INVARIANT DENSITIES OF RANDOM MAPS HAVE LOWER BOUNDS ON THEIR SUPPORTS

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 0 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.


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 1 and f 2 , we construct maps τ 1 and τ 2 which have f 1 and f 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 1 and f 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.

Preliminaries
Let (X,Ꮾ,λ) be a measure space, where λ is an underlying measure and τ k : X → X, k = 1,2,...,K are nonsingular transformations.A random map T with constant probabilities is defined as where {p 1 , p 2 ,..., p K } is a set of constant probabilities.For any x ∈ X, T(x) = τ k (x) with probability p k and, for any nonnegative integer and only if it satisfies the following condition [11]: for any E ∈ Ꮾ.
We now recall some definitions and results which will be needed to prove the main result in Section 3.

Paweł G óra et al. 3
Definition 2.1.Let τ : (X,Ꮾ,λ) → (X,Ꮾ,λ) be a nonsingular transformation and let μ be an acim with respect to Lebesgue measure λ possessing density function f .We define the support of μ as follows: Let -0 (I) denote the class of transformations τ : I → I that satisfies the following conditions: (i) τ is piecewise monotonic, that is, there exists a partition ᏼ for any i and for all x ∈ (a i−1 ,a i ); (ii) g(x) = 1/|τ i (x)| is a function of bounded variation, where τ i (x) is the appropriate one-sided derivative at the end points of ᏼ.We say that τ ∈ -1 (I) if τ ∈ -0 (I) and α > 1 in condition (2.4), that is, τ is piecewise expanding.

Support of invariant density of random maps
In this section we prove that the invariant density of an acim of the random map T = {τ 1 ,τ 2 ,...,τ K ; p 1 , p 2 ,..., p K }, τ 1 ,τ 2 ,...,τ K ∈ -1 is strictly positive on its support.For notational convenience, we consider K = 2, that is, we consider only two transformations τ 1 , τ 2 .The proofs for larger number of maps are analogous.We consider random maps with constant probabilities.Let ᏽ denote the set of endpoints of intervals of partition ᏼ except the points 0 and 1.
The main result of this paper applies to random maps, where each component map is in -1 (I), but the first two lemmas are proved under the more general assumptions of Theorem 2.6.
Proof.First, let us note that if ν is the normalized Lebesgue measure on J, then Integrating (3.1) over J, we obtain and, using (3.4), we obtain Thus, at least one of the numbers λ(τ 1 (J)), λ(τ 2 (J)) is larger than λ(J).
For any interval J contained in an element of partition ᏼ, we have which implies that at least one of the images is longer than interval J.
The density function f of the acim μ is a function of bounded variation by Theorem 2.6 and thus, by Lemma 2.4, f can be redefined on a countable set to become a lower semicontinuous function f and f = f almost everywhere.Thus, supp( f , where I i 's are open disjoint intervals.Without loss of generality, let us assume that λ(I i ) ≥ λ(I i+1 ) for i = 1,2,....We will prove that ᏽ ∩ I 1 = ∅.Suppose ᏽ ∩ I 1 = ∅.Then I 1 is contained in one of the subintervals, J * , of the partition ᏼ and τ 1 (I 1 ) and τ 2 (I 1 ) are both open intervals.Since f is an invariant density of the random map T, we have . It is clear that at least one element of {τ −1 1, * (x)} is in I 1 and since I 1 ⊂ supp( f ) we have f (x) > 0. Thus, τ 1 (I 1 ) is a subset of supp( f ).Similarly, τ 2 (I 1 ) is a subset of supp( f ).By Lemma 3.1, at least one of the intervals τ 1 (I 1 ), τ 2 (I 1 ) has larger length than the length of I 1 .This is a contradiction because supp( f ) does not contain an interval of length greater than λ(I 1 ).This proves that ᏽ ∩ I 1 = ∅.
Corollary 3.5.The number of different ergodic acim for the random map T satisfying the assumptions of Theorem 2.6 is at most equal to the cardinality of the partition ᏼ minus one.
Proof.Again we can assume that supp( f ) = ∪ ∞ i=1 I i , where I i 's are open disjoint intervals.Let Ᏸ = { j ≥ 1 : I j contains a discontinuity of τ 1 ,τ 2 , or both}.By Lemma 3.4, Ᏸ is not empty.Also, Ᏸ is a finite set.If j ∈ Ᏸ, then τ i (I j ), i = 1,2 is a finite union of intervals.Let J be the shortest interval of the family , where i is not necessarily in Ᏸ and let S is a finite union of open disjoint intervals since it is a family of disjoint intervals with length ≥ λ(J) > 0. For any j ∈ Ᏸ, I j ⊆ S.
Example 3.7.We present a random map satisfying the assumptions of Theorem 2.6, where the support of absolutely continuous T-invariant measure is an infinite countable union of disjoint intervals.In this case the invariant density cannot be bounded away from 0 on its support.Let us define the maps where E(t) denotes the integral part of t, as shown in Figure 3.1.Let us consider the random map T = {τ 1 ,τ 2 ;3/4,1/4}.It satisfies all the assumptions of Pelikan's Theorem 2.6.In particular, T admits an absolutely continuous invariant measure μ with density f of bounded variation.Let S denote the support of f .Lebesgue measure restricted to the interval L 0 = [0,1/4] is invariant for τ 1 .Since τ 1 ([0,1]) = L 0 and τ 1 is exact on L 0 , we have L 0 ⊂ S. Since the interval K 0 = (1/4,1/2) is not in the image of τ 1 or τ 2 , it is disjoint from S. Every image of L 0 is in S. Since τ 1 ([0,1]) = L 0 , we consider only images by τ 2 .We have 1 , and in general, for n+1 . (3.25) Let K (1)  n be the open interval between L (1)  n and L (1)  n+1 , and let K (2)  n be the open interval between L (2)  n and L (2)  n+1 , n ≥ 1.Then, obviously, we have for all n ≥ 1, Paweł G óra et al. 9 Since τ 2 is an injective map, we also have (3.27) We have Thus, We have proved that that is, the support of the absolutely continuous T-invariant measure is an infinite countable union of disjoint intervals.Since the invariant density f is of bounded variation, we have sup for i = 1,2, so the density f is not bounded away from 0.
In Lemma 3.8 and Theorem 3.9, we return to the assumption τ 1 ,τ 2 ∈ -0 , but we assume additionally that the support of the invariant density is a finite union of disjoint intervals.That was proved in Theorem 3.6 using the assumption τ 1 ,τ 2 ∈ -1 .
Proof.Since S = {x : f (x) > 0} is a finite union of open intervals, S = ∪ r i=1 I i , we can assume they are separated by intervals of positive measure.Then S = S \ {a 0 ,a 1 ,...,a q } is also a finite union of intervals: Let (c,d) be any interval in Ᏺ or Ꮿ.We associate with its endpoints two classes of standard intervals: We now define a relation → between elements of .For η,η ∈ : η → η if and only if τ j (U) ∈ η for at least one of j = 1,2 and for sufficiently small U ∈ η.The relation has the following two properties: (1) if η is associated with an endpoint of I i ∈ Ᏺ, then there exists at least one η such that η → η .To prove this, let us fix I i and η associated with one of its endpoints.We Paweł G óra et al. 11 claim that for each J k ∈ Ꮿ, either τ j (J k ) ⊆ I i , j = 1,2 or τ j (J k ) ∩ I i = ∅, j = 1,2.To show this, let us note that since τ j (J k ), j = 1,2 is contained in S = ∪ r i=1 I i and {I i } are separated, τ j (J k ), j = 1,2 is contained in one of I ij , j = 1,2.Now, since λ I ij \ τ j ( S) ≤ λ S \ τ j ( S) = 0, j = 1,2, (3.36) there must exist J k with τ j (J k ) ∈ η , j = 1,2 and this implies (1); (2) if c is an endpoint of I i such that lim x→c f (x) = 0, η is associated with c , η → η , and c is an endpoint of η, then for any U ∈ η, which is a contradiction.We make the following observations: (3) in the setting of (2) above, c / ∈ S. Therefore, c is an endpoint of an interval I i ∈ Ᏺ.Now we define 0 = η : η is associated with an endpoint c of I i ∈ Ᏺ, lim