arXiv subjects

Uri Gabor

Publications and source records attributed to Uri Gabor.

4 recordsLinked to original sources

arXiv2025

Moments of finitary factor maps between Bernoulli processes

The problem of what moments can exist for the coding radius of a finitary map between two i.i.d. processes, has been extensively studied in the case of $\mathbb{Z}$-processes. Here we treat this problem for factor maps between $\mathbb{Z}^{d}$-processes ($d>1$). By modeling the homomorphism with a map between spaces of finite sequences, we extend Harvey and Peres' result, showing that for a finitary homomorphism between two i.i.d. processes of equal entropy, if the coding radius of the map has a finite $\frac{d}{2}$-moment, then the two processes share the same informational variance. We use our modeling technique to prove a "Schmidt-type theorem" - that in case the above homomorphism has a coding radius of exponential tails, then the two processes are essentially the same. This result appears to be new even for the one-dimensional case, addressing a question of Angel and Spinka.

math.PR↗

arXiv2024

Moment-optimal finitary isomorphism for i.i.d. processes of equal entropy

The finitary isomorphism theorem, due to Keane and Smorodinsky, raised the natural question of how "finite" the isomorphism can be, in terms of moments of the coding radius. More precisely, for which values does there exist an isomorphism between any two i.i.d. processes of equal entropy, with coding radii exhibiting finite t-moments? [3, 4]. Parry [13] and Krieger [10] showed that those finite moments must be lesser than 1 in general, and Harvey and Peres [5] showed that they must be lesser than 1/2 in general. However, the question for the range between 0 and 1/2 remained open, and in fact no general construction of an isomorphism was shown to exhibit any non trivial finite moments. In the present work we settle this problem, showing that between any two aperiodic Markov processes (and i.i.d. processes in particular) of the same entropy, there exists an isomorphism f with coding radii exhibiting finite t-moments for all t in (0,1/2). The isomorphism is constructed explicitly, and the tails of the radii are shown to be optimal up to a poly-logarithmic factor.

math.DS↗

arXiv2019

On the failure of Ornstein theory in the finitary category

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].

math.DS↗

arXiv2019

Fluctuations of ergodic averages for amenable group actions

We show that for any countable amenable group action, along Følner sequences that have for any $c>1$ a two sided $c$-tempered tail, one have universal estimate for the probability that there are $n$ fluctuations in the ergodic averages of $L^{\infty}$ functions, and this estimate gives exponential decay in $n$. Any two-sided Følner sequence can be thinned out to satisfy the above property, and in particular, any countable amenble group admits such a sequence. This extends results of S. Kalikow and B. Weiss for $\mathbb{Z}^{d}$ actions and of N. Moriakov for actions of groups with polynomial growth.

math.DS↗