Anatole Katok

Lyapunov exponents, entropy and periodic orbits for diffeomorphisms

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Invariant Manifolds, Entropy and Billiards, Smooth Maps with Singularities

Existence of invariant manifolds for smooth maps with singularities.- Absolute continuity.- The estimation of entropy from below through Lyapunov characteristic exponents.- The estimation of entorpy from above through Lyapunov characteristic numbers.- Plane billiards as smooth dynamical systems with singularities.

Topological transitivity of billiards in polygons

Consider a billiard in a polygon Q⊂R2 having all angles commensurate with π. For the majority of initial directions, density of every infinite semitrajectory in configuration space is proved. Also proved is the typicality of polygons for which some billiard trajectory is dense in phase space.

First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity

This is the first in a series of papers exploring rigidity properties of hyperbolic actions ofZk orRk fork ≥ 2. We show that for all known irreducible examples, the cohomology of smooth cocycles over these actions is trivial. We also obtain similar Hölder and C1 results via a generalization of the Livshitz theorem for Anosov flows. As a consequence, there are only trivial smooth or Hölder time changes for these actions (up to an automorphism). Furthermore, small perturbations of these actions are Hölder conjugate and preserve a smooth volume.

Interval exchange transformations and some special flows are not mixing

An interval exchange transformation (I.E.T.) is a map of an interval into itself which is one-to-one and continuous except for a finite set of points and preserves Lebesgue measure. We prove that any I.E.T. is not mixing with respect to any Borel invariant measure. The same is true for any special flow constructed by any I.E.T. and any “roof” function of bounded variation. As an application of the last result we deduce that in any polygon with the angles commensurable with π the billiard flow is not mixing on two-dimensional invariant manifolds.

Local rigidity for certain groups of toral automorphisms

AbstractLet Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on

Four applications of conformal equivalence to geometry and dynamics

\mathbb{T}

Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dyanmical systems

n is locally rigid, i.e., every action of Γ on