A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps

2024/02/22 by Oscar F. Bandtlow, Bandtlow, O. F., Wolfram Just +3
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Primary 37D20 #Secondary 37E30

paper · pdf · doi:10.48550/arxiv.2402.14770

openalex publication_date 2024/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system.

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