Viana maps and limit distributions of sums of point measures

This thesis consists of five articles mainly devoted to problems in dynamical systems and ergodic theory. We consider non-uniformly hyperbolic two dimensional systems and limit distributions of point measures, which are absolutely continuous with respect to Lebesgue measure. Let fa0(x) = a0−x 2 be a quadratic map, where the parameter a0 ∈ (1, 2) is chosen such that the critical point 0 is pre-periodic (but not periodic). In Papers A and B, we study skew-products (θ, x) #→ F (θ, x) = (g(θ), fa0(x) + αs(θ)), (θ, x) ∈ S × R. The functions g : S → S and s : S → [−1, 1] are the base dynamics and the coupling functions, respectively, and α is a small, positive constant. Such quadratic skew-products are also called Viana maps. In Papers A and B, we show for several choices of the base dynamics and the coupling function that the map F has two positive Lyapunov exponents and for some cases we further show that F admits also an absolutely continuous invariant probability measure. In Paper C we consider certain Bernoulli convolutions. By showing that a specific transversality property is satisfied, we deduce absolute continuity of the distributions associated to these Bernoulli convolutions. In Papers D and E, we consider sequences of real numbers on the unit interval and study how they are distributed. The sequences in Paper D are given by the forward iterations of a point x ∈ [0, 1] under a piecewise expanding map Ta : [0, 1] → [0, 1] depending on a parameter a contained in an interval I. Under the assumption that each Ta admits a unique absolutely continuous invariant probability measure μa and that some technical conditions are satisfied, we show that the distribution of the forward orbit T j a (x), j ≥ 1, is described by the distribution μa for Lebesgue almost every parameter a ∈ I. In Paper E we apply the ideas in Paper D to certain sequences, which are equidistributed in the unit interval and give a geometrical proof of a well-known result by Koksma from 1935. iii te l-0 06 94 20 1, v er si on 1 4 M ay 2 01 2