Random perturbations of non-uniformly expanding maps

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1985

Let T and H r (0<r <r0) be continuous mappings of a compact metric space (M,d), d(x,H~(x))<r for any xsM. We consider Markov processes ~ with transition functions lit (x, A) = p xA(T(x)) + q xA(H~(x)), They are random compositions of T and H r. We study the existence, uniqueness and asymptotic (r~0) behaviour of Tr-invariant measures /2,.. We do this by converting the problem into the problem of small stochastic perturbations of the mapping T. The main result is that the weak limit points (for r~0) of the set {fi/ 0<r<r0} are measures concentrated on "attractors" of the mapping T. Our definition of "attractors" is based on ideas similar to those of Ruelle . The perturbations we deal with are nonlocal and singular, whereas so far most authors considered local and "absolutely continuous" stochastic perturbations (e.g. Ruelle [5]).

2011

Metastable dynamical systems were recently studied [Gonzáles-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 [Gonzáles-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, τ 1 , defined on two disjoint invariant sets of R N and the other is a deterministically perturbed version of τ 1 , τ 2 , which allows passage between the two disjoint invariant sets of τ 1. We model this system by a position dependent random map based on τ 1 and τ 2 , 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 [Gonzáles-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.

Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2013

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle-Perron-Frobenius operator acting on the space of Hölder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.

Journal of Applied Mathematics and Stochastic Analysis, 2006

A random map is a discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. The asymptotic properties of a random map are described by its invariant densities. If Pelikan&#39;s average expanding condition is satisfied, then the random map has invariant densities. For individual maps, piecewise expanding is sufficient to establish many important properties of the invariant densities, in particular, the fact that the densities are bounded away from 0 on their supports. It is of interest to see if this property is transferred to random maps satisfying Pelikan&#39;s condition. We show that if all the maps constituting the random map are piecewise expanding, then the same result is true. However, if one or more of the maps are not expanding, this may not be true: we present an example where Pelikan&#39;s condition is satisfied, but not all the maps are piecewise expanding, and show that the invariant dens...

arXiv: Dynamical Systems, 2020

In the context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. The technique consists in using an inducing scheme in a finite Markov structure with infinitely many symbols to code the dynamics to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map.

arXiv: Dynamical Systems, 2020

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our method merely requires that the expanding map satisfies Chernov&#39;s one-step expansion at $q$-scale and eventually covers a magnet interval. Therefore, our approach is particularly powerful for maps whose inverse Jacobian has low regularity and those who does not satisfy the big image property. The main ingredients of our coupling method are two crucial lemmas: the growth lemma in terms of the characteristic $\cZ$ function and the covering ratio lemma over the magnet interval. We first prove the existence of an absolutely continuous invariant measure. What is more important, we further show that the growth lemma enables the liftablity of the Lebesgue measure to the associated Hofbauer tower, and the resulting invariant measure on the tower admits a decomposition of Pesin-Sinai type. Furthermore, we obtain the exponential decay of ...

Annals of Mathematics

We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potential switching of "stable" and "unstable" directions. This paper shows that with the addition of (very) small random perturbations, one obtains with relative ease Lyapunov exponents reflecting the geometry of the deterministic maps.

Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2022

We study the effects of IID random perturbations of amplitude ǫ > 0 on the asymptotic dynamics of one-parameter families {fa : S 1 → S 1 , a ∈ [0, 1]} of smooth multimodal maps which "predominantly expanding", i.e., |f ′ a | ≫ 1 away from small neighborhoods of the critical set {f ′ a = 0}. We obtain, for any ǫ > 0, a checkable, finite-time criterion on the parameter a for random perturbations of the map fa to exhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponent comparable to S 1 log |f ′ a | dx. This stands in contrast with the situation for the deterministic dynamics of fa, the chaotic regimes of which are determined by typically uncheckable, infinite-time conditions. Moreover, our finite-time criterion depends on only k ∼ log(ǫ −1) iterates of the deterministic dynamics of fa, which grows quite slowly as ǫ → 0.

Ergodic Theory and Dynamical Systems, 2005

Random endomorphisms are studied in one dimension; we show that stable one dimensional random endomorphisms occur open and dense and that in one parameter families bifurcations are typically isolated. We classify codimension one bifurcations for one dimensional random endomorphisms; we distinguish three possible kinds, the random saddle node, the random homoclinic and the random boundary bifurcation. The theory is illustrated on families of random circle diffeomorphisms and random unimodal maps.

Journal of Difference Equations and Applications, 2007

In this paper, we consider a system whose state x changes to if a perturbation occurs at the time t, for and the state x changes to the new state at the time t, for . Here, and are logistic maps. We assume that the number of perturbations in the interval is a discrete random variable . We show that

Physica A: Statistical Mechanics and its Applications, 1996

We discuss the characterization of chaotic behaviours in random maps both in terms of the Lyapunov exponent and of the spectral properties of the Perron-Frobenius operator. In particular, we study a logistic map where the control parameter is extracted at random at each time step by considering finite dimensional approximation of the Perron-Frobenius operator.

Physical Review E, 2013

We investigate the effects of random perturbations on fully chaotic open systems. Perturbations can be applied to each trajectory independently (white noise) or simultaneously to all trajectories (random map). We compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time-dependent conditionally-map-invariant measure. For the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map. In random maps we show that the escape rate κ and dimensions D of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation. We discuss the accuracy (bias) and precision (variance) of finite-size estimators of κ and D, and show that the improvement of the precision of the estimations with the number of trajectories N is extremely slow (∝1/ ln N ). We also argue that the finite-size D estimators are typically biased. General theoretical results are combined with analytical calculations and numerical simulations in area-preserving baker maps.

Physical Review E, 2010

The dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science. Previous studies of escape in chaotic systems have rather focused on the case of unbounded noise, usually assumed to have Gaussian distribution. In this paper, we address the problem of escape induced by bounded noise. We show that the dynamics of escape from an attractor’s basin is equivalent to that of a closed system with an appropriately chosen “hole.” Using this equivalence, we show that there is a minimum noise amplitude above which escape takes place, and we derive analytical expressions for the scaling of the escape rate with noise amplitude near the escape transition. We verify our analytical predictions through numerical simulations of two well-known two-dimensional maps with noise.

Communications in Mathematical Physics

In this paper, we give a quantitative estimate for the sum of the first N Lyapunov exponents for random perturbations of a natural class 2N-dimensional volume-preserving systems exhibiting strong hyperbolicity on a large but noninvariant subset of phase space. Concrete models covered by our setting include systems of coupled standard maps, in both 'weak' and 'strong' coupling regimes.

Publications of the Research Institute for Mathematical Sciences, 1983

Introduction 83 § 2. Skew Product Transformation and Markov Chain 85 § 3. Ergodicity 86 § 4. Assumptions for Further Investigations 88 § 5. Exactness and Cluster Property 90 § 6. Asymptotic Behavior of w-Orbit and Unique Ergodicity 94 § 7. Stability of Invariant Measure under Random Perturbation 95 References 98 Xn(o>}=fjLn^^°fxn-^^° '•• "/J^CaoU) which is, of course, a radom point. It is easy to see that X%(a)) becomes a (time homogeneous) Markov chain