Ergodic Theory: Recurrence

Frantzikinakis, Nikos; McCutcheon, Randall

doi:10.1007/978-0-387-30440-3_184

Cited by 5 publications

(3 citation statements)

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“…More broadly, phenomenon of recurrence in dynamical systems appeared first in the work of Poincaré in the 19 th century (see [Fur3,Ber2]) and remains an active area of research in topological, measurable, homogeneous, and linear dynamics. One reason for the topic's longevity is its fruitful relationship with combinatorics and additive number theory, from Bogolyubov's work [Bog] linking the combinatorics of iterated difference sets with Bohr almost periodic structure to Furstenberg's proof [Fur2] of Szemerédi's theorem via recurrence in ergodic theory, and beyond [Fur4,FM2].…”

Section: Simultaneous Approximation In Topological Dynamical Systemsmentioning

confidence: 99%

Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets

Glasscock¹

2022

Preprint

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In the topological dynamical system (X, T ), a point x simultaneously approximates a point y if there exists a sequence n 1 , n 2 , . . . of natural numbers for which T ni x, T 2ni x, . . . , T kni x all tend to y. In 1978, Furstenberg and Weiss showed that every system possesses a point which simultaneously approximates itself (a multiply recurrent point) and deduced refinements of van der Waerden's theorem on arithmetic progressions. In this paper, we study the denseness of the set of points that are simultaneously approximated by a given point. We show that in a minimal nilsystem, all points simultaneously approximate a δ-dense set of points under a necessarily restricted set of powers of T . We tie this theorem to the multiplicative combinatorial properties of return-time sets, showing that all nil-Bohr sets and typical return-time sets in a minimal system are multiplicatively thick in a coset of a multiplicative subsemigroup of the natural numbers. This yields an inhomogeneous multiple recurrence result that generalizes Furstenberg and Weiss' theorem and leads to new enhancements of van der Waerden's theorem. This work relies crucially on continuity in the prolongation relation (the closure of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the theory of rational points and polynomials on nilmanifolds developed by Leibman, Green, Tao; and the machinery of topological characteristic factors developed recently by Glasner, Huang, Shao, Weiss, and Ye.

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Section: Simultaneous Approximation In Topological Dynamical Systemsmentioning

confidence: 99%

Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets

Glasscock¹

2022

Preprint

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“…The simplest (and the most known) instance of the finitely generated semigroup is a discrete time dynamical system, which corresponds to the case of a single generator. In this case the points of a realization {T n x} n∈Z+ are naturally ordered with respect to time n. In the case when the map T is continuous the situation with the recurrence is relatively simple and a number of results are already known (see [15,13]). On the other hand, in the measurable category the situation is more complicated and the constructions described in Section 2 are new even in the case of a single generator.…”

Section: Michael Blankmentioning

confidence: 99%

Recurrence for measurable semigroup actions

Blank¹

2021

Discrete &Amp; Continuous Dynamical Systems - A

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We study qualitative properties of the set of recurrent points of finitely generated free semigroups of measurable maps. In the case of a single generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but otherwise it turns out to be possible that almost all points are recurrent, while there is an wandering set of positive (non-invariant) measure. For a general semigroup the assumption about the common invariant measure for all generators looks somewhat unnatural (despite being widely used). Instead we give abstract conditions (of conservativity type) for this problem and propose a weaker version of the recurrent property. Technically, the problem is reduced to the analysis of the recurrence of a specially constructed Markov process. Questions of inheritance of the recurrence property from the semigroup generators to the entire semigroup and vice versa are studied in detail and we demonstrate that this inheritance might be rather unexpected.1. Introduction. Recall that a (discrete time) semigroup of maps T is a closed under composition collection of maps acting on the same topological space X. One says that T is finitely generated if there is a finite number of elements T := {T 1 , T 2 , . . . , T d } ⊂ T whose compositions generate T . Given a point x ∈ X one can define its trajectory under the action of the semigroup T in two very different ways. First of them, to which we will refer as a global trajectory, is the union of all images of the point x under the action of the maps from T , i.e. ∪ T ∈T T x. The main disadvantage of this definitions is that the points of the global trajectory are not ordered. To avoid this difficulty we consider another approach, to which we will refer as a realization (as in random processes), and which can be described as an infinite sequence of compositions of the generators of the semigroup evaluated at the pointThe union of all realizations is exactly the global trajectory of the point under the action of the semigroup.The idea of recurrence is very old and simple. Roughly speaking it means that a trajectory (or a realization of a random process) returns eventually arbitrary close to its initial point. From the point of view of dynamical systems theory, recurrence is a distant generalization of the notion of periodicity. In terms of the global trajectories this means that there is a sequence {τ j1 , . . . , τ jn , . . . } of elements τ j ∈ T such that

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“…For a comprehensive survey of the impact of the Poincaré recurrence principle in ergodic theory, especially as pertains to the field of ergodic Ramsey theory/additive combinatorics, see [9], [3] and [18].…”

Section: 1mentioning

confidence: 99%