Abundance of periodic orbits for typical impulsive flows

TL;DR

This paper investigates the prevalence of hyperbolic periodic orbits in impulsive flows, showing that for generic impulses, such orbits are dense among non-wandering points, with applications to billiard, Anosov, and Lorenz systems.

Contribution

It establishes that typical impulsive semiflows have dense hyperbolic periodic orbits among non-wandering points and provides conditions for invariance of the non-wandering set.

Findings

Hyperbolic periodic orbits are dense in non-wandering points meeting the impulsive region.

For generic impulses, the non-wandering set (excluding discontinuities) is invariant.

Applications include impulsive billiard, Anosov, and Lorenz flows.

Abstract

Impulsive dynamical systems, modeled by a continuous semiflow and an impulse function, may be discontinuous and may have non-intuitive topological properties, as the non-invariance of the non-wandering set or the non-existence of invariant probability measures. In this paper we study dynamical features of impulsive flows parameterized by the space of impulses. We prove that impulsive semiflows determined by a C1-Baire generic impulse are such that the set of hyperbolic periodic orbits is dense in the set of non-wandering points which meet the impulsive region. As a consequence, we provide sufficient conditions for the non-wandering set of a typical impulsive semiflow (except the discontinuity set) to be invariant. Several applications are given concerning impulsive semiflows obtained from billiard, Anosov and geometric Lorenz flows.