Michael Boshernitzan

Quantitative recurrence results

SummaryLetX be a probability measure spaceX=(X, Φ, μ) endowed with a compatible metricd so that (X,d) has a countable base. It is well-known that ifT∶X→X is measure-preserving, then μ-almost all pointsx∈X are recurrent, i.e.,

Ergodic averaging sequences

\lim \begin{array}{*{20}c} {\inf } \\ {n \geqq 1} \\ \end{array} d(x, T^n (x)) = 0

Periodic billiard orbits are dense in rational polygons

. We show that, under the additional assumption that the Hausdorff α-measureHα(X) ofX is σ-finite for some α>0, this result can be strengthened:

Uniform distribution and Hardy fields

\lim \begin{array}{*{20}c} {\inf } \\ {n \geqq 1} \\ \end{array} \left\{ {n^{1/\alpha } . d(x, T^n (x))} \right\}< \infty

Homogeneously distributed sequences and Poincaré sequences of integers of sublacunary growth

, for μ-almost all pointsx∈X. A number of applications are considered.

Generic Continuous Spectrum for Ergodic Schrödinger Operators

LetT be an interval exchange transformation onN intervals whose lengths lie in a quadratic number field. Let {Tn}n=1∞ be any sequence of interval exchange transformations such thatT1 =T andTn is the first return map induced byTn-1 on one of its exchanged intervals In-1. We prove that {Tn}n=1∞ contains finitely many transformations up to rescaling. If the interval In is chosen according to a consistent pattern of induction, e.g., the first interval is chosen, then there existk,n0 ∈ ℤ+, λ ∈R+ such that for alln ≥n0,In = λIn+k andTn,Tn+k are the same up to rescaling. Rephrased arithmetically, this says that a certain family of vectorial division algorithms, applied to quadratic vector spaces, yields sequences of remainders that are eventually periodic. WhenN = 2 the assertion reduces to Lagrange’s classical theorem that the simple continued fraction expansion of a quadratic irrational is eventually periodic. We also discuss the case of periodic induced sequences.These results have applications to topology. In particular, every projective measured foliation on Thurston’s boundary to Teichmüller space that is minimal and metrically ‘quadratic’ is fixed by a hyperbolic element of the modular group. Moreover, if the foliation is orientable, it covers (via a branched covering) an irrational foliation of the two-torus.We also obtain a new proof, for quadratic irrationals, of Boshernitzan’s result that a minimal rank 2 interval exchange transformation is uniquely ergodic.