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<p>We provide a stochastic characterization of the classical shadowing property of dynamical systems. Then, we reformulate stochastic stability for differentiable systems as well as topological dynamical systems. We also investigate the stability of the set of invariant measures for homeomorphisms on compact metric spaces under <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^0\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>-perturbations. We explore alternative distances, such as the random distance, and introduce the concept of random-stable homeomorphism. These results reveal that while every topologically stable homeomorphism of a compact metric space is also random-stable, the converse is not always true. We analyze the recurrent set and the chain recurrent set of a random-stable homeomorphism on compact manifolds.</p>

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Stochastic Shadowing and Stability

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • C.A Morales,
  • B. Shin

摘要

We provide a stochastic characterization of the classical shadowing property of dynamical systems. Then, we reformulate stochastic stability for differentiable systems as well as topological dynamical systems. We also investigate the stability of the set of invariant measures for homeomorphisms on compact metric spaces under \(C^0\) C 0 -perturbations. We explore alternative distances, such as the random distance, and introduce the concept of random-stable homeomorphism. These results reveal that while every topologically stable homeomorphism of a compact metric space is also random-stable, the converse is not always true. We analyze the recurrent set and the chain recurrent set of a random-stable homeomorphism on compact manifolds.