Browsing by Author "Boshernitzan, Michael"
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Item Borel-Cantelli Sequences(Springer, 2012-06) Boshernitzan, Michael; Chaika, JonItem Centralizers and Conjugacy Classes in the Group of Interval Exchange Transformations(2018-04-18) Bernazzani, Daniel; Boshernitzan, MichaelWe study the group G of interval exchange transformations. Firstly, we will classify, up to conjugation in G, those interval exchange transformations which arise as towers over rotations. Secondly, expanding on previous work of Novak and Li, we will classify, up to conjugation in G, minimal interval exchange transformations which exhibit bounded discontinuity growth. As an application of this classification, we will prove that no infinite order interval exchange transformation could be conjugate in G to one of its proper powers. Thirdly, we will develop broadly applicable, generic conditions which ensure that an interval exchange transformation has no roots in G and that its centralizer is torsion-free. By combining these results with a result of Novak, we will show that a typical interval exchange transformation does not commute with any interval exchange transformations other than its powers. Finally, we will completely describe the possible centralizers of a minimal three-interval exchange transformation.Item Dichotomy for arithmetic progressions in subsets of reals(American Mathematical Society, 2016) Boshernitzan, Michael; Chaika, JonLet H stand for the set of homeomorphisms φ:[0, 1] → [0, 1]. We prove the following dichotomy for Borel subsets A ⊂ [0, 1]: • either there exists a homeomorphism φ ∈ Hsuch that the image φ(A) contains no 3-term arithmetic progressions; • or, for every φ ∈ H, the image φ(A) contains arithmetic progressions of arbitrary finite length. In fact, we show that the first alternative holds if and only if the set A is meager (a countable union of nowhere dense sets).Item Ergodic properties of compositions of interval exchange maps and rotations(IOP Publishing, 2012) Athreya, Jayadev S; Boshernitzan, MichaelWe study the ergodic properties of compositions of interval exchange transformations (IETs) and rotations. We show that for any IET T, there is a full measure set of α ∈ [0, 1) so that T Rα is uniquely ergodic, where Rα is rotation by α.Item Ergodic Schrödinger operators in the infinite measure setting(EMS Press, 2021) Boshernitzan, Michael; Damanik, David; Fillman, Jake; Lukic, MilivojeWe develop the basic theory of ergodic Schrödinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur–Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite ergodic theory that may be of independent interest.Item Heaviness: An extension of a lemma of Yuval Peres(2008) Ralston, David; Boshernitzan, MichaelIn a 1988 paper, Y. Peres proved that in any probability-measure preserving system {X, mu, T}, where X is compact and T is continuous, coupled with a continuous function f, there is at least one point x (which we term heavy) such that 1ni=0 n-1f&j0;Ti&parl0;x&parr0;≥ Xfdm for all n ∈ N . We simplify and expand the proof to a more general result, and investigate the properties of the set of heavy points for a given T and f. The structure of the set of such points in irrational circle rotations is studied extensively, followed by a development of similar ideas in symbolic dynamics. Finally, the notion of heaviness is exported to arbitrary sequences, and a few results contrasting heaviness with equidistribution are developed. Open problems for future research are included throughout.Item Interval exchange transformations: Applications of Keane's construction and disjointness(2010) Chaika, Jon; Boshernitzan, MichaelThis thesis is divided into two parts. The first part uses a family of Interval Exchange Transformations constructed by Michael Keane to show that IETs can have some particular behavior including: (1) IETs can be topologically mixing. (2) A minimal IET can have an ergodic measure with Hausdorff dimension alpha for any alpha ∈ [0,1]. (3) The complement of the generic points for Lebesgue measure in a minimal non-uniquely ergodic IET can have Hausdorff dimension 0. Note that this is a dense Gdelta set. The second part shows that almost every pair of IETs are different. In particular, the product of almost every pair of IETs is uniquely ergodic. In proving this we show that any sequence of natural numbers of density 1 contains a rigidity sequence for almost every IET, strengthening a result of Veech.