We study ergodic properties of random maps defined as collections of nonsingular maps $f_1,\dots,f_w$ from a compact metric space $(X,\rho)$ into itself taken with probabilities $p_1,\dots,p_w$ ($\sum_i^wp_i=1$) and considered as Markov processes on $X$. For a random map $F$ we construct a transfer (Perron-Frobenius) operator $P_F$ acting in a certain Banach space. Two different possibilities are considered. In the first situation, when the random map is expanding on average, we mainly follow ideas borrowed from the theory of piecewise expanding maps and the corresponding Banach space is a version of the space of functions of bounded variation on $X$. In the second (opposite) situation, when the random map is contracting on average the Banach space is defined in terms of a Hutchinson-like metric in the space of generalized functions on $X$. In both cases we prove the quasi-compactness of the transfer operator $P_F$ and analyze its spectrum by means of Ionescu-Tulchea and Marinescu ergodic theorem. We show that it consists of at most a countable number of isolated eigenvalues of finite multiplicity (corresponding to the so called Ruelle resonances) and the remaining part (essential spectrum) is located inside a disk centered at origin with the radius strictly smaller than 1. Estimates of the essential spectrum radius for both cases are obtained.
If the map $F$ is expanding on average we give sufficient conditions of the existence of absolutely continuous invariant measures and study their ergodic properties.
Our interest to this type of systems is to a some extent due to the analysis of spectral properties of Anosov maps in [BKL], where the presence of stable foliations relates the situation to the case of contracting on average maps. Some of the methods and ideas used here were originated from the construction in [BKL] needed to study the behavior of the transfer operators `along the stable foliation' and are based technically on the establishing of the so called Lasota-Yorke type inequalities. The nature of the system under study gives some advantages compared to the Anosov maps, in particular, the functional Banach spaces in our case do not depend on the fine structure of the map, and the spectrum stability results are proven for a much broader class of random perturbations.
The original Ulam conjecture made 36 years ago about the finite state Markov chain approximation of Sinai-Bowen-Ruelle (or physical) measures of chaotic dynamical systems were verified for a number of interesting situations theoretically and is widely used numerically. In fact, this approach is the main numerical tool in modern applied chaotic dynamics. Despite a lot of efforts no situation when this approach does not hold has been found (even in the case of contractive maps). We give the first counterexample demonstrating some restrictions of this approach and possibly showing the way to prove the validity of the conjecture for a more general class of maps. Generalizations of Ulam conjecture for the analysis of the dynamical spectrum and its stochastic stability will be discussed as well.
[BK97] M.L.Blank, G.Keller, Random perturbations of chaotic dynamical systems: stability of the spectrum, Nonlinearity 11 (1998), 81-107.
[Bl00] M.L. Blank, Stability and localization in chaotic dynamics, Monograph, MCCME, Moscow, 2001.
[Bl1] M.Blank, Perron-Frobenius spectrum for random maps and its approximation, Moscow Math. J., (to appear).
[BKL] M.Blank, G.Keller, C.Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, nlin.CD/0104031, preprint IHES/M/01/31, Juin 2001, 58pp.
