arXiv subjects

Matan Tal

Publications and source records attributed to Matan Tal.

6 recordsLinked to original sources

arXiv2026

On a Constraint on Invariant Measures of Certain Cellular Automata

In [7], a constraint on invariant measures of bi-permutative cellular automata has been observed: fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional probability at the zero index. When the alphabet is a finite group and the automaton is multiplication of two neighbors, that set is in fact a coset of some subgroup. In the present paper, we strengthen the formulations in [7] and investigate further the implications of this constraint. In the finite group case mentioned above, relations between some attributes of the group structure and the invariant measures are examined. We also inspect a factor, with respect to the shift, that this constraint induces, and analyze the special case in which it has zero measure-theoretical entropy, thus observing an interplay between existence of zero entropy invariant measures on that factor and existence of positive entropy measures corresponding to them on the original system. Then, we leave the setting of bi-permutative cellular automata and generalize our results to a wider class which we named RLP subshifts. The peculiar situation is that although this class may be much larger than the class of bi-permutative cellular automata, we were able to prove only for essentially one other example - the symbolic coding of the times 2 times 3 system on the circle (and its generalizations) - that it belongs to it.

math.DS↗

arXiv2025

The Return Times Theorem, Auto-Correlation and Sequences with an Empty Fourier-Bohr Spectrum

This paper explores the proof by J. Bourgain, H. Furstenberg, Y. Katznelson and D.S. Ornstein of their return times theorem [2] and lights a corner in it regarding the role of auto-correlation. As for pointwise convergence, this was already observed in [5], and here we exploit the opportunity to write down the proof. This yields a more intrinsic characterization of the sequences satisfying the pointwise theorem. Then we proceed and obtain a characterization linked to auto-correlation also to sequences satisfying the mean theorem - by that theorem those were already known to be exactly the sequences with an empty Fourier-Bohr spectrum. Some further investigation is done and examples are provided regarding generic sequences satisfying the pointwise theorem for which the measure on the circle that the auto-correlation function represents (by Fourier transform) is not atomless, and also regarding the existence of sequences that satisfy the mean theorem but not the pointwise one.

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arXiv2025

Parsings of Stationary Processes, Stopping Times and the Fundamental Pointwise Convergence Theorems of Ergodic Theory

The idea of a parsing of a stationary process according to a collection of words is introduced, and the basic framework required for the asymptotic analysis of these parsings is presented. We demonstrate how the pointwise ergodic theorem and the Shannon-McMillan-Breiman theorem can be deduced from their respective weaker convergence in probability versions combined with our observations regarding parsings, where the parsings are done according to collections that originate in stopping times tailored for that purpose.

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arXiv2021

Furstenberg's Times 2, Times 3 Conjecture (a Short Survey)

The following is a concise exposition of the conjecture and three of its proofs for the case of positive entropy by D. Rudolph [23] , B. Host [15] and W. Parry [22]. A simpler theorem of R. Lyons [20] - preceding them - is also presented and proved. This is a survey, no new results are introduced.

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arXiv2021

Some Remarks on the Notion of Bohr Chaos and Invariant Measures

The notion of Bohr chaos was introduced in [3, 4]. We answer a question raised in [3] of whether a non uniquely ergodic minimal system of positive topological entropy can be Bohr chaotic. We also prove that all systems with the specification property are Bohr chaotic, and by this line of thought give an independent proof (and stengthening) of theorem 1 of [3] for the case of invertible systems. In addition, we present an obstruction for Bohr chaos: a system with fewer than a continuum of ergodic invariant probability measures cannot be Bohr chaotic.

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arXiv2019

Conic Representations of Topological Groups

We define basic notions in the category of conic representations of a topological group and prove elementary facts about them. We show that a conic representation determines an ordinary dynamical system of the group together with a multiplier, establishing facts and formulae connecting the two categories. The topic is also closely related to the affine representations of the group. The central goal was attaining a better understanding of irreducible conic representations of a group, and - particularly - to determine whether there is a phenomenon analogous to the existence of a universal irreducible affine representation of a group in our category (the general answer is negative). Then we inspect embeddings of irreducible conic representations of semi-simple Lie groups in some "regular" conic representation they possess. We conclude with what is known to us about the irreducible conic representations of $SL_{2}\left(\mathbb{R}\right)$.

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