2019/09/25 by Uri Gabor, Gabor, Uri
Computer Science · Mathematics · #37A35 #37A50 #60G10 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1909.11453
openalex publication_date 2019/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show the invalidity of finitary counterparts for three classification theorems: The preservation of being a Bernoulli shift through factors, Sinai's factor theorem, and the weak Pinsker property. We construct a finitary factor of an i.i.d. process which is not finitarily isomorphic to an i.i.d. process, showing that being finitarily Bernoulli is not preserved through finitary factors. This refutes a conjecture of M. Smorodinsky [11], which was first suggested by D. Rudolph [7]. We further show that any ergodic system is isomorphic to a process none of whose finitary factors are i.i.d. processes, and in particular, there is no general finitary Sinai's factor theorem for ergodic processes. An immediate consequence of this result is the invalidity of a finitary weak Pinsker property, answering a question of G. Pete and T. Austin [1].