SRB Measures For Certain Markov Processes

Abstract

We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $\delta_0$ and $\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.

Full text

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SRB MEASURES FOR CERTAIN MARKOV PROCESSES
WAEL BAHSOUN AND PAWE? L G´ORA
Abstract. We study Markov processes generated by iterated function sys-
tems (IFS). The constituent maps of the IFS are monotonic transformations
of the interval. We first obtain an upper bound on the number of SRB (Sinai-
Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have
common fixed points at 0 and 1, theorems are given to analyze properties of
the ergodic invariant measures δ0 and δ1. In particular, sufficient conditions
for δ0and/or δ1to be, or not to be, SRB measures are given. We apply some
of our results to asset market games.
1. Introduction
In the 1970’s, Sinai, Ruelle and Bowen studied the existence of an important class
of invariant measures in the context of deterministic dynamical systems. These in-
variant measures are nowadays known as SRB (Sinai-Ruelle-Bowen) measures [14].
SRB measures are distinguished among other ergodic invariant measures because
of their physical importance. In fact, from ergodic theory point of view, they are
the only useful ergodic measures. This is due to the fact that SRB measures are
the only ergodic measures for which the Birkhoff Ergodic Theorem holds on a set
of positive measure of the phase space. In this note, we study SRB measures in a
stochastic setting—Markov processes generated by iterated function systems (IFS).
An IFS1is a discrete-time random dynamical system [1, 10] which consists of
a finite collection of transformations and a probability vector {τs;ps}L
time step, a transformation τsis selected with probability ps> 0 and applied to the
process. IFS has been a very active topic of research due to its wide applications in
fractals and in learning models. The survey articles [5, 13] contain a considerable
list of references and results in this area.
s=1. At each
The systems which we study in this note do not fall in the category of the IFS2
considered in [5, 13] and references therein. Moreover, in general, our IFS do not
satisfy the classical splitting3condition of [7]. In fact, our aim in this note is to
depart from the traditional goal of finding sufficient conditions for an IFS to admit
a unique attracting invariant measure [7, 5, 13]. Instead, we study cases where an
IFS may admit more than one invariant measure and aim to identify the physically
relevant ones; i.e., invariant measures for which the Ergodic Theorem holds on a
1991 Mathematics Subject Classification. Primary 37A05, 37E05, 37H99.
Key words and phrases. Iterated Function System, SRB-Measures.
1In some of the literature an IFS is called a random map or a random transformation.
2In most articles about IFS, the constituent maps are assumed to be contracting or at least
contracting on average. Here we do not impose any assumption of this type. In fact the class of
IFS which we study in Section 4 cannot satisfy such assumptions.
3In particular, when all the maps have common fixed points at 0 and 1. See Section 4.
1
arXiv:0907.3372v2 [math.DS] 30 Dec 2009

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2 WAEL BAHSOUN AND PAWE? L G´ORA
set of positive measure of the ambient phase space. We call such invariant measures
SRB.
Physical SRB measures for random maps have been studied by Buzzi [3] in the
context of random Lasota-Yorke maps. However, Buzzi’s definition of a basin of an
SRB measure is different from ours. We will clarify this difference in Section 2. A
general concept of an SRB measure for general random dynamical systems can be
found in the survey article [11]. In this note we study physical SRB measures for IFS
whose constituent maps are strictly increasing transformations of the interval. We
obtain an upper bound on the number of SRB measures for the IFS. Moreover, when
all the constituent maps have common fixed points at 0 and 1, we provide sufficient
conditions for δ0and/or δ1to be, or not to be, SRB measures. To complement our
theoretical results, we show at the end of this note that examples of IFS of this
type can describe evolutionary models of financial markets [4].
In Section 2 we introduce our notation and main definitions. In particular, Sec-
tion 2 includes the definition of an SRB measure for an IFS. In Section 3 we identify
the structure of the basins of SRB measures and we obtain a sharp upper bound
on the number of SRB measures. Section 4 contains sufficient conditions for δ0and
δ1, the delta measures concentrated at 0 and 1 respectively, to be SRB. It also con-
tains sufficient conditions for δ0and δ1not to be SRB measures. Our main results
in this section are Theorems 4.3 and Theorem 4.7. In Section 5 we study ergodic
properties of δ0and δ1without having any information about the probability vector
of the IFS. In Section 6 we apply our results to asset market games. In particular,
we find a generalization of the famous Kelly rule [9] which expresses the principle
of “betting your beliefs”. The importance of our generalization lies in the fact that
it does not require the full knowledge of the probability distribution of the random
states of the system. Section 7 contains an auxiliary result which we use in the
proof of Theorem 4.7.
2. Preliminaries
2.1. Notation and assumptions. Let ([0,1],B) be the measure space where B
is the Borel σ-algebra on [0,1]. Let λ denote Lebesgue measure on ([0,1],B) and
δr denote the delta measure concentrated at point r ∈ [0,1]. Let S = {1,...,L}
be a finite set and τs, s ∈ S, be continuous transformations from the unit interval
into itself. We assume:
(A) τsare strictly increasing.
Let p = (ps)L
The collection
s=1be a probability vector on S such that for all s ∈ S, ps> 0.
F = {τ1,τ2,...,τL;p1,p2,...,pL}
is called an iterated function system (IFS) with probabilities.
We denote the space of sequences ω = {s1,s2,...}, sl∈ S, by Ω. The topology
on Ω is defined as the product of the discrete topologies on S. Let πpdenote the
Borel measure on Ω defined as the product measure pN. Moreover, we write
st:= (s1,s2,...,st)

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SRB MEASURES FOR CERTAIN MARKOV PROCESSES3
for the history up to time t, and for any r0∈ [0,1] we write
rt(st) := τst◦ τst−1◦ ··· ◦ τs1(r0).
Finally, by E(·) we denote the expectation with respect to p, by E(·|st) the con-
ditional expectation given the history up to time t and by var(·) the variance with
respect to p.
2.2. Invariant measures. F is understood as a Markov process with a transition
function
L
?
where A ∈ B and χA is the characteristic function of the set A. The transition
function P induces an operator P on measures on ([0,1],B) defined by
?1
L
?
Following the standard notion of an invariant measure for a Markov process, we
call a probability measure µ on ([0,1],B) F-invariant probability measure if and
only if
Pµ = µ.
Moreover, it is called ergodic if it cannot be written as a convex combination of
other invariant probability measures.
P(r,A) =
s=1
psχA(τs(r)),
Pµ(A) =
0
P(r,A)dµ(r)
=
s=1
psµ(τ−1
s A).
(2.1)
2.3. SRB measures. Let µ be an ergodic probability measure for the IFS. Suppose
there exists a set of positive Lebesgue measure in [0,1] such that
(2.2)
1
T
T−1
?
t=0
δrt(st)
weakly
→
µ
with πp-probability one.
Then µ is called an SRB (Sinai-Ruelle-Bowen) measure. The set of points r0∈ [0,1]
for which (2.2) is satisfied will be called the basin4of µ and it will be denoted by
B(µ). Obviously, if λ(B(µ)) = 1 then µ is the unique SRB measure of F.
3. Number of SRB measures and their basins
The basin of an SRB measure for the systems we are dealing with is described
by the following two propositions.
Proposition 3.1. Let µ be an SRB measure and B(µ) be its basin. Let r0, ¯ r0∈
B(µ), r0> ¯ r0. Then [¯ r0,r0] ⊆ B(µ).
4Our definition of a basin is different from Buzzi’s definition [3]. In his definition he defines
random basins Bω(µ) for an SRB measure. In particular, according to Buzzi’s definition, for
the same SRB measure, basins corresponding to two different ω’s may differ on a set of positive
lebsegue measure of [0,1]. See [3] for more detials.

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4 WAEL BAHSOUN AND PAWE? L G´ORA
Proof. When weak convergence is considered on an interval, then µn
only if µn(f) → µ(f) for any C1function5. Since every C1function is a difference
of two continuous increasing functions, this means that µn
µn(f) → µ(f) for any continuous increasing function.
weakly
→
µ if and
weakly
→
µ if and only if
Let r0, ¯ r0∈ B(µ) and ¯ r0< r?
that f is continuous and increasing. Let us fix an stfor which
0< r0. We will show that r?
0∈ B(µ). Let assume
lim
T→∞
1
T
T−1
?
t=0
f(¯ rt(st)) = lim
T→∞
1
T
T−1
?
t=0
f(rt(st)) = µ(f).
We have ¯ rt(st) < r?
t(st) < rt(st) (since all τsare increasing) and
1
T
T−1
?
t=0
f(¯ rt(st)) ≤1
T
T−1
?
t=0
f(r?
t(st)) ≤1
T
T−1
?
t=0
f(rt(st)).
The averages on the left and on the right have common limit µ(f). Thus,
1
T
T−1
?
t=0
δr?
t(st)(f) =1
T
T−1
?
t=0
f(r?
t(st)) → µ(f).
Since the event
{ lim
T→∞
1
T
T−1
?
t=0
f(¯ rt(st)) = lim
T→∞
1
T
T−1
?
t=0
f(rt(st)) = µ(f)}
occurs with πp-probability 1, the event
{1
T
T−1
?
t=0
f(r?
t(st)) → µ(f)}
also occurs with πp-probability 1.
?
Proposition 3.2. Let µ be an SRB measure and B(µ) = ?a,b? be its basin, where
?a,b? denotes an interval closed or open at any of the endpoints. Then,
τs(a) ≥ a , s = 1,...,L, and if a ?= 0 then τs(a) = a for at least one s;
τs(b) ≤ b , s = 1,...,L, and if b ?= 1 then
Proof. We will prove only the second claim with b ?= 1. The first claim is proven
analogously.
τs(b) = b for at least one s.
5Here is a sketch of the proof of this claim: Assume
µn(f) → µ(f)
for any f ∈ C1([0,1]). Let g be a continuous function and let {fk}k≥1be a sequence of C1
functions converging to g in C0norm. We have
|µn(g) − µ(g)| ≤ |µn(g) − µn(fk)| + |µn(fk) − µ(fk)| + |µ(fk) − µ(g)|
≤ 2?fk− g?C0 + |µn(fk) − µ(fk)| .
Now, for any ε > 0, we can find k0 such that 2?fk0− g?C0 < ε/2 and then we can find n0 such
that for any n ≥ n0we have |µn(fk0) − µ(fk0)| < ε/2.

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SRB MEASURES FOR CERTAIN MARKOV PROCESSES5
Assume that τs0(b) > b, for some 1 ≤ s0 ≤ L. Then, we can find r0 ∈ (a,b)
such that τs0(r0) > b. For arbitrary continuous function f, for ω ∈ A ⊂ Ω with
πp(A) = 1, we have
1
T
t=0
The set As0= {(s1,s2,...) : (s0,s1,s2,...) ∈ A} is also of πp-probability 1. Let
r?
Then,
lim
T→∞
T−1
?
f(rt(st)) = µ(f).
0= τs0(r0) and let (st)?denote the initial subsequences of length t of ω ∈ As0.
1
T
T−1
?
t=0
f(r?
t((st)?)) =1
T
T−1
?
t=0
f(rt(st)) −1
Tf(r0) +1
Tf(r?
T−1((sT−1)?)) −→
T→+∞µ(f).
This shows that τs0(r0) ∈ Bω(µ) and contradicts the assumptions.
Now, we assume that τs(b) < b, s = 1,...,L. Then, we can find r0> b such that
τs(r0) ∈ (a,b) for all s. Let
1
T
t=0
We have πp(As) = 1 for each s. Hence, πp(A) = 1, where A = ∪1≤s≤L(s,As)
and (s,As) = {(s,s1,s2,s3,...) : (s1,s2,s3,...) ∈ As}. For arbitrary continuous
function f, for ω ∈ A, if ω1= s we have
1
T
t=0
?
T
t=0
where r?
implies that r0∈ B(µ). Since r0> b, this leads to a contradiction.
We now state the main result of this section. Firstly, we recall that ?·,·? denotes
an interval which is closed or open at any of the endpoints. Secondly, we define a
set BS whose elements are intervals of the form ?·,·? with the following property:
?a,b? ∈ BS
if and only if
τs(a) ≥ a , s = 1,...,L and τs(a) = a for at least one s;
and
τs(b) ≤ b , s = 1,...,L and τs(b) = b for at least one s.
Theorem 3.3. The number of SRB measures of F is bounded above by the cardi-
nality of the set BS. In particular, if 0 and 1 are the only fixed points of some τs0,
s0∈ S, then F admits at most one SRB measure.
Proof. The fact that number of SRB measures of F is bounded above by the cardi-
nality of the set BS is a direct consequence of Proposition 3.2. To elaborate on the
second part of the theorem, assume without loss of generality that τs0(r) > r for all
r ∈ (0,1). Obviously, by Proposition 3.2, if all the other maps τs, s ∈ S \ {s0} has
no fixed points in (0,1), then F admits at most one SRB measure. So let us assume
As= {ω : lim
T→∞
T−1
?
f(r?
t(st)) = µ(f), for r?
0= τs(r0)} , s = 1,...,L .
lim
T→∞
T−1
?
f(rt(st))
= lim
T→∞
0= τs(r0) and (st)?are the initial subsequences of length t of ω ∈ As. This
1
T−1
?
f(r?
t((st)?)) +1
Tf(r0) −1
Tf(r?
T−1((sT−1)?))
?
= µ(f),
?

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6 WAEL BAHSOUN AND PAWE? L G´ORA
that there exists an s∗∈ S \ {s0} such that τs∗ has a finite or infinite number of
fixed points in [0,1]. In the case of finite number of fixed points, denote the fixed
points of τs∗ in [0,1] by r∗
Since τs(r∗
would be either ?r∗
let
¯ r = sup{r ∈ (0,1) : τs∗(r) = r}.
If ¯ r < 1, then τs0(¯ r) > ¯ r. By Proposition 3.2, ?¯ r,1? is the only possible basin for an
SRB measure. If ¯ r = 1, let¯J denote the closure of the set of fixed points of τs∗ and
let¯J0⊆¯J be the minimal closed subset of¯J which contains the point 1.¯J0is the
only possible basin for an SRB measure. Moreover, it cannot be decomposed into
basins of different SRB measures. Indeed, let J1∪ J2=¯J0such that J1= ?a,b?
with b < 1. Since τs0(b) > b, by Proposition 3.2, J1cannot be a basin of an SRB
measure. Thus, F admits at most one SRB measure.
i, i = 1,...,q, such that 0 ≤ r∗
ifor all r∗
q−1,1? or ?r∗
1< r∗
2< ··· < r∗
q≤ 1.
i) > r∗
i∈ (0,1), the only possible basin for an SRB measure
q,1?. In the case of infinite number of fixed points,
?
The following example shows that Proposition 3.2 can be used to identify inter-
vals which are not in the basin of an SRB measure. In particular, it shows that the
bound obtained on the number of SRB measures in Theorem 3.3 is really sharp.
Example 3.4. Let
τ1(r) =
?
3r2
1 −3
?
, for 0 ≤ r ≤ 1/3;
, for 1/3 < r ≤ 1;
2(r − 1)2
,
and
τ2(r) =
3
2r2
1 − 3(r − 1)2
, for 0 ≤ r ≤ 2/3;
, for 2/3 < r ≤ 1.
The graphs of the above maps are shown in Figure 1.
we see that the points of the interval (1/3,2/3) do not belong to a basin of any
SRB measure. Moreover, by Theorem 3.3, F admits at most two SRB measures.
Indeed, one can easily check that δ0and δ1are the only SRB measures with basins
B(δ0) = [0,1/3] and B(δ1) = [2/3,1] respectively. For any r ∈ [0,1/3) for all ω’s
the averages
T
which this does not happen is ω = {1,1,1,...} so again the averages converge
weakly to δ0with πp-probability 1. Similarly, we can show that B(δ1) = [2/3,1].
If r ∈ (1/3,2/3), then with positive πp-probability the averages converge to δ0and
with positive πp-probability the averages converge to δ1. Thus, these points do not
belong to a basin of any SRB measure and there are only two SRB measures.
Using Proposition 3.2,
1
?T−1
t=0δrt(st)converge weakly to δ0. For r = 1/3 the only ω for
4. Properties of δ0and δ1
In addition to condition (A), we assume in this section that for all s ∈ S:
(B) τs(0) = 0 and τs(1) = 1;
Obviously by Condition (B) the delta measures δ0and δ1are ergodic probability
measures for the IFS. We will be mainly concerned with the following question:
When does F have δ0 and/or δ1 as SRB measures? We start our analysis by

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SRB MEASURES FOR CERTAIN MARKOV PROCESSES7
Figure 1. Maps τ1and τ2in Example 3.4
proving a lemma which provides a sufficient condition for δx, the point measure
concentrated at x ∈ [0,1], to be an SRB measure.
Lemma 4.1. Suppose that τs(x) = x for all s ∈ {1,...,L} and that there exists
an initial point of a random orbit r0, r0 ?= x, for which limt→∞rt(st) = x with
probability πp= 1. Then δxis an SRB measure for F and Bω(δx) ⊇ [x,r0]6.
Proof. Let f be a continuous function on [0,1]. Let r0?= x and fix a history stfor
which limt→∞rt(st) = x. Then
lim
t→∞f(rt(st)) = f(x).
Consequently
lim
T→∞
1
T
T−1
?
t=0
f(rt(st)) = f(x).
Since the event
{ lim
t→∞rt(st) = x}
appears with probability one, the event
{ lim
T→∞
1
T
T−1
?
t=0
f(rt(st)) = f(x)}
6The notation here is for the case when r0> x.

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8 WAEL BAHSOUN AND PAWE? L G´ORA
also appears with probability one. Thus, by Proposition 3.1, δxis an SRB measure
for F and Bω(δx) ⊇ [x,r0].
The following lemma, which is easy to prove, is a key observation for our main
results in this section.
Lemma 4.2. Each constituent map of the IFS can be represented as follows:
τs(r) = rβs(r),
with βs(r) satisfying:
(1) βs(r) > 0 in (0,1) ;
(2) (lnr)βs(r) increasing;
(3) limr→0(lnr)βs(r) = −∞;
(4) limr→1(lnr)βs(r) = 0.
In the rest of this section, the following notation will be used:
?
αt
def
:= βs(rt−1) with probability ps, t = 1,2,...
Theorem 4.3. Let F = {τs;ps}s∈S be an IFS such that τs(r) = rβs(r). Assume
that 0 < bs≤ βs(r) ≤ Bs< ∞ for all r ∈ [0,1].
(1) If E(lnαt|st−1) ≤ 0 a.s., then limt→∞rt(st) ?= 0 a.s.
(2) If limsupT→∞
(3) If liminfT→∞
T
Proof. Let us consider the sequence of random exponents
α(t) = αtαt−1···α2α1,
where αi= βs(ri−1) with probability ps, and observe that
rt(st) = rα(t).
We have
lnα(t + 1) = lnαt+1+ lnα(t),
and, with probability one,
E(lnα(t + 1)|st) − ln(α(t)) = E(lnαt+1|st) ≤ 0.
Therefore, lnα(t) is a supermartingale. Moreover, because 0 < bs≤ βs(rt) ≤ Bs<
∞, |lnα(t + 1) − lnα(t)| = |lnαt+1| < ∞. Hence lnα(t) is a supermartingale
with bounded increments. Thus, using Theorem 5.1 in Chapter VII of [12], with
probability one lnα(t) does not converge to +∞. Consequently, with probability
one, rt(st) = rα(t)does not converge to zero.
We now prove the second statement of the theorem. Again we consider the
sequence of random exponents
α(t) = αtαt−1···α2α1.
Let Mtdenote the martingale difference
Mt:= lnαt− E(lnαt|st−1).
We have E(Mt) = 0 and lnαtis uniformly bounded. Therefore, by the strong law
of large numbers (see Theorem 2.19 in [8]), with probability one
1
T
1
?T
t=1E(lnαt|st−1) < 0 a.s., then limt→∞rt(st) = 1 a.s.
?T
t=1E(lnαt|st−1) > 0 a.s., then limt→∞rt(st) = 0 a.s.
(4.1) lim
T→∞
1
T
T
?
t=1
Mt= 0.

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SRB MEASURES FOR CERTAIN MARKOV PROCESSES9
Therefore, with probability one,
limsup
T→∞
1
Tlnα(T) = limsup
T→∞
1
T
T
?
T
?
t=1
lnαt
= limsup
T→∞
1
T
t=1
Mt+ limsup
T→∞
1
T
T
?
t=1
E(lnαt|st−1) < 0.
From this we can conclude that for T large enough there is a positive random
variable η such that
α(T) ≤ e−Tηa.s.
Thus, since r ∈ [0,1], for T large enough we obtain
rT+1= rα(T)≥ re−Tηa.s.
By taking the limit of T to infinity we obtain
lim
T→∞rT+1= lim
T→∞rα(T)≥ lim
T→∞re−Tη= 1 a.s.
The proof of the third statement is very similar to the proof of the second one with
slight changes. In particular, using (4.1), we see that, with probability one,
1
Tlnα(T) > 0.
From this we can conclude that for T large enough there is a positive random
variable η such that
α(T) ≥ eTηa.s.
Thus, since r ∈ [0,1], for T large enough we obtain
rT+1= rα(T)≤ reTηa.s.
By taking the limit of T to infinity we obtain
liminf
T→∞
lim
T→∞rT+1= lim
T→∞rα(T)≤ lim
T→∞reTη= 0 a.s.
?
Corollary 4.4. Let F = {τs;ps}s∈Sbe an IFS such that τs(r) = rβs(r). Assume
that 0 < bs≤ βs(r) ≤ Bs< ∞ for all r ∈ [0,1].
(1) If limsupT→∞
measure of F with B(δ1) = (0,1] .
(2) If liminfT→∞
T
measure of F with B(δ0) = [0,1).
1
T
?T
?T
t=1E(lnαt|st−1) < 0 a.s., then δ1 is the unique SRB
1
t=1E(lnαt|st−1) > 0 a.s., then δ0 is the unique SRB
Proof. The proof is a consequence of statements (2) and (3) of Theorem 4.3 and
Lemma 4.1.
?
Remark 4.5. Observe that:
(1)
(2)
(3)
Thus, the conditions in the statements of Theorem 4.3 and Corollary 4.4 are very
easy to check for certain systems.
?
?
spslnBs≤ 0 =⇒ E(lnαt|st−1) ≤ 0 a.s.
?
spslnBs< 0 =⇒ limsupT→∞
spslnbs> 0 =⇒ liminfT→∞
1
T
?T
t=1E(lnαt|st−1) < 0 a.s.
t=1E(lnαt|st−1) > 0 a.s.
1
T
?T

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10 WAEL BAHSOUN AND PAWE? L G´ORA
Remark 4.6. In the proof of statement (1) of Theorem 4.3, we have with probability
πp= 1, limt→∞lnα(t) ?= ∞. In general, it is not clear that this implies that δ0is
not an SRB measure. However, in the following theorem under additional natural
assumption on the variance of lnαtwe show that δ0is indeed not an SRB measure.
Theorem 4.7. If E(lnαt|st−1) ≤ 0 and var(lnαt|st−1) ≥ d > 0, for all t ≥ 1, then
δ0is not an SRB measure of F.
Proof. Consider the sequence of random exponents
α(t) = αtαt−1···α2α1,
where αi= βs(rt−1) with probability ps, and observe that
rt(st) = rα(t).
Observe that
lnα(T) =
T
?
t=1
lnαt.
Since
E(lnα(t)|st−1) − lnα(t − 1) = E(lnαt|st−1) ≤ 0,
and
0 < bs≤ βs(rt) ≤ Bs< ∞.
the sequence ZT = lnα(T) forms a supermartingale with bounded increments.
Doob’s decomposition theorem gives the representation
ZT= WT+ ST,
where WT=?T
t=1E(lnαt|st−1) is a decreasing predictable sequence and
T
?
is a 0 mean martingale with bounded increments. By Theorem 5.1 (Ch. VII) of [12]
with probability 1 process ST either converges to finite limit or limsupT→∞ST =
−liminfT→∞ST = ∞. In the first case the process ZT is bounded from above.
We will consider only the second case to show that with positive probability the
process ZT is bounded from above for a set of indices T which has positive density
in N, i.e, there exist M > 0, 0 < a,b < 1 such that
#{t ≤ T : Zt≤ M}
T
ST=
t=1
[lnαt− E(lnαt|st−1)],
(4.2)
πp(limsup
T→∞
≥ a) > b.
Let us denote
Xt= lnαt− E(lnαt|st−1) , t ≥ 1.
This sequence satisfies assumptions of Theorem 7.1, with At= σ(st). We have
E(X2
?
s=1
t|st−1) = E((lnαt− E(lnαt|st−1))2|st−1)
L
?
=
s=1
ps(lnβs(r))2−
L
?
pslnβs(r)
?2
= var(lnαt|st−1) ≥ d > 0.

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SRB MEASURES FOR CERTAIN MARKOV PROCESSES 11
Thus, the sequence Xtsatisfies assumptions of Proposition 7.2. In particular, (7.1)
holds, i.e., if PosT is the number of times lnα(t) > 0 for t ≤ T, then
limsup
T→∞
where a,b are some numbers in (0,1). This means that if NTis the number of times
lnα(t) ≤ 0 for t ≤ T, then
limsup
T→∞
Now, we we show that
NT
T
For T > T0we have πp(NT
set which contains points from infinitely many ATis A = ∩i∪T>iATand since the
sequence (∪T>iAT)iis decreasing we have
πp(A) = lim
[πp(PosT
T
≤ a)] = b > 0,
[πp(NT
T
≥ 1 − a)] = b > 0.
πp(limsup
T→∞
T≥ 1 − a) > b/2. Let AT= {NT
≥ 1 − a) ≥ b/2 > 0.
T≥ 1 − a}, T ≥ T0. The
i→∞πp(∪T>iAT) ≥ b/2 .
Thus, with a positive probability b/2 > 0, there exist a sequence Tn→ ∞ such that
NTn
Tn≥ 1 − a or
πp(limsup
T→∞
Thus, lnα(T) is negative with positive density, i.e.,
1
T#{t ≤ T : lnα(t) ≤ 0} ≥ 1 − a > 0,
with positive probability b/2. This implies that rT(sT) ≥ ¯ r > 0 with positive
density 1−a and positive probability b/2. We can construct a continuous function
f which is 0 around 0 and 1 above ¯ r. The averages of this function satisfy
1
T
T?≤T
with nonzero probability b/2 which proves there is no weak convergence to δ0.
NT
T
≥ 1 − a) ≥ b/2 > 0.
lim
T→∞
limsup
T→∞
?
f(rT?(sT?)) ≥ 1 − a,
?
Remark 4.8. If E(lnαt|st−1) ≥ 0 and var(lnαt|st−1) ≥ d > 0, for all t ≥ 1, using
essentially the proof of Theorem 4.7, we obtain that δ1is not an SRB measure of
F. In particular, if E(lnαt|st−1) = 0 and var(lnαt|st−1) ≥ d > 0, for all t ≥ 1, we
obtain that neither δ0nor δ1is an SRB measure.
5. Properties of δ0and δ1: The case when p is unknown
In general, one cannot decide whether δ0or δ1is the unique SRB measure without
having information about p. We illustrate this fact in the following example.
Example 5.1. Let F = {τ1,τ2;p1,p2} where τ1= r2, τ2=√r and p1,p2are un-
known. Observe that the exponents, which are explicit in this case and independent
of r, are β1(r) = 2 and β2(r) = 1/2. Then
p1lnB1(r) + p2lnB2(r) = (2p1− 1)ln2.
By Corollary 4.4, if p1 < 1/2 the measure δ1 is the unique SRB measure of F;
however, if p1> 1/2 the measure δ0is the unique SRB measure of F. Thus, for

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12WAEL BAHSOUN AND PAWE? L G´ORA
this example, without having information about p, no information about the nature
of δ0or δ1can be obtained.
Although Example 5.1 shows that the analysis cannot be definitive in some cases
without knowing the probability distribution on S, our aim in this section is to find
situations when δ0and/or δ1are not SRB. Moreover, in addition to studying the
properties of δ0 and δ1, we are going to study the case when the IFS admit an
invariant probability measure whose support is separated from zero and is not
necessarily concentrated at one. The definition of such a measure is given below.
Definition 5.2. Let µ be a probability measure on ([0,1],B), where B is the Borel
σ-algebra. We define the support of µ, denoted by supp(µ), as the smallest closed
set of full µ measure. We say that supp(µ) is separated from zero if there exists an
η > 0 such that µ([0,η]) = 0.
In addition to properties (A) and (B), we assume in this section that:
(C) Every τshas a finite number of fixed points.
In this section, we use a graph theoretic techniques to analyze ergodic properties
of δ0and δ1. This approach is inspired by the concept of a Markov partition used
in the dynamical systems literature. For instance, in [2], the ergodic properties of
a deterministic system which admits a Markov partition is studied via a directed
graph and an incidence matrix. In our approach we construct a partition for our
random dynamical system akin to that of a Markov partition and use two directed
graphs to study ergodic properties of the system.
We now introduce the two graphs, Gdand Gu, which we will use in our analysis.
(1) Both Gdand Guhave the same vertices;
(2) For s ∈ {1,...,L}, an interval Js,m= (as,m,as,m+1) is a vertex in Gdand
Guif and only if τs(as,m) = as,m, τs(as,m+1) = as,m+1and τs(r) ?= r for
all r ∈ (as,m,as,m+1);
(3) Let Js,mand Jl,jbe two vertices of Gd. There is a directed edge connecting
Js,mto Jl,jif and only if ∃ an r ∈ Js,m, r > al,j+1, and a t ≥ 1 such that
τt
(4) Let Js,mand Jl,jbe two vertices of Gu. There is a directed edge connecting
Js,mto Jl,j if and only if ∃ an r ∈ Js,m, r < al,j, and a t ≥ 1 such that
τt
(5) By the out-degree of a vertex we mean the number of outgoing directed
edges from this vertex in the graph, and by the in-degree of a vertex we
mean the number of incoming directed edges incident to this vertex in the
graph.
(6) A vertex is called a source if it is a vertex with in-degree equals to zero. A
vertex is called a sink if it is a vertex with out-degree equals to zero.
For the above graphs, one can identify two types of vertices: let (as,m,as,m+1) be
a vertex. If τs(r) > r for all r ∈ (as,m,as,m+1), then the vertex (as,m,as,m+1)
will be denoted byˆJs,m. If τs(r) < r for all r ∈ (as,m,as,m+1), then the vertex
(as,m,as,m+1) will be denoted byˇJs,m. When we prove a statement for a vertex
Js,m(without a label), this means that the result holds for both types of vertices.
The following lemma contains some properties of Gdand Gu.
s(r) ∈ Jl,j.
s(r) ∈ Jl,j.

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SRB MEASURES FOR CERTAIN MARKOV PROCESSES 13
Lemma 5.3. Let Gdand Gube defined as above.
(1) IfˆJs,mis a vertex in Gd, thenˆJs,mis a sink in Gd.
(2) LetˇJs,mand Jl,jbe two vertices in Gd. There is a directed edge connecting
ˇJs,mto Jl,j in Gdif and only if as,m< al,j+1< as,m+1. In particular for
all s ∈ S there is no directed edge in Gdconnecting Js,mto Js,jfor any m
and j.
(3) IfˇJs,mis a vertex in Gu, thenˇJs,mis a sink in Gu.
(4) LetˆJs,mand Jl,jbe two vertices in Gu. There is a directed edge connecting
ˆJs,mto Jl,jin Guif and only if as,m< al,j< as,m+1. In particular for all
s ∈ S there is no directed edge in Guconnecting Js,mto Js,jfor any m and
j.
Proof. The proof of the first statement is straight forward. Indeed, let Jl,jbe any
vertex in Gd and r ∈ˆJs,m such that r > al,j+1. Then for all t ≥ 1 τt
τt−1
s
(r) > ...τs(r) > r > al,j+1. The proof of the second statement follows from
the fact that if r > as,m≥ al,j+1then for t ≥ 1 we have τt
r > al,j+1> as,m, then there exits a t ≥ 1 such that as,m< τt
of the third and fourth statements are similar to the first two.
s(r) >
s(r) > as,m≥ al,j+1. If
s(r) < al,j+1. Proofs
?
For our further analysis we introduce the following notion.
Definition 5.4. We say that a random orbit of F stays above a point c if all the
points of the infinite orbit are bigger than or equal to c with probability πp= 1.
Lemma 5.5. Let Jl,j be a vertex in Gd such that al,j+1?= 1. If Jl,j is a source
in Gd, then the random orbit of F starting from r > al,j+1stays above al,j+1with
probability πp= 1.
Proof. Suppose Jl,jis a source in Gd. Then for all r > al,j+1, we have τt
for all s ∈ S and t ≥ 1. This means that if r > al,j+1we have τs1(r) > al,j+1and
τs2◦ τs1(r) > al,j+1and so on.
Theorem 5.6. Let F be an IFS whose transformations satisfy the properties (A),
(B) and (C).
(1) If for all s ∈ S there is a vertexˇJs,min Gdwith as,m= 0, then δ0is an
SRB measure, B(δ0) ⊇ [0,a), where a = mins{as,m+1}. In particular, for
any r0∈ [0,a), limtrt(st) = 0 a.s.
(2) If for all s ∈ S there is a vertexˆJs,m in Gd with as,m+1= 1, then δ1 is
an SRB measure. Moreover, B(δ1) ⊇ (b,1], where b = maxs{as,m}. In
particular, for any r0∈ (b,1], limtrt(st) = 1 a.s.
(3) Let Jl,j be a vertex in Gdsuch that al,j+1?= 1. If Jl,j is a source in Gd7
then F preserves a probability measure whose support is separated from 0
8.
(4) Let Jl,j be a vertex in Gu such that al,j+1?= 0. If Jl,j is a source in Gu
then F preserves a probability measure whose support is separated from 1.
s(r) > al,j+1
?
7In the case where al,j= 0, even if otherˆ Js,m, with as,m = 0, receives a directed edge, the
result still holds. Thus, to know the existence of an invariant probability measure whose support
is separated from 0, it is enough to check that one vertex Jl,jwith al,j= 0 which is a source in
Gd. Statements of Lemma 5.3 can be useful to visualize cases of this type.
8The invariant measure here is not necessarily δ1.

Page 14

14 WAEL BAHSOUN AND PAWE? L G´ORA
(5) LetˆJs∗,mbe a vertex with as∗,m= 0 whose out-degree in Guis at least one.
If ∃ a vertex Js0,jin Gd, as0,j= 0 and as0,j+1< as∗,m+1, which is a source
in Gd, then for any r0∈ (0,1], limtrt(st) ?= 0 a.s. Moreover, δ0is not an
SRB measure for F.
(6) LetˇJs0,j be a vertex in Gdsuch that as0,j+1= 1 and whose out-degree in
Gdis at least one. If ∃ a Js∗,min Gu, as∗,m+1= 1 and as∗,m> as0,j, which
is a source in Gu, then for any r0∈ [0,1), limtrt(st) ?= 1 a.s. Moreover,
δ1is not an SRB measure for F.
(7) If for all s ∈ S the vertices whose as,m= 0 are of the formˆJs,mand their
as,m+1 ≡ a are identical, then for any r0 in (0,a], with probability one,
limtrt(st) = a. In particular, δa is an SRB measure with B(δa) = (0,a]
and δ0is not an SRB measure.
(8) If for all s ∈ S the vertices whose as,m+1 = 1 are of the formˇJs,m and
their as,m≡ b are identical, then for any r0in [b,1), with probability one,
limtrt(st) = b. In particular, δbis an SRB measure with B(δb) = [b,1) and
δ1is not an SRB measure.
Proof. We only prove the odd numbered statements in the theorem. Proofs of the
even numbered statements are very similar.
(1) For any r0∈ [0,a), any random orbit of F will converge to zero. Using Lemma
4.1, this shows that δ0is an SRB measure with B(δ0) ⊇ [0,a).
(3) Let r0 > al,j+1. Since [0,1] is a compact metric space and for all s ∈ S τs
is continuous, the average
T
the weak* topology to an F invariant probability measure9. By Lemma 5.5, this
measure is supported on [al,j+1,1].
(5) Let D = {Js,m\ {0} : as,m = 0}. For any r0 ∈ D, there exists a finite
t ≥ 1 such that τt
τt
with positive probability, the random orbit of r0is bounded away from 0. Let us
consider now the case of a starting point r?
are homeomorphisms and 0 is a common fixed point, for any r?
t ≥ 0, with positive probability, rt(st) > as0,m+1.
with strictly positive probability, limt→∞rt(st) ≥ as0,m+1. Moreover, with strictly
positive probability, for any r ∈ (0,1], there exists a T − 1 > t0≥ 1 such that
1
T
t=0
T
t=0
Therefore, with strictly positive probability, for any r ∈ (0,1],
1
T
t=0
Now, to show that δ0 is not an SRB measure, it is enough to find a continuous
function f on [0,1] such that with positive probability, for any r ∈ (0,1],
1
T
t=0
1
?T−1
t=0Ptδr0of the probability measures converges in
s∗(r0) > as0,j+1. Since Js0,j is a source in Gd, by Lemma 5.5,
s∗(r0) stays above as0,j+1 with probability πp = 1. Therefore, for any r0 ∈ D,
0> as0,m+1. Since all the transformations
0> as0,m+1and any
Hence, for any r ∈ (0,1],
T−1
?
rt(st) ≥1
t0−1
?
rt(st) −(t0+ 1)
T
as0,m+1+ as0,m+1.
(5.1) lim
T→∞
T−1
?
rt(st) ≥ as0,m+1.
{ lim
T→∞
T−1
?
f(rt(st)) ?= f(0)}.
9This follows from a random version of the Krylov-Bogoliubov Theorem [1].

Page 15

SRB MEASURES FOR CERTAIN MARKOV PROCESSES15
Indeed, this is the case if we use the function f(r) = r and (5.1). Thus, δ0is not
an SRB measure..
(7) Obviously, for any r0∈ (0,a], the random orbit of F starting at r0will converge
to a. Using Lemma 4.1, this implies that δais an SRB measure with B(δa) = (0,a].
Moreover, since all the transformations are homeomorphisms with common fixed
point at a, for any r?
SRB measure.
0> a, the random orbit of F stays above a. Thus, δ0is not an
?
6. Asset Market Games
In this section, we apply our results to evolutionary models of financial markets.
In particular, we will focus on the model introduced by [4]. First, we recall the
model of [4].
6.1. The Model. Let S is a finite set and st∈ S, t = 1,2,..., be the “state of the
world” at date t. Let p be a probability distribution on S such that for all s ∈ S
p(s) > 0. We also assume that stare independent and identically distributed.
In this model there are K “short-lived” assets k = 1,2,...,K (live one period
and are identically reborn every next period). One unit of asset k issued at time t
yields payoff Dk(st+1) ≥ 0 at time t + 1. It is assumed that
?K
and
EDk(st) > 0
for each k = 1,2,...,K , where E is the expectation with respect to the underlying
probability p. The total amount of asset k available in the market is Vk= 1.
k=1Dk(s) > 0 for all s ∈ S
In this model there are I investors (traders) i = 1,...,I. Every investor i at each
time t = 0,1,2,... has a portfolio
xi
t= (xi
t,1,...,xi
t,K),
where xi
(s1,...,st). We assume that for each moment of time t ≥ 1 and each random
situation st, the market for every asset k clears:
t,kis the number of units of asset k in the portfolio xi
t= xi
t(st), st=
(6.1)
I
?
i=1
xi
t,k(st) = 1.
Each investor is endowed with initial wealth wi
time t + 1 can be computed as follows:
0> 0. Wealth wi
t+1of investor i at
(6.2)
wi
t+1=
K
?
k=1
Dk(st+1)xi
t,k.
Total market wealth at time t + 1 is equal to
(6.3)
wt+1=
I
?
i=1
wi
t+1=
K
?
k=1
Dk(st+1).
Investment strategies are characterized in terms of investment proportions:
Λi= {λi
0, λi
1, λi
2,...}