2022/09/19 by Pablo G. Barrientos, Barrientos, Pablo G., Fumihiko Nakamura +5 · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2209.08714
openalex publication_date 2022/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For random compositions of independent and identically distributed measurable maps on a Polish space, we study the existence and finitude of absolutely continuous ergodic stationary probability measures (which are, in particular, physical measures) whose basins of attraction cover the whole space almost everywhere. We characterize and hierarchize such random maps in terms of their associated Markov operators, as well as show the difference between classes in the hierarchy by plenty of examples, including additive noise, multiplicative noise, and iterated function systems. We also provide sufficient practical conditions for a random map to belong to these classes. For instance, we establish that any continuous random map on a compact Riemannian manifold with absolutely continuous transition probability has finitely many physical measures whose basins of attraction cover Lebesgue almost all the manifold.