SRB-like Measures for C 0 Dynamics

New Trends in One-Dimensional Dynamics

2019

Self Cite

We define the empiric stochastic stability of an invariant measure in the finite-time scenario, the classical definition of stochastic stability. We prove that an invariant measure of a continuous system is empirically stochastically stable if and only if it is physical. We also define the empiric stochastic stability of a weak * -compact set of invariant measures instead of a single measure. Even when the system has not physical measures it still has minimal empirically stochastically stable sets of measures. We prove that such sets are necessarily composed by pseudo-physical measures. Finally, we apply the results to the one-dimensional C1-expanding case to conclude that the measures of empirically stochastically sets satisfy Pesin Entropy Formula. Eleonora Catsigeras formula (1) below). We call ε the noise level, or also the amplitude of the random perturbation. To define the empiric stochastic stability we will take ε → 0 + . In the stochastic system (M, f , P ε ), the symbol P ε denotes the family of probability distributions, which are called transition probabilities, according to which the noise is added to f (x) for each x ∈ M. Precisely, each transition probability is, for all n ∈ N, the distribution of the state x n+1 of the noisy orbit conditioned to x n = x, for each x ∈ M. As said above, the transition probability is supported on the ball with center at f (x) and radius ε > 0. So, the zero-noise system (M, f ) is recovered by taking ε = 0; namely, (M, f ) = (M, f , P 0 ). The observer naturally expects that if the amplitude ε > 0 of the random perturbation were small enough, then the ergodic properties of the stochastic system "remembered" those of the zero-noise system.The foundation and tools to study the random perturbations of dynamical systems were early provided in [28], [4], [18]. The stochastic stability appears in the literature mostly defined through the stationary meaures µ ε of the stochastic system (M, f , P ε ) Classically, the authors prove and describe, under particular conditions, the existence and properties of the f -invariant measures that are the weak * -limit of ergodic stationary measures as ε → 0 + . See for instance the early results of [30], [20], [8], [21], [19]), and the later works of [24], [2], [1], [3]. For a review on stochastic and statistical stability of randomly perturbed dynamical systems, see for instance [29] and Appendix D of [7].The stationary measures of the ramdom perturbations provide the probabilistic behaviour of the noisy system asymptotically in the future. Nevertheless, from a rather practical or experimental point of view the concept of stochastic stability should not require the knowledge a priori of the limit measures of the perturbed system as n → +∞ . For instance [14] presents numerical experiments on the stability of one-dimensional noisy systems in a finite time. The ergodic stationary measure is in fact substituted by an empirical (i.e. obtained after a finite-time observation of the system) probability. Also in other applications of the theor...

Section: Resultsmentioning

confidence: 99%

“…Hence, we deduce that, for our map f , µ is the unique pseudo-physical measure. Besides in [10], it is proved that if the set of pseudo-physical or SRB-like measures is finite, then all the pseudo-physical measures are physical. We deduce that our map f has a unique physical measure µ.…”

Section: We Constructmentioning

confidence: 99%

Empiric Stochastic Stability of Physical and Pseudo-physical Measures

New Trends in One-Dimensional Dynamics

2019

Self Cite

“…Proof. From [CE1,Theorem 1.3] the set O f is closed. Let us prove that its interior in M f is empty.…”

Section: Consider the Setmentioning

confidence: 99%

Invariant measures for typical continuous maps on manifolds

Troubetzkoy

2019

Nonlinearity

Self Cite

We study the invariant measures of typical C 0 maps on compact connected manifolds with or without boundary, and also of typical homeomorphisms. We prove that the weak * closure of the set of ergodic measures coincides with the weak * closure of the set of measures supported on periodic orbits and also coincides with the set of pseudo-physical measures. Furthermore, we show that this set has empty interior in the set of invariant measures.

“…Theorems 1.3 and 1.5 of [CE11] prove that, for any continuous map f : M → M the set of SRB-like measures is nonempty, it contains pω(x) for Lebesgue-almost all x ∈ M , and it is the minimal weak * -compact set K ⊂ P such that pω(x) ⊂ K for Lebesgue-almost all x ∈ M . Therefore, the set of SRB-like measures minimally describes the asymptotic statistics of Lebesgue-almost all orbits.…”

Section: Definition 14 (Srb Physical and Srb-like Measures)mentioning

confidence: 99%

“…That is, we would like to know when an invariant measure under f ∈ Diff 1 (M ) satisfies Pesin Entropy Formula. We first recall some definitions and previous results taken from [CE11].…”

Section: Introductionmentioning

confidence: 99%

The Pesin entropy formula for diffeomorphisms with dominated splitting

Cerminara

Enrich

2014

Ergod. Th. Dynam. Sys.

Self Cite

For any C 1 diffeomorphism with dominated splitting we consider a nonempty set of invariant measures which describes the asymptotic statistics of Lebesguealmost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating subbundle. As a consequence, if those exponents are non negative, and if the exponents on the dominated subbundle are non positive, those measures satisfy the Pesin Entropy Formula.