Topological methods in surface dynamics

We study unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DTn(Tc)|≧Kλcn for some constantsK>0 and λc>1 (c is the critical point ofT). We prove exponential mixing properties of the unique invariant probability density ofT, describe the long term behaviour of typical (in the sense of Lebesgue measure) trajectories by Central Limit and Large Deviations Theorems for partial sum processes of the form\(S_n = \Sigma _{i = 0}^{n - 1} f(T^i x)\), and study the distribution of “typical” periodic orbits, also in the sense of a Central Limit Theorem and a Large Deviations Theorem.

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Spectral Theory, Zeta Functions and the Distribution of Periodic Points for Collet-Eckmann Maps

We show that if a homeomorphism f of the torus, isotopic to the identity, has three or more periodic orbits with non-collinear rotation vectors, then it has positive topological entropy. Furthermore, for each point ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p/q, p ∈ ℤ2, q ∈ ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q, and for every compact connected subset C of Δ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’.

Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity

Let T be a rational function of degree 2 on the Riemann sphere. Denote L the transfer operator of a HH older-continuous function on its Julia set J = J(T) satisfying P(T;) > sup z2J (z). We study the behavior of fL n : n 1g for HH older-continuous functions and show that the sequence is (uniformly) norm-bounded in the space of HH older-continuous functions for suuciently small exponent. As a consequence we obtain that the density of the equilibrium measure for with respect to the exppP(T;) ? ]-conformal measure is HH older-continuous. We also prove that the rate of convergence of L n to this density in sup-norm is O ? exp(? p n). >From this we deduce the central limit theorem for .

On the transfer operator for rational functions on the Riemann sphere

We prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients. We also develop techniques of Markov approximations for dynamical systems. We apply our results to expanding interval maps, Axiom A diffeomorphisms, chaotic billiards and hyperbolic attractors.

Limit theorems and Markov approximations for chaotic dynamical systems

Rotation Sets for Maps of Tori

Rotation sets and ergodic measures for torus homeomorphisms

We prove an almost sure invariance principle and a central limit theorem for the process , where f is a map of an interval with a non-positive Schwarzian derivative whose trajectories of critical points stay far from the critical points, and F is a measurable function with bounded p-variation ( p ≥ 1). The almost sure invariance principle implies the Log-log laws, integral tests and a distributional type of invariance principle for the process .

Krystyna Ziemian

Papers

Rotation Sets for Maps of Tori

Rotation sets and ergodic measures for torus homeomorphisms

Almost sure invariance principle for some maps of an interval