INVARIANT DENSITIES OF RANDOM MAPS HAVE LOWER BOUNDS ON THEIR SUPPORTS

BOUNDS ON THEIR SUPPORTS

PAWEŁ G ´ORA, ABRAHAM BOYARSKY, AND MD SHAFIQUL ISLAM Received 8 February 2005; Revised 2 October 2005; Accepted 4 October 2005

A random map is a discrete-time dynamical system in which one of a number of transfor- mations 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 indi- vidual 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.

Copyright © 2006 Paweł G ´ora et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A fundamental problem in ergodic theory is to describe the asymptotic behavior of tra- jectories 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]).

Hindawi Publishing Corporation

Journal of Applied Mathematics and Stochastic Analysis Volume 2006, Article ID 79175, Pages1–13

DOI10.1155/JAMSA/2006/79175

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, f1and f2, we construct maps τ1andτ2which have f1and f2as 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 f1and f2. 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 mea- sures. 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 sup- port. 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.

InSection 2we present the notation and summarize the results we will need in the sequel. InSection 3we prove the main result.

2. Preliminaries

Let (X,Ꮾ,λ) be a measure space, whereλis an underlying measure andτk:X→X,k= 1, 2,. . .,Kare nonsingular transformations. A random mapTwith constant probabilities is defined as

T=

τ1,τ2,. . .,τK;p1,p2,. . .,pK

, (2.1)

where{p1,p2,. . .,pK}is a set of constant probabilities. For anyx∈X,T(x)=τk(x) with probabilityp_kand, for any nonnegative integerN,T^N(x)=τ_kN◦τ_kN−1◦ ··· ◦τ_k1(x) with probabilityΠ^Nj=1p_kj. A measureμisT-invariant if and only if it satisfies the following condition [11]:

μ(E)= K k=1

p_kμτ_k⁻¹(E), (2.2)

for anyE∈Ꮾ.

We now recall some definitions and results which will be needed to prove the main result inSection 3.

Definition 2.1. Letτ: (X,Ꮾ,λ)→(X,Ꮾ,λ) be a nonsingular transformation and letμbe an acim with respect to Lebesgue measureλpossessing density function f. We define the support ofμas follows:

supp(μ)=supp(f)=

x∈X:f(x)>0. (2.3) Definition 2.2. A function f :R→Ris said to be a lower semicontinuous function if and only if f(y)≤lim infx→yf(x) for anyy∈R.

Theorem 2.3 [5]. If f is lower semicontinuous onI=[a,b]⊂R, then it is bounded below and assumes its minimum value. For anyc∈R, the set{x: f(x)> c}is open.

Lemma 2.4 [5]. Iff is of bounded variation onI, then it can be redefined on a countable set to become a lower semicontinuous function.

Let᐀0(I) denote the class of transformationsτ:I→Ithat satisfies the following con- ditions:

(i)τis piecewise monotonic, that is, there exists a partitionᏼ= {Ii=[ai−1,ai], i= 1, 2,. . .,q}ofIsuch thatτ_i=τ|I_iisC¹, and

τ_i(x)≥α >0, (2.4)

for anyiand for allx∈(a_i−1,a_i);

(ii)g(x)=1/|τ_i(x)|is a function of bounded variation, whereτ_i(x) is the appropri- ate one-sided derivative at the end points ofᏼ.

We say thatτ∈᐀1(I) ifτ∈᐀0(I) andα >1 in condition (2.4), that is,τ is piecewise expanding.

Theorem 2.5 [8,9]. Letτ∈᐀1(I) and f be aτ-invariant density which can be assumed to be lower semicontinuous. Then there exists a constantβ >0 such that f|supp(f)≥β.

Theorem 2.6 [1,11]. LetT= {τ1,τ2,. . .,τK;p1,p2,. . .,pK}be a random map, whereτk∈

᐀0(I), with the common partitionᏼ= {J1,J2,. . .,J_q},k=1, 2,. . .,K. If, for allx∈[0, 1], the following Pelikan’s condition

K k=1

pk

τ_k(x)^≤γ <1 (2.5)

is satisfied, then for all f ∈L¹=L¹([0, 1],λ):

(i) the limit

nlim→∞

1 n

n−1 i=1

Pⁱ_T(f)=f^∗ exists inL¹; (2.6) (ii)PT(f^∗)=f^∗;

(iii)V[0,1](f^∗)≤C· f1, for some constantC >0, which is independent of f ∈L¹.

3. Support of invariant density of random maps

In this section we prove that the invariant density of an acim of the random mapT= {τ1,τ2,. . .,τK;p1,p2,. . .,pK},τ1,τ2,. . .,τK∈᐀1is strictly positive on its support. For no- tational convenience, we considerK=2, that is, we consider only two transformations τ1,τ2. The proofs for larger number of maps are analogous. We consider random maps with constant probabilities.

Letᏽdenote the set of endpoints of intervals of partitionᏼexcept 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.

Lemma 3.1. Let the random mapT= {τ1,τ2;p1,p2}satisfy the assumptions ofTheorem 2.6.

In particular,

p1

τ1(x)+ p2

τ2(x)^≤γ <1, (3.1)

for allx∈I\ᏽ. Then, for any intervalJ disjoint withᏽ, at least one of the imagesτ1(J), τ2(J) is longer thanJ.

Proof. First, let us note that ifνis the normalized Lebesgue measure onJ, then 1=

J1dν 2

= J

τ(x)· 1

τ(x)dν(x) 2

≤ J

τ(x)dν(x)·

J

τ1(x)dν(x),

(3.2)

1

Jτ(x)dν(x)^≤ J

τ1(x)dν(x), (3.3)

or

1

1/λ(J)_Jτ(x)dx ^≤ 1 λ(J) J

τ1(x)dx. (3.4)

Integrating (3.1) overJ, we obtain p1

J

τ11(x)dx+p2 J

τ21(x)dx≤γ·λ(J), (3.5)

and, using (3.4), we obtain

p1λ(J)

λτ1(J)+ p2λ(J)

λτ2(J)^≤γ. (3.6)

Thus, at least one of the numbersλ(τ1(J)),λ(τ2(J)) is larger thanλ(J).

Corollary 3.2. Under assumptions ofLemma 3.1, ifJis an interval disjoint with endpoints of the partitionᏼ⁽ⁿ⁾=ᏼ∨

τ⁻_jn−¹₁τ⁻_jn¹₋₂···τ⁻_j1¹ᏼforn >1, then

(jn,jn−1,...,j1)

p_jnp_jn−1···p_j1 λ(J)

λτ_jnτ_jn−1···τ_j1(J)^≤γⁿ, (3.7) wherejn,jn−1,. . .,j1∈ {1, 2}. In particular, we have

Jmax∈ᏼ⁽ⁿ⁾diam(J)≤γⁿ, n≥1. (3.8) Proof. It follows from the fact that Pelikan’s condition (2.5) implies

sup

x

(jn,jn−1,...,j1)

p_jnp_jn−1···p_j1

τjn◦τjn−1◦ ··· ◦τj1

(x)^≤γⁿ, (3.9)

n≥1, j_n,j_n−1,. . .,j1∈ {1, 2}.

Remark 3.3. Instead of Pelikan’s condition (2.5) we can use throughout the paper a weaker condition

p1lnτ₁(x)+p2lnτ₂(x)≥β >0, x∈[0, 1], (3.10) which one could call Morita’s condition [10]. Condition (3.10) implies the existence of absolutely continuous invariant measure but its density may not be of bounded variation, see [3, Example 1.2]. If we assume additionally that this density is of bounded variation or at least is a lower semicontinuous function, then we can reprove the results of this paper under assumption of condition (3.10).

Condition (3.10) implies

(jn,jn−1,...,j1)

p_jnp_jn−1···p_j1lnτ_jn◦τ_jn−1◦ ··· ◦τ_j1

(x)≥nβ >0, x∈[0, 1], (3.11) n≥1,j_n,j_n−1,. . .,j1∈ {1, 2}. Using condition (3.10) we can prove an analogue ofLemma 3.1.

For any intervalJcontained in an element of partitionᏼ, we have p1lnλτ1(J)

λ(J) +p2lnλτ2(J)

λ(J) ^≥β >0, (3.12)

which implies that at least one of the images is longer than intervalJ.

Analogously, for any intervalJcontained in an element of partitionᏼ⁽ⁿ⁾, we have

(jn,jn−1,...,j1)

pjnpjn−1···pj1lnλτjn◦τjn−1◦ ··· ◦τj1(J)

λ(J) ^≥nβ >0, (3.13)

n≥1, j_n,j_n−1,. . .,j1∈ {1, 2}. This, in particular, implies that

Jmax∈ᏼ⁽ⁿ⁾diam(J)≤e^−nβ, n≥1. (3.14)

Lemma 3.4. LetT= {τ1,τ2;p1,p2}be a random map on [0, 1] satisfying the conditions of Theorem 2.6. Then, the support of the invariant density ofT contains an interval which is not disjoint withᏽ.

Proof. Let supp(f)= {x∈[0, 1] :f(x)>0}. The density function f of the acimμis a function of bounded variation byTheorem 2.6and thus, byLemma 2.4, f can be rede- fined on a countable set to become a lower semicontinuous function f and f = f al- most everywhere. Thus, supp(f)=supp(f)= {x: f >0}is an open set byTheorem 2.3.

Thus, supp(f)= ∪^∞_i=1Ii, whereIi’s are open disjoint intervals. Without loss of generality, let us assume thatλ(Ii)≥λ(Ii+1) fori=1, 2,. . . .We will prove thatᏽ∩I1= ∅. Suppose ᏽ∩I1= ∅. ThenI1is contained in one of the subintervals,J_∗, of the partitionᏼand τ1(I1) andτ2(I1) are both open intervals. Since f is an invariant density of the random mapT, we have

f(x)=p1· q i=1

fτ_1,i⁻¹(x) τ1

τ_1,i⁻¹(x)χτ1(Ii)(x) +p2· q i=1

fτ_2,i⁻¹(x) τ2

τ_2,i⁻¹(x)χτ2(Ii)(x). (3.15)

Let x∈τ1(I1). It is clear that at least one element of {τ_1,⁻_∗¹(x)}is in I1 and sinceI1⊂ supp(f) we have f(x)>0. Thus,τ1(I1) is a subset of supp(f). Similarly,τ2(I1) is a subset of supp(f). ByLemma 3.1, at least one of the intervalsτ1(I1),τ2(I1) has larger length than the length ofI1. This is a contradiction because supp(f) does not contain an interval of length greater thanλ(I1). This proves thatᏽ∩I1= ∅. Corollary 3.5. The number of different ergodic acim for the random mapTsatisfying the assumptions ofTheorem 2.6is at most equal to the cardinality of the partitionᏼminus one.

ForTheorem 3.6we assume thatτ1,τ2∈᐀1. After the theorem we give an example showing that it fails if we assume onlyτ1,τ2∈᐀0.

Theorem 3.6. LetT= {τ1,τ2;p1,p2}be a random map on [0, 1], whereτ1,τ2∈᐀1and have a common partitionᏼ= {J1,J2,. . .,Jq}. Then the support of the invariant density f of T, supp(f) is a finite union of open intervals almost everywhere.

Proof. Again we can assume that supp(f)= ∪^∞i=1Ii, whereIi’s are open disjoint intervals.

Let Ᏸ= {j≥1 :I_jcontains a discontinuity ofτ1,τ2, or both}. ByLemma 3.4,Ᏸ is not empty. Also,Ᏸis a finite set. If j∈Ᏸ, thenτi(Ij),i=1, 2 is a finite union of intervals. Let Jbe the shortest interval of the family

Ij

j∈Ᏸ∪

I:Iis a connected component ofτi

Ij

,i=1, 2, j∈Ᏸ. (3.16)

LetᏲ= {i≥1 :λ(I_i)≥λ(J)}, whereiis not necessarily inᏰand let

S= ∪i∈ᏲIi⊂ ∪iIi=supp(f). (3.17) Sis a finite union of open disjoint intervals since it is a family of disjoint intervals with length≥λ(J)>0. For anyj∈Ᏸ,Ij⊆S.

We will prove thatτi(S)⊆S,i=1, 2.LetIk⊂S.Ifk /∈Ᏸ, thenτi(Ik) is contained in an intervalIki,i=1, 2 and

λIki

≥λτi

Ik

=

Ik

τ_i(x)dλ > inf

x∈[0,1]

τ_i(x)·λIk

, i=1, 2. (3.18)

Since inf_x∈[0,1]|τ_i(x)|>1 fori=1, 2, we have

λI_ki≥λI_k> λ(J). (3.19)

Thus, by the definition ofS, we getIki⊆S,i=1, 2 and henceτi(Ik)⊂Iki⊆S,i=1, 2. If k∈Ᏸ, then by the definition ofS,τ_i(I_k)⊂S,i=1, 2. Thus,τ_i(S)⊆S,i=1, 2.

Now we will prove that supp(f)⊆S. Suppose not. Let Is be the largest interval of supp(f)\S. Thus,s /∈Ᏸand

λτi Is

= Is

τ_i(x)dλ > inf

x∈[0,1]

τ_i(x)·λIs

> λIs

, i=1, 2. (3.20) Then τi(Is)⊂S,i=1, 2. Thus, Is⊂τ_i⁻¹(S), i=1, 2. ButIs⊂S, so Is⊂τ_i⁻¹(S)\S, i= 1, 2. Letμ= f ·λbe theT-invariant absolutely continuous measure. We will show that μ(τ_i⁻¹(S)\S)=0,i=1, 2. Sinceτ_i(S)⊆S,i=1, 2, we have

S⊆τ_i⁻¹τ_i(S)⊆τ_i⁻¹(S), i=1, 2. (3.21) Thus,

0=μ(S)−μ(S)=

p1μτ₁⁻¹S+p2μτ₂⁻¹S−

p1μ(S) +p2μ(S)

=p1

μτ₁⁻¹S−μ(S)+p2

μτ₂⁻¹S−μ(S)

=p1μτ₁⁻¹S\S+p2μτ₂⁻¹S\S.

(3.22)

Thus, bothμ(τ₁⁻¹S\S)=0 andμ(τ₂⁻¹S\S)=0 if p1,p2>0. Thus,μ(I_s)=0, which con-

tradicts the fact thatIs⊂supp(f).

Example 3.7. We present a random map satisfying the assumptions of Theorem 2.6, where the support of absolutely continuousT-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

τ1(x)=

8x−E(8x)

4 ,

τ2(x)=

⎧⎪

⎪⎨

⎪⎪

⎩ 3 4+x

2 for 0≤x≤1 2, x

2 for1

2< x≤1,

(3.23)

whereE(t) denotes the integral part oft, as shown inFigure 3.1.

0 1/4 1/2 1 1/4

1/2 1

(a)

0 1/2 1

1/4 1/2 3/4 1

(b) Figure 3.1. The graphs ofτ₁andτ₂.

Let us consider the random mapT= {τ1,τ2; 3/4, 1/4}. It satisfies all the assumptions of Pelikan’sTheorem 2.6. In particular,Tadmits an absolutely continuous invariant mea- sureμwith density f of bounded variation. LetS denote the support of f. Lebesgue measure restricted to the interval L0=[0, 1/4] is invariant for τ1. Since τ1([0, 1])=L0

andτ1is exact onL0, we haveL0⊂S. Since the intervalK0=(1/4, 1/2) is not in the image ofτ1orτ2, it is disjoint fromS. Every image ofL0is inS. Sinceτ1([0, 1])=L0, we consider only images byτ2. We have

τ2

L0

=[1−1/4, 1−1/8]=L⁽¹⁾₁ , τ2

L⁽¹⁾₁ =[1/2−1/8, 1/2−1/16]=L⁽²⁾₁ ,

τ2

L⁽²⁾₁ =[1−1/16, 1−1/32]=L⁽¹⁾₂ ,

τ2

L⁽¹⁾2

=[1/2−1/32, 1/2−1/64]=L⁽²⁾2 ,

(3.24)

and in general, forL⁽¹⁾n =[1−1/4ⁿ, 1−1/(2·4ⁿ)] andL⁽²⁾n =[1/2−1/(2·4ⁿ), 1/2−1/(4· 4ⁿ)],

τ2

L⁽¹⁾_n =L⁽²⁾_n , τ2

L⁽²⁾_n =L⁽¹⁾_n+1. (3.25) LetKn⁽¹⁾ be the open interval betweenL⁽¹⁾n and L⁽¹⁾_n+1, and let Kn⁽²⁾ be the open interval betweenL⁽²⁾n andL⁽²⁾_n+1,n≥1. Then, obviously, we have for alln≥1,

τ2

K_n⁽¹⁾=K_n⁽²⁾, τ2

K_n⁽²⁾=K_n+1⁽¹⁾. (3.26)

Sinceτ2is an injective map, we also have τ2⁻¹

K_n⁽²⁾=K_n⁽¹⁾,

τ₂⁻¹K_n+1⁽¹⁾=K_n⁽²⁾, n≥1. (3.27) We have

τ₂⁻²K₁⁽¹⁾=K0 (3.28)

andτ₁⁻¹(Kn⁽ⁱ⁾)= ∅fori=1, 2,n≥1. Thus,

K_n⁽ⁱ⁾∩S= ∅, i=1, 2;n≥1. (3.29)

We have proved that

S=L0∪

i=1,2

n≥1

L⁽ⁱ⁾_n, (3.30)

that is, the support of the absolutely continuousT-invariant measure is an infinite count- able union of disjoint intervals. Since the invariant density f is of bounded variation, we have

sup

x∈L⁽ⁱ⁾n

f(x)−→0, n−→+∞, (3.31)

fori=1, 2, so the density f is not bounded away from 0.

InLemma 3.8andTheorem 3.9, we return to the assumptionτ1,τ2∈᐀0, but we as- sume additionally that the support of the invariant density is a finite union of disjoint intervals. That was proved inTheorem 3.6using the assumptionτ1,τ2∈᐀1.

Lemma 3.8. LetT= {τ1,τ2;p1,p2}be a random map on [0, 1], whereτ1,τ2∈᐀0and have a common partitionᏼ= {J1,J2,. . .,J_q}. Let f be the invariant density of an acimμof the random map T andS=supp(f)= {x:f(x)>0}.We assume thatSis a finite union of disjoint intervals. Then

(i)τ_i(S\ {a0,a1,. . .,a_q})⊆S,i=1, 2;

(ii)λ(S\τi(S\ {a0,a1,. . .,aq}))=0,i=1, 2;

where{a0,a1,. . .,aq}are the endpoints of the intervals in the partitionᏼ.

Proof. We assume that f is lower semicontinuous. If it is not, we modify it on at most a countable set. By assumption,S= ∪^r_i=1I_i. Letx∈S\ {a0,a1,. . .,a_q}.Thenx∈IntI_k, for somek∈ {1, 2,. . .,r}and there exists>0 such thatB(x,)⊂Ikand f(y)>(1/2)f(x)>

0 for ally∈B(x,) since f is lower semicontinuous. We may assume thatτi|Ik,i=1, 2 is

increasing and that f(τi(x))=limy→τi(x)⁺f(y),i=1, 2. Now, for anyδ >0,τi([x,x+δ))= [τi(x),τi(x) +δ),i=1, 2 andδ→0 asδ→0. Then, fori=1, 2, we have

[τi(x),τi(x)+δ)f dλ=μτi

[x,x+δ)≥μ[x,x+δ)

= [x,x+δ)f dλ≥1

2f(x)λ[x,x+δ)

≥1

2f(x) 1

maxτ_iλτi(x),τi(x) +δ.

(3.32)

Since f is lower semicontinuous, fτi(x)=lim

δ→0

1

λτi(x),τi(x) +δ [τi(x),τi(x)+δ)f dλ

≥1

2f(x) 1

maxτ_i>0, i=1, 2.

(3.33)

Hence,τi(x)∈S,i=1, 2, and part (i) is proved.

Part (ii) is proved using reasoning similar to the end ofTheorem 3.6(3.22), which

does not depend on the expanding property ofτ1,τ2.

Theorem 3.9. LetT= {τ1,τ2;p1,p2}be a random map on [0, 1], whereτ1,τ2∈᐀1and have a common partitionᏼ= {J1,J2,. . .,Jq}. Let f be the invariant density of an acimμof the random map T and letS=suppf = {x:f(x)>0}.We assume thatSis a finite union of disjoint intervals. Then there exists a constanta >0 such that f_|S≥a.

Proof. SinceS= {x: f(x)>0}is a finite union of open intervals,S= ∪^r_i=1Ii, we can as- sume they are separated by intervals of positive measure. Then S=S\ {a0,a1,. . .,aq} is also a finite union of intervals: S= ∪^s_i=1J_i. Let Ᏺ= {I_i}^r_i=1, and letᏯ= {J_j}^s_j=1. For anyJk∈Ꮿ,τj|Jk, j=1, 2 is of classC¹. Therefore, there existIij∈Ᏺ, j=1, 2 such that τj(Jk)⊆Iij,j=1, 2.

Let (c,d) be any interval inᏲ orᏯ.We associate with its endpoints two classes of standard intervals:

η_c=

(c,c+) :>0, η_d=

(d−,d) :>0. (3.34) The pointscanddare referred to as the endpoints of the classesηcandηd, respectively.

Let

᏷=

ηc,ηd:c,dare endpoints of intervals ofᏲ,Ꮿ. (3.35) We now define a relation→between elements of᏷.Forη,η∈᏷:η→ηif and only ifτj(U)∈ηfor at least one ofj=1, 2 and for sufficiently smallU∈η. The relation has the following two properties:

(1) ifηis associated with an endpoint ofIi∈Ᏺ, then there exists at least oneηsuch thatη→η. To prove this, let us fixIi andηassociated with one of its endpoints. We

claim that for eachJk∈Ꮿ, eitherτj(Jk)⊆Ii, j=1, 2 orτj(Jk)∩Ii= ∅,j=1, 2. To show this, let us note that sinceτj(Jk), j=1, 2 is contained inS= ∪^ri=1Iiand{Ii}are separated, τ_j(J_k), j=1, 2 is contained in one ofI_ij,j=1, 2. Now, since

λIij\τj(S)≤λS\τj(S)=0, j=1, 2, (3.36) there must existJkwithτj(Jk)∈η,j=1, 2 and this implies (1);

(2) ifcis an endpoint ofI_isuch that lim_x→cf(x)=0,ηis associated withc,η→η, andcis an endpoint ofη, then for anyU∈η,

limx→c

x∈U

f(x)=0. (3.37)

To prove this let us suppose that limx∈U,x→cf(x)=a >0, for someU∈η. By the def- inition of the relation→, for at least one of j=1, 2, say j=1, ifη= {(c,c+ε)}, then we haveτ1(c,c+ε)=(c,c+ε), forεsmall enough. Then,

xlim→c

x∈U

f(x)=lim

ε→0

1

λ(c,c+ε) (c,c+ε)f(t)dt=lim

ε→0

μ(c,c+ε) λ(c,c+ε)

≥lim

ε→0p1· μ(c,c+ε) maxτ₁·λ(c,c+ε)⁼

p1

maxτ₁a >0,

(3.38)

which is a contradiction.

We make the following observations:

(3) in the setting of (2) above,c /∈S. Therefore,cis an endpoint of an intervalIi∈Ᏺ.

Now we define

᏷0=

η:ηis associated with an endpointcofIi∈Ᏺ, lim_x→c

x∈Ii

f(x)=0

. (3.39)

From (2) and (3) we obtain what follows:

(4) ifη∈᏷0andη→η, thenη∈᏷0.

We note that by (1) and (4), for eachη∈᏷0there exists at least oneη∈᏷0such that η→η.

Now letη∈᏷0. For any n≥1, anyη such that η→η inn-steps also belongs to

᏷0. ChooseU∈ηto be sufficiently small, that is, completely contained in an element of partitionᏼ. Then, all the preimagesτ⁻_jn¹◦τ⁻_jn¹₋₁◦ ··· ◦τ⁻_j2¹◦τ⁻_j1¹(U), ji∈ {1, 2}touch an endpoint of someηin᏷0. LetMbe the number of elements in᏷0and letγ <1 be the constant from Pelikan’s condition (2.5). Then,

μ(U)=

(jn,jn−1,...,j1)

p_jnp_jn−1···p_j1μτ⁻_jn¹◦τ⁻_jn−¹₁◦ ··· ◦τ⁻_j1¹(U)

≤supf·

(jn,jn−1,...,j1)

p_jnp_jn−1···p_j1λτ⁻_jn¹◦τ⁻_jn¹₋₁◦ ··· ◦τ⁻_j1¹(U).

(3.40)

Eachτ⁻_jn¹◦τ⁻_jn¹₋₁◦ ··· ◦τ⁻_j1¹(U) consists of intervals contained in elements of partitionᏼ⁽ⁿ⁾ and touches one ofMendpoints of classes in᏷0. Thus, usingCorollary 3.2, we have

λτ⁻_jn¹◦τ⁻_jn−¹₁◦ ··· ◦τ⁻_j2¹◦τ⁻_j1¹(U)≤M·γⁿ, μ(U)≤supf·

(jn,jn−1,...,j1)

p_jnp_jn−1···p_j1Mγⁿ=supf·Mγⁿ. (3.41)

Thus,μ(U)=0 which implies thatλ(U)=0 sinceU⊆S. This contradicts the fact that U∈ηis an open, nonempty interval. Hence,᏷0= ∅and limx∈U,x→cf(x)>0 for each of finitely many endpoints of intervalsIi∈Ᏺ.On the other hand, since f is lower semi- continuous, it assumes its infimum on any closed interval. Hence, there existsa >0 such

that f(x)≥afor allx∈S.

Acknowledgments

The authors are grateful to Martijn de Vries who suggested the problem of strict positivity of invariant density of random map. P. G ´ora is grateful for valuable comments to the participants of International Conference in Dynamical Systems, in memory of Wiesław Szlenk, June 6–11, 2005, Banach Center, Warsaw, Poland. The research was supported by NSERC Grant. M. S. Islam was supported by the Department of Mathematics and Statistics, Concordia University.

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Paweł G ´ora: Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec, Canada H4B 1R6

E-mail address:[email protected]

Abraham Boyarsky: Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec, Canada H4B 1R6

E-mail address:[email protected]

Md Shafiqul Islam: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, Canada T1K 3M4

E-mail address:[email protected]