Class Of Non-uniformly Hyperbolic Systems Research Articles

Convergence to a Lévy Process in the Skorohod $${{\mathcal {M}}}_1$$ and $${{\mathcal {M}}}_2$$ Topologies for Nonuniformly Hyperbolic Systems, Including Billiards with Cusps

We prove convergence to a Levy process for a class of dispersing billiards with cusps. For such examples, convergence to a stable law was proved by Jung & Zhang. For the corresponding functional limit law, convergence is not possible in the usual Skorohod J_1 topology. Our main results yield elementary geometric conditions for convergence (i) in M_1, (ii) in M_2 but not M_1. In general, we show for a large class of nonuniformly hyperbolic systems how to deduce functional limit laws once convergence to the corresponding stable law is known.

The thermodynamic approach to multifractal analysis

AbstractMost results in multifractal analysis are obtained using either a thermodynamic approach based on the existence and uniqueness of equilibrium states or an orbit-gluing approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions, and obtain new results for a number of examples that have already been studied using the orbit-gluing approach. We compare the thermodynamic and orbit-gluing approaches and give a survey of many of the multifractal results in the literature.

Large deviations for nonuniformly hyperbolic systems

We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case of polynomial decay of correlations, we obtain polynomial large deviation estimates, and exhibit examples where these estimates are essentially optimal. In contrast with many treatments of large deviations, our methods do not rely on thermodynamic formalism. Hence, for Holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton, as well as polynomial estimates in situations where there is not a unique equilibrium measure.

Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities

AbstractWe show the existence of countable Markov partitions for a large class of non-uniformly hyperbolic systems with singularities including dispersing billiards in any dimension.