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

2025/11/12 by Magdalena Foryś-Krawiec, Jana Hantáková, Foryś-Krawiec, Magdalena +5
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DS

paper · pdf · doi:10.48550/arxiv.2511.09718

openalex publication_date 2025/11/12 · openalex created_date 2025/11/15 · openalex updated_date 2026/08/01

Abstract

We study observable dynamics of generic continuous interval maps. We prove that, for a residual subset of C([0,1]), the global Milnor, statistical, and physical attractors coincide with the non-wandering set. Thus, for a typical interval map, the asymptotic dynamics detected by Lebesgue-almost every initial condition recovers all recurrent dynamics. The common attractor is a robust Cantor set of zero Hausdorff dimension. Moreover, every invariant probability measure has a basin of zero Lebesgue measure. We also present examples showing that, outside the generic setting, positive topological entropy may occur outside the global Milnor attractor, and the three attractors need not coincide.

Related

• Topological and statistical attractors for interval maps
• Ergodic Formalism for topological Attractors and historic behavior
• Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e.
• On the Asymptotic Properties of Piecewise Contracting Maps
• Observable invariant measures