Limit Theorems for Random Expanding or Anosov Dynamical Systems and Vector-Valued Observables
- PDF / 729,583 Bytes
- 49 Pages / 439.37 x 666.142 pts Page_size
- 86 Downloads / 221 Views
Annales Henri Poincar´ e
Limit Theorems for Random Expanding or Anosov Dynamical Systems and Vector-Valued Observables Davor Dragiˇcevi´c and Yeor Hafouta Abstract. The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic (Anosov) dynamics developed by the first author et al. to establish quenched versions of the large deviation principle, central limit theorem and the local central limit theorem for vector-valued observables. We stress that the previous works considered exclusively the case of scalarvalued observables. In another direction, we show that this method can be used to establish a variety of new limit laws (either for scalar or vectorvalued observables) that have not been discussed previously in the literature for the classes of dynamics we consider. More precisely, we establish the moderate deviations principle, concentration inequalities, Berry– Esseen estimates as well as Edgeworth and large deviation expansions. Although our techniques rely on the approach developed in the previous works of the first author et al., we emphasize that our arguments require several nontrivial adjustments as well as new ideas. Mathematics Subject Classification. Primary 37D20, 60F05.
1. Introduction The so-called spectral method represents a powerful approach for establishing limit theorems. It has been introduced by Nagaev [40,41] in the context of Markov chains and by Guivarc’h and Hardy [27] as well as Rousseau-Egele [45] for the deterministic dynamical systems. We refer to [33] for a detailed presentation of this method. In the case of deterministic dynamics, we have a map T on the state space X which preserves a probability measure μ on X. Then, for a suitable class of observables g, we want to obtain limit laws for the process (g ◦ T n )n∈N . In other words, we wish to study the distribution of Birkhoff sums
D. Dragiˇcevi´c and Y. Hafouta
Ann. Henri Poincar´e
n−1 Sn g = i=0 g ◦ T i , n ∈ N. Let L be the transfer operator (acting on a suitable Banach space B) associated with T and for each complex parameter θ, let Lθ be the so-called twisted transfer operator given by Lθ f = L(eθg · f ), f ∈ B. The core of the spectral method consists of the following steps: • rewriting the characteristic function of Sn g in terms of the powers of the twisted transfer operators Lθ ; • applying the classical Kato’s perturbation theory to show that for θ sufficiently close to 0, Lθ inherits nice spectral properties from L. More precisely, usually one works under assumptions which ensure that L is a quasi-compact operator of spectral radius 1 with the property that 1 is the only eigenvalue on the unit circle with multiplicity one (and with the eigenspace essentially corresponding to μ). Then, for θ sufficiently close to 0, Lθ is again a quasi-compact operator with an isolated eigenvalue of multiplicity one such that both the eigenvalue and the corresponding eigenspace (as well as other related objects) depend analytically on θ. This method has been used to esta
Data Loading...
