# Spectral properties of Schrodinger operators with substitution potentials

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# Partial hyperbolicity: a brief discourse

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# Partial hyperbolicity: a brief discourse

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# Partial hyperbolicity: a brief discourse

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This is the first in a series of two (or three) talks on partial (and normal) hyperbolicity. Partial hyperbolicity is in a sence a generalization of the notion of uniform hyperbolicity -- a well developed branch of smooth dynamical systems. In this talk we will begin with a motivation, definitions and some basic examples, laying the ground for the subsequent discussion of more advanced topics (mainly questions concerning generalization of resulrts of hyperbolic dynamics to partially hyperbolic systems).

# Pugh-Shub stable ergodicity theory and Lyapunov exponents

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I will describe recent advances in the Pugh-Shub stable ergodicity theory for partially hyperbolic diffeomorphisms. In particular, I consider two "competing" methods to show that a given partially hyperbolic diffeomorphism is stably ergodic (i.e., it is ergodic along with any of its sufficiently small perturbations). One of them relates the problem to to the global estimates of the action of the system along its central direction while another one deals with a more delicate estimates using Lyapunov exponents in the central direction.

# On the Dynamics of the Period Doubling Trace Map

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We will be looking at the trace map of the discrete Schrodinger operator with potential given by the period doubling sequence. It is known that for any positive coupling constant the spectrum of the corresponding operator is a Cantor set of Lebesgue measure zero. We are interested in the structure of the spectrum for small coupling constant, specifically the Hausdorff dimension and thickness.

# Lagrange Spectrum, Sum of Cantor Sets, Self-Similarity, and Measure

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In this talk, we will begin with simple questions of the Diophantine Approximation Theory, for instance, how closely can a given irrational number x be approximated by a rational number r with denominator no larger than a fixed number? This will lead us to talk about the set known as the Lagrange Spectrum whose structure closely resembles the structure of the sum of dynamically defined Cantor sets, which are defined by an iterate system of expanding differentiable functions on intervals. We will construct two Cantor sets whose arithmetic sum is a uniformly contracting self-similar set. A local result on a sufficient condition for a uniformly contracting self-similar set to be of Lebesgue measure zero will be proven.

# On symbolic representation of Anosov-Katok example

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The structure of Anosov-Katok example (in fact, this is a series of examples that can be constructed using similar techniques) will be presented. This is a way to build a smooth realization for several classes of measure preserving transformations.

# Piecewise Translations

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We will review the recent (and not so recent) results on dynamics of piecewise isometries (especially piecewise translations), both in one and in higher dimensional case. Some interesting results (by Suzuki, Goetz, Zhuravlev, Boshernitzan, Bruin, Troubetzkoy, Buzzi) are known, but most of natural questions are still open. The main goal of the talk is to expose these open questions to potential researchers.