generic maps.

The unified attractor governs the long-term statistical behavior of most initial conditions.

Such attractors are not Lyapunov stable and lack dense orbits, indicating the absence of Palis attractors.

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.