Mathematics Department
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Browsing Mathematics Department by Author "Boshernitzan, Michael"
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Item Borel-Cantelli Sequences(Springer, 2012-06) Boshernitzan, Michael; Chaika, JonItem 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.