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COPYRIGHT 2006 Hindawi Publishing Corp.
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'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 on their supports. It is of interest to see if this property is transferred to random maps satisfying Pelikan'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's condition is satisfied, but not all the maps are piecewise expanding, and show that the invariant density is not separated from 0.
1. Introduction
A fundamental problem in ergodic theory is to describe the asymptotic behavior of trajectories defined by a dynamical system. In general, the long term behavior of trajectories of a chaotic dynamical system is unpredictable. Therefore, it is natural to describe the behavior of the system by statistical means. In this approach, one attempts to prove the existence of meaningful invariant measures and determine their ergodic properties. For a single transformation, much is known about the densities of the absolutely continuous invariant measures (acim). For example, it is known that the densities inherit smoothness properties from the map itself (Halfant [7]), that the supports consist of a finite union of intervals, and that the densities are bounded below on their supports (Keller [8] and Kowalski [9]).
Random dynamical systems provide a useful framework for modeling and analyzing various physical, social, and economic phenomena [4, 12]. A random dynamical system of special interest is a random map where the process switches from one map to another according to fixed probabilities [11] or, more generally, position-dependent probabilities [2, 6]. In [4] we model the two-slit experiment of quantum mechanics by a random map. More specifically, given two probability density functions, [f.sub.1] and [f.sub.2], we construct maps [[tau].sub.1] and [[tau].sub.2] which have [f.sub.1] and [f.sub.2] as their respective invariant probability density functions. We then define a random map based on these two maps; that is a discrete-time random process which at each time chooses one map or the other with specified probability. This process is referred to as a random map and possesses an invariant probability density function which is a "combination" of [f.sub.1] and [f.sub.2]. Computer experiments on this random map confirm the similarity to the interference results of the two-slit experiment. Random maps are also a convenient framework for modeling processes with randomly changing environment, for example, stock market. We used it in [2] to replace the binomial model applied to determine option prices.
The existence and properties of invariant measures for random maps reflect their long time behavior and play an important role in understanding their chaotic nature. It is, therefore, important to establish properties of their absolutely continuous invariant measures. In this paper we generalize to random maps results of Keller [8] and Kowalski [9], which state that the density of an acim of a nonsingular map is strictly positive on its support. Our main results are proven under the assumption that the individual maps used to construct the random map are piecewise expanding. We also give an example satisfying Pelikan's condition (2.5), showing that the assumption of expanding cannot be removed.
In Section 2 we present the notation and summarize the results we will need in the sequel. In Section 3 we prove the main result.
2. Preliminaries
Let (X, B, [lambda]) be a measure space, where [lambda] is an underlying measure and [[tau].sub.k] : X [right arrow] X, k = 1, 2, ..., K are nonsingular transformations. A random map T with constant probabilities is defined as
T = {[[tau].sub.1], [[tau].sub.2], ..., [[tau].sub.k]; [p.sub.1], [p.sub.2], ..., [p.sub.K]}, (2.1)
where {[p.sub.1], [p.sub.2], ..., [p.sub.K]} is a set of constant probabilities. For any x [member of] X, T(x) = [[tau].sub.k](x) with probability [p.sub.K] and, for any nonnegative integer N, [T.sup.N](x) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (x) with probability [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE...
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