The gluing orbit property, uniform hyperbolicity and large deviations principles for semiflows

Abstract

In this article we introduce a gluing orbit property, weaker than specification, for both maps and flows. We prove that flows with the C 1 -robust gluing orbit property are uniformly hyperbolic and that every uniformly hyperbolic flow satisfies the gluing orbit property. We also prove a level-1 large deviations principle and a level-2 large deviations lower bound for semiflows with the gluing orbit property. As a consequence we establish a level-1 large deviations principle for hyperbolic flows and every continuous observable, and also a level-2 large deviations lower bound. Finally, since many non-uniformly hyperbolic flows can be modeled as suspension flows we also provide criteria for such flows to satisfy uniform and non-uniform versions of the gluing orbit property.

The weak specification property for geodesic flows on CAT(–1) spaces

“…Large deviations in dynamical systems were first developed by Orey and Pelikan [38] in analogy to results in Probability Theory, see [20]. Large deviations results for flows and semi-flows with weak specification have also been announced in the preprint [2].…”

Minimality and gluing orbit property

“…Example 6.1. In [2], it is shown that a topologically transitive subshift of finite type has gluing orbit property. Note that such a system has periodic points.…”

“…The notion of gluing orbit property was introduced in [12], [5] and [2]. As a weaker form of the well-studied specification properties, it turns out to be a more general property which still captures crucial topological features of the systems, especially the non-hyperbolic ones.…”

“…It is called weak specification in [5] in a slightly generalized form. It is in [2] that the name gluing orbit is called to indicate its dissimilarity with specification like properties.…”

“…Here we attempt to reformulate the definitions of specification and gluing orbit properties to make our argument more clear and more convenient. We follow the names called in [2], [10] and [13]. Note that in our definitions of periodic specification and periodic gluing orbit properties the gap G may be ∅.…”