161 research outputs found
Escape Rates Formulae and Metastability for Randomly perturbed maps
We provide escape rates formulae for piecewise expanding interval maps with
`random holes'. Then we obtain rigorous approximations of invariant densities
of randomly perturbed metabstable interval maps. We show that our escape rates
formulae can be used to approximate limits of invariant densities of randomly
perturbed metastable systems.Comment: Appeared in Nonlinearity, May 201
Pseudo-Orbits, Stationary Measures and Metastability
We study random perturbations of multidimensional piecewise expanding maps.
We characterize absolutely continuous stationary measures (acsm) of randomly
perturbed dynamical systems in terms of pseudo-orbits linking the ergodic
components of absolutely invariant measures (acim) of the unperturbed system.
We focus on those components, called least-elements, which attract
pseudo-orbits. We show that each least element admits a neighbourhood which
supports exactly one ergodic acsm of the random system. We use this result to
identify random perturbations that exhibit a metastable behavior.Comment: To appear in Dynamical System
Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems
We consider a simple class of fast-slow partially hyperbolic dynamical
systems and show that the (properly rescaled) behaviour of the slow variable is
very close to a Friedlin--Wentzell type random system for times that are rather
long, but much shorter than the metastability scale. Also, we show the
possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon
that turns out to be related to the lack of absolutely continuity of the
central foliation.Comment: To appear in Journal of Statistical Physic
METASTABLE SYSTEMS AS RANDOM MAPS
Metastable dynamical systems were recently studied [GonzĂĄlez-Tokman et al., 2011] in the framework of one-dimensional piecewise expanding maps on two disjoint invariant sets, each possessing its own ergodic absolutely continuous invariant measure (acim). Under small deterministic perturbations, holes between the two disjoint systems are created, and the two ergodic systems merge into one. The long term dynamics of the newly formed metastable system is defined by the unique acim on the combined ergodic sets. The main result of [GonzĂĄlez-Tokman et al., 2011] proves that this combined acim can be approximated by a convex combination of the disjoint acims with weights depending on the ratio of the respective measures of the holes. In this note we present an entirely different approach to metastable systems. We consider two piecewise expanding maps: one is the original map, Ï1, defined on two disjoint invariant sets of âN and the other is a deterministically perturbed version of Ï1, Ï2, which allows passage between the two disjoint invariant sets of Ï1. We model this system by a position dependent random map based on Ï1 and Ï2, to which we associate position dependent probabilities that reflect the switching between the maps. A typical orbit spends a long time in one of the ergodic sets but eventually switches to the other. Such behavior can be attributed to physical holes as between adjoining billiard tables or more abstract situations where balls can "leap" from one table to the other. Using results for random maps, a result similar to the one-dimensional main result of [GonzĂĄlez-Tokman et al., 2011] is proved in N dimensions. We also consider holes in more than two invariant sets. A number of examples are presented
Dynamical problems and phase transitions
Issued as Financial status report, Technical reports [nos. 1-12], and Final report, Project B-06-68
Asymptotic escape rates and limiting distributions for multimodal maps
Funding: MD is partially supported by NSF grant DMS 1800321.We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and non-periodic points, as the diameter of the hole goes to zero.PostprintPeer reviewe
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