On the dynamics of dominated splitting

Abstract

Let f : M → M be a C 2 diffeomorphism of a compact surface. We give a complete description of the dynamics of any compact invariant set having dominated splitting. In particular, we prove a Spectral Decomposition Theorem for the limit set L(f ) under the assumption of dominated splitting. Moreover, we describe all the bifurcations that these systems can exhibit and the different types of dynamics that could follow for small C r −perturbations.

Pinball billiards with dominated splitting

“…Theorem B ( [PS09]): Let M be a compact 2-manifold and f a C 2 -diffeomorphism defined as before. Assume that L(f ) has a dominated splitting.…”

Two-dimensional Blaschke products: degree growth and ergodic consequences

“…Since there is a global dominated splitting for f on all of T 2 , Theorem 2.3 from [22] applies for δ-E-arcs for any δ > 0. This gives that ω(I ′ ) is either a periodic closed curve, a periodic closed arc J (with I ′ ⊂ W s (J)), or a periodic point which is either a sink or a saddle-node.…”

Limit sets of convex non-elastic billiards

“…Some important consequences of this property on the dynamics were given by Pujals and Sambarino in [20]. A spectral decomposition theorem was obtained for C 2 compact surface diffeomorphisms having dominated splitting over the limit set L(f ) = x∈M (ω(x) ∪ α(x)) where ω(x) and α(x) are the ω and α-limit sets of x, respectively:…”

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Inspired by the work of Pujals and Sambarino on dominated splitting [20], we present billiards with a modified reflection law which constitute simple examples of dynamical systems with limit sets having dominated splitting and where the dynamics is a rational or irrational rotation.

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