Consider a permutation $\tau$ of the set $\{1,2,\dots,n,\}$.  If we divide the unit interval $[0,1)$ into $n$ half-open subintervals, we can consider the map $f$ which rearranges the subinterval according to the permutation $\tau$.  Such maps are called interval exchange transformations (IETs) and are the order preserving piecewise isometries of intervals, and preserve the Lebesgue measure.  IETs were first studied by Sinai in 1973, and then Keane in 1977, who showed that each minimal IET had a finite number of ergodic measures and conjectured that the Lebesgue measure was in fact the only ergodic invariant measure for such maps.  Much of the following research on IETs was based around proofs of this conjecture and will be discussed in the talk.

A convolution of two singular continuous measures can be singular continuous or absolutely continuous (or of a mixed type). It is usually hard to determine which case is present for a specific pair of measures. It turnes out that for measures of maximal entropy of large Hausdorff dimension supported on dynamically defined Cantor sets generically the convolution is a.c. (this is a joint result with D.Damanik and B.Solomyak). This is in a sense a measure-theoretical counterpart of the claim (known as Newhouse Gap Lemma) that the sum of two sufficiently thick Cantor sets must contain an interval. 

Finite-time Lyapunov exponents and vectors are used to define and diagnose boundary-layer type, two-timescale behavior and to determine the associated manifold structure in the flow. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting, defined by finite-time Lyapunov exponents and vectors, is interpreted in relation to the asymptotic theory of partially hyperbolic sets. The finite-time Lyapunov approach relies more heavily on the Lyapunov vectors due to their relatively fast convergence compared to that of the corresponding exponents. Examples of determining slow manifolds and solving Hamiltonian boundary-value problems associated with optimal control are described.

I will discuss a general approach for constructing SRB measures for diffeomorphisms possessing chaotic attractors (i.e., attractors with nonzero Lyapunov exponents). I introduce a certain recurrence condition on the iterates of Lebesgue measure called “effective hyperbolicity” and I will show that if the asymptotic rate of effective hyperbolicity is exponential on a set of positive Lebesgue measure, then the system has an SRB measure. Along the way a new notion of hyperbolicity -- "effective hyperbolicity'' will be introduced and a new example of a chaotic attractor will be presented. This is a joint work with V. Climenhaga and D. Dolgopyat.

A substitution rule is an algorithm for replacing a symbol with a finite string of symbols (for example, replace 0 by 01 and replace 1 by 0) and a substitution sequence is a sequence obtained from repeated applications of the substitution rule. A Sturmian sequence is a non-periodic sequence of minimal complexity. Both of these objects that are central to the study of mathematical models of quasicrystals. We will discuss interesting dynamical, algebraic, and combinatorial properties of these two families of sequences, as well as their relation to each other.