A one-dimensional confined Nonlinear Random Walk is a tuple of N diffeomorphisms of the unit interval driven by a probabilistic Markov chain. For generic such walks, we obtain a geometric characterization of their ergodic stationary measures and prove that all of them have negative Lyapunov exponents.These measures appear to be probabilistic manifestations of physical measures for certain deterministic dynamical systems. These systems are step skew products over transitive subshifts of finite type (topological Markov chains) with the unit interval fiber.For such skew products, we show there exist only finite collection of alternating attractors and repellers; we also give a sharp upper bound for their number. Each of them is a graph of a continuous map from the base to the fiber defined almost everywhere w.r.t. any ergodic Markov measure in the base. The orbits starting between the adjacent attractor and repeller tend to the attractor as t → +∞, and to the repeller as t → −∞. The attractors support ergodic hyperbolic physical measures.

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite type contains an open, dense, and full Lebesgue measure subset of the space of ITMs of three intervals. For this, we show that any ITM of three intervals can be reduced either to a rotation or to a double rotation.1991 Mathematics Subject Classification. Primary: 37C05, 37C20, 37C70, 37D20, 37D45.

We consider a minimal action of a finitely generated semigroup by homeomorphisms of a circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random dynamics of circle homeomorphisms: Antonov's theorem and its corollaries.

For a smooth manifold of any dimension greater than one, we present an open set of smooth endomorphisms such that any of them has a transitive attractor with a non-empty interior. These maps are $m$-fold non-branched coverings, $m \ge 3$. The construction applies to any manifold of the form $S^1 \times M$, where $S^1$ is the standard circle and $M$ is an arbitrary manifold.Comment: 13 pages. Added more references, updated existing one

Abstract. In this paper we consider a class of skew products over transitive subshifts of finite type with interval fibers. For a natural class of 1-parameter families we prove that for all but countably many parameter values the nonwandering set (in particular, the union of all attractors and repellers) has zero measure. As a consequence, the same holds for a residual subset of the space of skew products.