Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus

Abstract

In this paper we address the existence and ergodicity of nonuniformly hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems have formulation as a skew product system defined by planar diffeomorphisms, with average contraction condition, forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one physical measure. In our approach, these skew product systems arising from iterated function systems which are generated by finitely many weak contractive diffeomorphisms. Under some conditions including negative fiber Lyapunov exponents, we prove the existence of unique non-uniformly hyperbolic attracting invariant graphs for these systems which attract positive orbits of almost all initial points. Also, we prove that these systems are Bernoulli and therefore they are mixing. Moreover, these properties remain true under small perturbations in the space of endomorphisms on the solid torus.

Invariant Gragh and Random Bony Attractors

“…We mention that in skew product systems with uniformly contracting fiber maps, there exist continuous invariant attracting sets for the overall dynamics, see [14], Theorem 6.1a, [15]. Results in the non-uniform case, when the fiber map possesses negative Lyapunov exponents in the fibre [3,11,12,29,30], are very recent and invariant graphs are very sensitive to perturbations. This work is organized as follows: In Subsections 1.1 and 1.2 we recall some standard definitions.…”