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The existence of absolutely continuous invariant measures for piecewise expanding operators and random maps
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The existence of absolutely continuous invariant measures for piecewise expanding operators and random maps
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Content
The Existence of Absolutely Continuous Invariant Measures for
Piecewise Expanding Operators and Random Maps
by
Bining Jia
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(APPLIED MATHEMATICS)
August 2024
Copyright 2024 Bining Jia
Table of Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter 2: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Measure theory and Dynamical systems . . . . . . . . . . . . . . . . . . . . . . 4
2.2 FrobeniusPerron (FP) operator . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Piecewise expanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Chapter 3: The existence of absolutely continuous invariant measures . . . . . . . . . . . . 12
Chapter 4: The existence of ACIMs on random maps . . . . . . . . . . . . . . . . . . . . . 18
Chapter 5: Further research directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
ii
List of Figures
2.1 Piecewise monotonic function y= 4x mod 1 . . . . . . . . . . . . . . . . . . . . . 10
2.2 Another piecewise monotonic transformation . . . . . . . . . . . . . . . . . . . . 10
iii
Abstract
This thesis investigates discrete time dynamical systems on the unit interval driven by piecewise expanding continuous transformations. Our main objective is to establish the existence of absolutely
continuous invariant measures (ACIMs) in these systems. We further investigate the existence of
ACIMs in systems of random maps, where the transformations Ti are randomly selected with probabilities pi
. This work provides a foundational basis for further research on this specific class of
dynamical systems.
iv
Chapter 1
Introduction
This chapter provides the background and motivation for our study. We will discuss the significance of absolutely continuous invariant measures (ACIMs) and review the key literature and
recent advancements to set the stage for our investigation.
1.1 Background and Motivation
A dynamical system describes the time dependent behavior of points in an ambient space X governed by deterministic functions. In practical applications, such as dynamical systems described
by a system of differential equations, X is generally a manifold. The existence of fixed points, periodic orbits, chaotic behavior and strange attractors are properties of the system that only depend
on the corresponding topological structure of X.
However, when the focus shifts to longterm statistical properties of the system, we need to
impose a measure on X, transforming it into a measure space. For example, Boltzmann’s famous
ergodic hypothesis states that the time and space averages of an ideal gas are equal. The notion of
a space average only makes sense when there is a measure on X.
There are different measures we could impose on X and we wouldn’t expect a statement such
as the ergodic hypothesis to be true for all of the measures. First of all, we want the measure
theoretic structure to reflect the topological structure. For that we only consider Borel measure on
X. Secondly, we are interested in probability measure µ where µ(X) = 1. For such measures, the
1
measure of a set is the proportion of X that is filled by the set, and we might therefore expect it to
be the proportion of time points spend in the set under the dynamics. Thirdly, we are interested
in measures that are invariant under the dynamics, since these measures model the proportion of
time points spend in invariant subsets of X. Finally, we are interested in measures that are absolutely continuous with respect to Riemannian measure. If an invariant Borel probability measure
is absolutely continuous with respect to Riemannian measure, then it follows that the whole of X
is needed to study the longterm behavior of the system.
For these reasons, it is important to investigate whether invariant measures exist for a given
transformation T, and whether such measures are absolutely continuous with respect to the natural
measure (e.g., Lebesgue measure on R
n or Riemannian measure on manifolds) . In this thesis, we
aim to identify conditions on T under which an ACIM necessarily exists. We will concentrate our
attention to cases where X = [a,b] (denote as I) is an interval.
1.2 Literature Review
Research on ACIMs in dynamical systems has progressed significantly. Ulam and von Neumann
gave examples of transformations that possess an ACIM. Renyi further advanced the field by identifying classes of transformations that ensure the existence of ACIMs.
Focusing on piecewise expanding C
1
transformations, Lasota & Yorke [1] and Boyarsky &
Gora [2] both demonstrated the existence of ACIMs using the FrobeniusPerron (FP) operator
under conditions of bounded variation. However, their methodologies differed: Lasota & Yorke
incorporated the YosidaKakutani Theorem, whereas Boyarsky & Gora opted for an intuitive proof
technique.
The complexity increases when considering random map systems, where transformations Ti are
selected randomly based on a probability distribution pi
. Unlike deterministic C
1
transformations,
the random maps typically require T to be piecewise C
2
. Keller [3] demonstrated that ACIMs
2
persistently exist when each Ti
is piecewise expanding. Further, Pelikan [4] introduced a more
generalized set of conditions under which ACIMs can be proven to exist.
3
Chapter 2
Preliminaries
This chapter will review some definitions, propositions, and lemmas that will be used in Chapters
3 and 4.
2.1 Measure theory and Dynamical systems
A σalgebra B is a collection of subsets of X that includes X itself, is closed under complementation, and is closed under countable unions. A measure µ on a σalgebra B is a function
µ : B → [0,∞], that assigns zero to the empty set and is countably additive. We call the triplet
(X,B,µ) a measure space. A probability measure is a measure µ such that µ(X) = 1. If X is a
topological space, a Borel measure is a measure where B contains the Borel σalgebra.
Definition (Invariant measure). Suppose (X,B,µ) is a measure space with µ(X) = 1 and T : X →
X is a transformation. T is called measurable if for all B ∈ B,
T
−1
(B) ∈ B.
Furthermore, if µ(T
−1
(B)) = µ(B) for all measurable sets B, we say that the measure µ is Tinvariant or T preserves measure µ.
Definition (Dynamical system). Suppose (X,B,µ) is a measure space and T : X → X is a transformation that preserves µ. Then the quadruple (X,B,µ,T) is called a dynamical system.
4
Observation: Given (X,B) and T : X → X, the set of invariant probability measures form a
convex set in the space of all probability measures. This is because given two invariant probability
measures µ1 and µ2, we have (αµ1 + (1 − α)µ2)(T
−1A) = αµ1(T
−1A) + (1 − α)µ2(T
−1A) =
αµ1(A) + (1−α)µ2(A) = (αµ1 + (1−α)µ2)(A).
Definition (Absolutely continuous). Let ν and µ be two measures on (X,B), if whenever µ(B) =
0, it follows that ν(B) = 0, we say that ν is absolutely continuous with respect to µ, denoted as
ν ≪ µ.
If ν ≪ µ, then there exists a unique f ∈ L
1
, such that for every A ∈ B,
ν(A) = Z
A
f dµ.
f is called the RadonNikodym derivative of ν with respect to µ, denoted dν
dµ
.
The existence and nature of invariant Borel probability measures depend on the map T. The
follwing are four examples where X = [0,1].
Examples: Let X = [0,1] and B be the Borel σalgebra.
(1): T(x) = 1
Suppose µ is an invariant probability measure. Consider a set A ∈ B. If 1 ∈ A then T
−1
(A) = X,
so µ(A) = µ(T
−1A) = µ(X) = 1. If 1 ∈/ A then T
−1
(A) = /0, so µ(A) = µ(T
−1A) = µ(/0) = 0. Thus
there is a unique invariant probability measure for this transformation, namely the Dirac measure
concentrated at 1. Mathematically, this can be expressed as:
µ(A) =
1 if 1 ∈ A,
0 if 1 ∈/ A.
Notice this measure is not absolutely continuous w.r.t. Lebesgue measure.
(2): T(x) = x
Every measure is invariant for this transformation. Since T
−1
(A) = A, µ(A) = µ(T
−1A) holds
for all measures.
5
(3): T(x) = 2x mod 1
An invariant measure for this transformation is the Lebesgue measure λ. This means that the
measure of any interval [a,b] ⊆ [0,1] is simply its length:
µ([a,b]) = b−a.
We can see this is invariant, since for any interval [a,b], T
−1
([a,b]) = [ a
2
,
b
2
] ∪ [
a+1
2
,
b+1
2
], so
µ(T
−1
([a,b])) =
b
2 −
a
2
+
b+1
2 −
a+1
2
= b−a = µ([a,b]).
(4): T(x) = x
2
This map has two fixed points x = 0 and x = 1. Dirac measure on 0 and Dirac measure on 1
are both invariant probability measures. Any linear combinations of these are also invariant. For
∀ 0 < a < 1, T
−1
[0,a] = [0,
√
a]. So, if µ is invariant, µ[0,a] = µ[0,
√
a]. Thus, µ(a,
√
a) = 0. It
follows that µ(0,1) = 0. So these are the only invariant measures.
2.2 FrobeniusPerron (FP) operator
In dynamical systems, the FrobeniusPerron operator is a mathematical tool that describes the
evolution of probability density functions under transformations.
Definition. Let I = [a,b] and let λ be the normalized Lebesgue measure on I. Let T : I → I be a
nonsingular transformation. Then, the unique FrobeniusPerron operator PT associated with T is
defined for any integrable function f : I → R such that for all measurable sets A,
Z
A
PT f dλ =
Z
T −1(A)
f dλ.
Observation: Consider a random variable X on I with probability density function f ∈ L
1
. In
other words, P(X ∈ A) = R
A
f dλ. We have,
P(TX ∈ A) = P(X ∈ T
−1A) = Z
T −1A
f dλ =
Z
A
PT f dλ.
Then PT f is the probability density function of TX .
Let µ(A) = R
A
f dλ. Let T
∗µ(A) := µ(T
−1
(A)) = R
A PT f dλ. We have f = dµ/dλ and PT f =
dT∗µ/dλ.
Following are properties of the FP operator PT . (Proofs that are omitted can be found in [2].)
• PT is linear.
• PT is nonnegative. (f ≥ 0 =⇒ PT f ≥ 0.)
• ∥PT f ∥1 ≤ ∥ f ∥1.
Proof. Let f ∈ L
1
. Let f
+ = max(f,0) and f
− = −min(0, f), then f = f
+ − f
− and  f  =
f
+ + f
−.
PT f  = PT (f
+ − f
−) = PT f
+ −PT f
− ≤ PT f
++PT f
− = PT  f .
Then,
∥PT f ∥1 =
Z
I
PT f dλ ≤
Z
I
PT  f dλ =
Z
T −1(I)
 f dλ =
Z
I
 f dλ = ∥ f ∥1.
• PT
N f = P
N
T
f.
• Suppose f = dµ/dλ ∈ L
1
(I) with ∥ f ∥1 = 1. Then f is an eigenfunction of PT with eigenvalue 1 (i.e. PT f = f) if and only if µ is an absolutely continuous invariant Borel probability
measure on I.
7
2.3 Bounded Variation
Definition. A function f defined on the interval I = [a,b] is said to have bounded variation if
there exists a constant M such that for any partition P = {[xi−1, xi
],i = 1,...,n} of [a,b], where
a = x0 < x1 < ··· < xn = b, the variation is bounded by M:
n
∑
i=1
 f(xi)− f(xi−1) ≤ M.
For a function of bounded variation, we define the total variation to be
V(f) = sup
P
n
∑
i=1
 f(xi)− f(xi−1),
where the supremum is taken over all possible partitions P.
If f is of bounded variation on [a,b], the following are properties of V[a,b]
f : (Proofs that are omitted
can be found in [2].)
• V[a,b]
f = V[a,c]
f +V[c,b]
f , where c ∈ (a,b).
• V[a,b]
(f ±g) ≤ V[a,b]
f +V[a,b]g.
•  f(x) ≤  f(a)+V[a,b]
f , for all x ∈ [a,b].
• If f ∈ L
1
,  f(x) ≤ V[a,b]
f +
∥ f ∥1
b−a
, for all x ∈ [a,b].
Proof. We claim there exists y ∈ [a,b] such that  f(y) ≤ ∥ f ∥1
b−a
. Suppose not, then for any
x ∈ [a,b],
(b−a) f(x) > ∥ f ∥1.
Thus,
∥ f ∥1 =
Z b
a
 f(x)dλ(x) >
Z b
a
∥ f ∥1
b−a
dλ(x) = ∥ f ∥1,
which is a contradiction. Hence, there must exist y ∈ [a,b] such that  f(y) ≤ ∥ f ∥1
b−a
.
8
Since
 f(x) ≤  f(x)− f(y)+ f(y),
 f(x)− f(y) ≤ V[a,b]
f,
we have
 f(x) ≤ V[a,b]
f +
∥ f ∥1
b−a
.
• If f is C
1
in [a,b], V[a,b]
f =
R b
a
 f
′
(x)dλ(x).
Theorem (Helly’s Selection Theorem). (see [2]) Let F = { f } be an infinite family of functions
defined on an interval [a,b]. If all the functions in the family and the total variation of all the
functions in the family are bounded, i.e.,  f(x) ≤ K and V[a,b]
f ≤ K for all f ∈ F, then there
exists a sequence { fn} ⊂ F that converges at every point of [a,b] to some function f
∗ of bounded
variation.
2.4 Piecewise expanding
In the following, we will define a class of transformations on the interval [a,b]. Later chapters will
be dedicated to investigating transformations in this class.
Definition (Piecewise monotonic). Let I = [a,b] and T : I → I be an operator. We say T is piecewise monotonic with respect to a partition P = {Ii = [ai−1,ai
], i = 1,...,q} of I, if T
(ai−1,ai)
is C
1
and can be extended to a C
1
function on Ii
, and for any i and for all x ∈ Ii
, T
′
(x) > 0. We say T is
piecewise monotonic if there exists a partition with respect to which it is piecewise monotonic.
Note: If T is piecewise monotonic and we set φi = (TIi
)
−1
, by the definition of PT f , we have
PT f =
q
∑
i=1
f(φi)
T
′(φi)
χT(Ii)
. (2.1)
9
(See [2].)
Definition (Piecewise expanding). If T is piecewise monotonic with respect to a partition P =
{Ii = [ai−1,ai
], i = 1,...,q} of I and ∃α > 1 such that for all i and for all x ∈ Ii
, T
′
(x) ≥ α, then
we say T is piecewise expanding with respect to the partition P.
Examples of such transformations are shown in Figure 2.1 and Figure 2.2. We observe that example
1 is piecewise expanding and example 2 is piecewise monotonic but not piecewise expanding.
Figure 2.1: Piecewise monotonic function y= 4x mod 1
Figure 2.2: Another piecewise monotonic transformation
10
Note: Suppose T is piecewise expanding. For each n ≥ 1, let T
n
:= T ◦ ··· ◦ T and define the
partition P
(n)
as:
P
(n) =
n_−1
k=0
T
−k
(P) = {Ii0 ∩T
−1
(Ii1
)∩...∩T
−n+1
(Iin
) : Iij ∈ P}.
Since
(T
n
)
′
(x) =
n−1
∏
k=0
T
′
(T
k
(x)) ≥ α
n > 1,
we see that T
n
is also piecewise expanding on I, with partition P
(n)
.
11
Chapter 3
The existence of absolutely continuous invariant measures
In this chapter we will concentrate on piecewise expanding deterministic transformations and explore the existence of absolutely continuous invariant measures (ACIMs).
Definition. We consider the interval I = [a,b] with normalized Lebesgue measure λ. Let T (I)
denote the class of transformations T : I → I that satisfy,
• T is piecewise expanding;
• g(x) =
1
T
′(x)
is a bounded variation function.
Theorem. For all T ∈ T (I), there exists an absolutely continuous invariant measure.
Before proving this theorem, we introduce several useful results.
Lemma. 3.1 (see [2]) Let T ∈ T be given with partition P = {Ii = [ai−1,ai
], i = 1,...,q}. Let
gn(x) = 1
(T
n(x))′

. We define
W1 := max
1≤i≤q
VIi
g;
and
Wn := max
J∈P(n)
VJgn.
Then we have
Wn ≤
n
αn−1W1. (3.1)
12
Proof. We prove the lemma by mathematical induction. First, for n = 1, (3.1) holds obviously.
Assume that (3.1) is true for some fixed k ≥ 1. We need to prove that Wk+1 ≤
k+1
αk W1. Let
J ∈ P
(k+1) be given and let {[xi−1, xi
],i = 1,2,...,m} be any partition of interval J. Let
gn(x) =
(T
n
(x))′
−1 = T
′
(T
n−1
)·T
′
(T
n−2
)···T
′
(T)·T
′

−1
.
Notice that gn+1(x) = gn(T(x))g(x), so
m−1
∑
j=0
gk+1(x j+1)−gk+1(x j)
=
m−1
∑
j=0
gk(T(x j+1))g(x j+1)−gk(T(x j))g(x j)
≤
m−1
∑
j=0
(gk(T(x j+1))−gk(T(x j)))g(x j+1)
+
gk(T(x j))(g(x j+1)−g(x j))
≤
1
α
Wk +
1
αkW1
≤ (k +1)
1
αkW1.
Since J ∈ P
(k+1)
and the partition of J are arbitrary, we have Wk+1 ≤
k+1
αk W1. Then, (3.1) is true for
all natural numbers n.
Theorem (Mazur Theorem). (see [5], VIII 5.3) Let F be a Banach space and let A ⊂ F be a
relatively compact subset of F. Then the convex hull of A is also relatively compact.
Theorem (YosidaKakutani Theorem). (see [5], VIII 5.3) Let F be a Banach space and let P :
F → F be a bounded linear operator. Assume there exists c > 0 such that for ∀n ∈ N, ∥P
n∥ ≤ c.
Suppose A ⊂ F has the property that ∀ f ∈ A, the sequence { fn}, where
fn =
1
n
n
∑
k=1
P
k
f,
1
contains a subsequence { fnk
} which converges weakly in F. Then for any f ∈ A,
1
n
n
∑
k=1
P
k
f → f
∗ ∈ F
(norm convergence) and P(f
∗
) = f
∗
.
We are now ready to prove the existence of ACIM. Both Lasota & York [1] and Boyarsky &
Gora [2] give the proof in the deterministic case. In this thesis, we combine their proofs.
Proof. We will prove that for any f ∈ BV(I),
1
n
n−1
∑
k=0
P
n
T
f converges to an L
1
function f
∗
for which
PT f
∗ = f
∗
. As we have seen, any such function is the density function of an ACIM. First, we show
for any f ∈ BV(I), the family of functions P
n
T
f all lie in BV(I) and there is a uniform bound on
their total variation. Let T ∈ T is piecewise expanding with respect to P = {Ii
, i = 1,...,q}. Let
f ∈ BV(I). Let φi = (TIi
)
−1
. By equation (2.1),
PT f(x) =
q
∑
i=1
f(T
−1
Ii
(x))
T
′(T
−1
Ii
(x))
χT(Ii)
(x)
=
q
∑
i=1
f(φi(x))g(φi(x))χT(Ii)
(x).
Let a = x0 < x1 < ... < xr = b be an arbitrary partition of I = [a,b]. we have,
r
∑
j=1
PT f(x j)−PT f(x j−1)
=
r
∑
j=1
q
∑
i=1
g(φi(x j))f(φi(x j))χT(Ii)
(x j)−
q
∑
i=1
g(φi(x j−1))f(φi(x j−1))χT(Ii)
(x j−1)
≤
q
∑
i=1
r
∑
j=1
g(φi(x j))f(φi(x j))χT(Ii)
(x j)−g(φi(x j−1))f(φi(x j−1))χT(Ii)
(x j−1)
.
Let J
i
0
denote the set of j’s such that T(Ii)∩(x j−1, x j) = /0, J
i
1
denote the set of j’s where (x j−1, x j) ⊆
T(Ii), and J
i
2
denote the set of j’s where (x j−1, x j)∩T(Ii) ̸= /0 but (x j−1, x j) ⊈ T(Ii).
14
If j ∈ J
i
0
, then χT(Ii)
(x j) = χT(Ii)
(x j−1) = 0, so
q
∑
i=1
∑
j∈J
i
0
g(φi(x j))f(φi(x j))χT(Ii)
(x j)−g(φi(x j−1))f(φi(x j−1))χT(Ii)
(x j−1)
= 0.
If j ∈ J
i
1
, then χT(Ii)
(x j) = χT(Ii)
(x j−1) = 1. In this case,
q
∑
i=1
∑
j∈J
i
1
g(φi(x j))f(φi(x j))χT(Ii)
(x j)−g(φi(x j−1))f(φi(x j−1))χT(Ii)
(x j−1)
=
q
∑
i=1
∑
j∈J
i
1
g(φi(x j))f(φi(x j))−g(φi(x j−1))f(φi(x j−1))
≤
q
∑
i=1
∑
j∈J
i
1
f(φi(x j))
g(φi(x j))−g(φi(x j−1))
+
q
∑
i=1
∑
j∈J
i
1
g(φi(x j−1))
f(φi(x j))− f(φi(x j−1))
≤
q
∑
i=1
(sup
Ii
 f )VIi
g+ (sup
I
g)
q
∑
i=1
VIi
f
≤ max
1≤i≤q
(VIi
g)
q
∑
i=1
VIi
 f +
1
λ(Ii)
Z
Ii
 f dλ
+sup
I
g
q
∑
i=1
VIi
f
≤ max
1≤i≤q
(VIi
g)
VI
f +
1
δ
∥ f ∥1
+
1
α
VI
f,
where δ = min
1≤i≤q
λ(Ii).
If j ∈ J
i
2
, then either x j ∈ T(Ii) and x j−1 ∈/ T(Ii) or x j ∈/ T(Ii) and x j−1 ∈ T(Ii). For each i,
there are at most two j’s in J
i
2
; one where x j ∈ T(Ii) and x j−1 ∈/ T(Ii) and another where x j ∈/ T(Ii)
and x j−1 ∈ T(Ii). Call these j
′
and j
′′, we have
q
∑
i=1
∑
j∈J
i
2
g(φi(x j))f(φi(x j))χT(Ii)
(x j)−g(φi(x j−1))f(φi(x j−1))χT(Ii)
(x j−1)
=
q
∑
i=1
g(φi(x j
′))f(φi(x j
′))
+
g(φi(x j
′′−1)f(φi(x j
′′−1))
≤
1
α
q
∑
i=1
f(φi(x j
′))
+
f(φi(x j
′′−1))
.
Since si = φi(x j
′) and ri = φi(x j
′′−1) both lie in Ii
, we write
q
∑
i=1
( f(si)+ f(ri)) ≤
q
∑
i=1
(2 f(vi)+ f(vi)− f(ri)+ f(vi)− f(si)),
where vi ∈ Ii satisfies  f(vi) ≤ 1
λ(Ii)
R
Ii
 f λ(dx). Then we obtain
1
α
q
∑
i=1
( f(si)+ f(ri)) ≤
1
α
q
∑
i=1
VIi
f +
2
λ(Ii)
Z
Ii
 f λ(dx)
≤
1
α
VI
f +
2
αδ ∥ f ∥1.
Combining we have
VI(PT f) ≤
2
α
+ max
1≤i≤q
VIi
g
VI
f +B∥ f ∥1, (3.2)
where B =
2
αδ +
1
δ max1≤i≤qVIi
g.
Next, we consider T
n
:= T ◦ ··· ◦ T and use (3.2) to find a bound for VIPT
n f . Recall, T
n
is
piecewise expanding with partition P
(n)
.
Let gn(x) = 1
(T
n(x))′

. Recall gn−1(x) = gn−1(T x)g(x) = ... = g(T
n
x)g(T
n−1
x)...g(x). Since
g(x) ≤
1
α
, then gn(x) ≤
1
αn . Let W1 := max1≤i≤qVIi
g and Wn := maxJ∈P(n)VJgn. Apply (3.2) on
PT
n f by replacing T with T
n
, we obtain
VI(PT
n f) ≤
2
αn
+Wn
VI
f +B1∥ f ∥1, (3.3)
where B1 =
2
αnδ +
1
δ Wn. By lemma 3.1, we know that Wn ≤
n
αn−1W1. Since α > 1, there exists some
N such that 2
αn +Wn < 1 for all n ≥ N. For such n’s, we have VI(PT
n f) ≤ α
′VI
f +K∥ f ∥1, where
0 ≤ α
′ < 1 and K > 0 are constants that are independent of f .
Finally, we will complete the proof by using Helly’s Selection Theorem and YK theorem. We
have proved that for some N, VI(PT
N f) ≤ α
′VI
f +K∥ f ∥1. Notice that VIP
N
T
f = VIPT
N f , consider
the sequence {P
Nk
T
f }
∞
k=0
,
VIP
Nk
T
f = VIP
N
T
(P
N(k−1)
T
f) ≤ α
′VIP
N(k−1)
T
f +K∥P
N(k−1)
T
f ∥1 ≤ α
′VIP
N(k−1)
T
f +K∥ f ∥1.
16
By iterating, we get,
VIP
Nk
T
f ≤ α
′
h
α
′VIP
N(k−2)
T
f +K∥ f ∥1
i
+K∥ f ∥1 ≤ ··· ≤ α
′kVI
f +
k−1
∑
i=0
α
′iK∥ f ∥1.
Since 0 < α
′ < 1,
limsup
k→∞
VIP
Nk
T
f ≤ K(1−α
′
)
−1
∥ f ∥1.
In other words, the total variation of the sequence {P
Nk
T
f }
∞
k=0
is uniformly bounded. Combined
with the fact that for ∀k > kg and for ∀x ∈ [a,b], P
Nk
T
f(x) ≤ VI(P
Nk
T
f)+ ∥P
Nk
T
f ∥1
b−a ≤
K∥ f ∥1
1−α′ +
∥ f ∥1
b−a
, it
follows by Helly’s Selection Theorem that {P
Nk
T
f }
∞
k=0
has a subsequence that converges at every
point of [a,b]. By Dominated convergence theorem, since for ∀k > kg, P
Nk
T
f(x) is bounded by a
constant, convergence pointwise implies convergence in L
1 norm. Hence, {P
Nk
T
f }
∞
k=0
is relatively
compact in L
1
space. Since
{P
k
T
f }
∞
k=0 =
N
[−1
i=0
P
i
T {P
Nk
T
f }
∞
k=0
is a finite union of relatively compact sets, {P
k
T
f }
∞
k=0
is also relatively compact in L
1
. This is due to
the fact that in a finite union of relatively compact sets any sequence has infinitely many elements
in at least one of the sets. These elements form a subsequence which in turn has a convergent
subsequence. Since for all n,
1
n
n−1
∑
j=0
P
j
T
f lies in the convex hull of {P
k
T
f }
∞
k=0
, it follows by Mazur’s
theorem that n
1
n ∑
n−1
j=0
P
j
T
f
o∞
n=1
has a convergent subsequence. By YosidaKakutani theorem, we
obtain 1
n
n−1
∑
j=0
P
j
T
f converges to a function f
∗
and f
∗
satisfies PT f
∗ = f
∗
.
It follows that for all T ∈ T , there exists an absolutely continuous invariant measure whose
density function is f
∗
.
17
Chapter 4
The existence of ACIMs on random maps
In the previous chapters, we explored the existence of absolutely continuous invariant measures
for deterministic systems. We now turn our attention to the realm of random maps with constant
probabilities. This chapter will delve into Pelikan’s work [4] on random maps, examining sufficient
conditions under which ACIMs exist.
Consider an ambient space X, and for i = 1,...,m, let Ti
: X → X be transformations and
pi be probabilities with
m
∑
i=1
pi = 1. In a random dynamical system, the map at each iteration
is chosen at random from T1,...,Tm with probabilities p1,..., pm respectively, independently of
the maps chosen previously. We call this a random dynamical system acting on X and denote it
(X,{T1,...,Tm},{p1,..., pm}).
Definition (Invariant measure). Let(X,B,µ) be a measure space and (X,{T1,...,Tm},{p1,..., pm})
a random dynamical system on X. The measure µ is said to be invariant if for all measurable sets
A ∈ B,
µ(A) =
m
∑
i=1
piµ(T
−1
i
(A)).
The FrobeniusPerron operator PT f in random case represents the evolution of the probability
density function f of the random variable X under the randomly selected map Ti
. It combines the
effects of all possible maps Ti by taking a weighted average of them.
18
Definition (FrobeniusPerron operator). In random case, the FrobeniusPerron operator of T is
defined as,
PT f =
m
∑
i=1
piPTi
f.
Before the main result, we introduce several useful lemmas. Kella [3] gives the proof of lemma
4.1 and lemma 4.2.
Lemma. 4.1 (See [3]) Let I = [0,1]. Let f ∈ BV(I) and h ∈ L
1
. Set H(x) = R x
0
h(t)dt. Then
Z 1
0
f(t)h(t)dt
≤
VI
f +sup
x∈I
 f(x)
∥H∥∞.
Lemma. 4.2 (See [3]) Let I = [0,1]. Let f ∈ BV(I). Then VI
f = sup
h
R 1
0
f(t)h(t)dt
, where the
sup is taken over all h ∈ L
1 with R 1
0
hdt = 0 and sup
x

R x
0
h(t)dt ≤ 1.
Lemma. 4.3 (See [4]) Let I = [a,b] and T : I → J be a C
1
function (T
′
(x) ̸= 0), where J = T(I).
We set φ = T
−1
and σ =
dφ(x)
dx
. For all f ∈ BV, we obtain
VJ (f(φ(x))σ(x)) ≤
VI
f +sup
I
 f 
VJσ +sup
J
σ
.
Proof. By lemma 4.2,
VJ (f(φ(x))σ(x)) = sup
h
Z
J
(f ◦ φ)·σ · hdx
,
where the sup is taken over all h ∈ L
1 with R
J
hdt = 0 and ∥H∥∞ ≤ 1. Let zh = h ◦ T, then h =
zh ◦T
−1 = zh ◦ φ. By performing a change of variables and applying Lemma 4.1, we obtain
Z
J
(f ◦ φ)·σ · hdx
=
Z
J
(f(φ(x)))·zh(φ(x))d(φ(x))
=
Z
I
(f(t)·zh(t)d(t)
≤
VI
f +sup
I
 f 
sup
x
Z x
a
zh(t)dt
,
19
where
Z x
a
zh(t)dt
=
Z x
a
h(T x)dx
=
Z x
a
h(T x)σ(T x)T
′
(x)dx
=
Z
T[a,x]
h(t)σ(t)dt
≤
VJσ +sup
J
σ(x)
∥H∥∞
≤
VJσ +sup
J
σ
.
Hence, for all given h,
Z
J
(f ◦ φ)·σ · hdx
≤
VI
f +sup
I
 f 
VJσ +sup
J
σ
.
We complete the proof.
Lemma. 4.4 (See [4]) If we add a condition that let T : I → J be a C
2
function, then the following
equation holds for all f ∈ BV(I) for some constant β > 0 and K > 0,
VJ(f ◦ φ)σ ≤ βVI
f +K∥ f ∥1. (4.1)
Proof. Let J1, J2,..., Jk be an arbitrary partition of J. Let Ii = φ(Ji), supJi
σ = αi and supJ σ = α.
By the previous lemma, we obtain
VJ(f ◦ φ)σ ≤
k
∑
i=1
VIi
f +sup
Ii
 f 
! VJiσ +sup
Ji
σ
!
.
Notice that sup
Ii
 f  ≤ VIi
f +inf
Ii
 f  ≤ VIi
f +
R
Ii
 f dx, then
VJ(f ◦ φ)σ ≤
k
∑
i=1
2VIi
f +
Z
Ii
 f dx
(VJiσ +αi)
20
≤ 2αVI
f +α∥ f ∥1 +
k
∑
i=1
2VIi
fVJiσ +
k
∑
i=1
VJiσ
Z
Ii
 f dx.
Since T is C
2
, the derivative of σ exists, then
VJiσ =
Z
Ji
σ
′
dx ≤
supσ
′

infσ
Z
Ji
σdx.
Set K1 =
supσ
′

infσ
. We have VJiσ ≤ K1
R
Ji
d(φ(x)) = K1Ii
. Then
k
∑
i=1
2VIi
fVJiσ ≤ 2K1max
i
Ii
VI
f.
Since we can select small enough partition Ji
, then this part can be arbitrarily small, we have
VJ(f ◦ φ)σ ≤ 2αVI
f + (α +K1)∥ f ∥1. (4.2)
Thus, (4.1) holds and we take β = 2supJ σ and K = supJ σ +
supσ
′

infσ
.
Now, we give the mean theorem by Pelikan[4].
Theorem. Let (X,{T1,...,Tm},{p1,..., pm}) be a random dynamical system on I = [a,b]. Let T
denote the random map. In addition, suppose each Ti (i = 1,...,m) is piecewise monotone, C
2
and
nonsingular. If for all x ∈ I,
m
∑
i=1
pi
T
′
i
(x)
≤ γ < 1,
then there exists an absolutely continuous invariant measure for the random dynamical system.
Remark: Keller [3] demonstrated that ACIMs persistently exist when each Ti
is piecewise expanding. But here we do not assume that each Ti
is piecewise expanding.
21
To simplify the proof, we will consider the interval [0,1] instead of [a,b]. Consider a partition Ij
,
j = 1,2,...,l, such that for all Ti
, Ti
Ij
is monotone. Let gi(x) = 1
T
′
i
(x)
, H
j
i = Ti(Ij), φ
j
i = (Ti
Ij
)
−1
,
σ
j
i =
d
dxφ
j
i
=
1
T
′
i
(φ
j
i
)
= gi(φ
j
i
). Then by equation (2.1), we have
PT (f)(x) = ∑
i, j
pi
f(φ
j
i
(x))σ
j
i
(x)χ
H
j
i
(x).
Proof. First, similar to the deterministic case, we bound VIPT
N f . Consider the random map
T
N(x) = Tω(x) = TωN
◦ ... ◦Tω1
(x) with pω. Here, ω ∈ {1,2,...,m}
N.
∑
ω
pω
T
′
ω(x)
= ∑
ω∈{1,2,...,m}
N−1
pω
T
′
ω
(x)
m
∑
i=1
pi
T
′
i
(Tω(x))
≤ γ∑
ω
pω
T
′
ω
(x)
≤ ··· ≤ γ
N
.
Choose N ≥ 1,such that γ
N <
1
3
. Thus, it is equivalent to assume that γ <
1
3
and estimate VIPT f.
By equation (2.1) and notice that Ti(aj−1) and Ti(aj) are two endpoints of the interval H
j
i
, we have
VIPT f = VI
m
∑
i=1
piPT f = VI
m
∑
i=1
pi
l
∑
j=1
(f ◦ φ
j
i
)σ
j
i
χ
H
j
i
≤ ∑
j
∑
i
piVI
(f ◦ φ
j
i
)σ
j
i
χ
H
j
i
≤ ∑
j
∑
i
pi
h
f(aj−1)σ
j
i
(Ti(aj−1))
+
f(aj)σ
j
i
(Ti(aj))
+VH
j
i
((f ◦ φ
j
i
)σ
j
i
)
i
≤ γ
l
∑
j=1
 f(aj)+ f(aj−1)
+∑
i, j
piVH
j
i
((f ◦ φ
j
i
)σ
j
i
).
aj−1 and aj are two endpoints of the interval Ij
.  f(aj) +  f(aj−1) ≤ VIj
f + 2inf
Ij
 f  ≤ VIj
f +
2
R
Ij
 f dx. Then,
VIPT f ≤ γ(VI
f +2∥ f ∥1) +∑
i, j
piVH
j
i
((f ◦ φ
j
i
)σ
j
i
). (4.3)
22
Let us now estimate the final term of (4.3). Let partition {Lk} be a refinement of partition {Ij},
where the maximum length of the intervals in the partition Lk
is given by maxk
Lk
. By lemma 4.4,
VH
j
i
((f ◦ φ
j
i
)σ
j
i
) ≤ ∑
∀k,Lk⊆Ij
VTi(Lk)
((f ◦ φ
j
i
)σ
j
i
)
≤ ∑
∀k,Lk⊆Ij
2 sup
Ti(Lk)
σ
j
i VLk
f +Ki,k
Z
Lk
 f dx!
≤ ∑
∀k,Lk⊆Ij
2
1
inf
Lk
T
′
i
(x)
VLk
f +Ki,k
Z
Lk
 f dx
,
where Ki,k = sup
Ti(Lk)
σ
j
i +
sup(σ
j
i
)
′

infσ
j
i
.
∑
i, j
piVH
j
i
((f ◦ φ
j
i
)σ
j
i
) ≤ ∑
k
m
∑
i=1
2pi
inf
Lk
T
′
i
(x)
VLk
f +∑
i,k
Ki,k∥ f ∥1
≤ 2γ
1VI
f +K∥ f ∥1,
where K = ∑i,kKi,k and
γ
1 = max
k
m
∑
i=1
pi
inf
Lk
T
′
i
(x)
≥ γ.
Combined with (4.3), we get
VIPT f ≤ 3γ
1VI
f + (2γ +K)∥ f ∥1. (4.4)
Finally, we will show that by selecting a partition {Lk}, 3γ
1 may less than 1. We have
∑
i
pi
T
′
i
(x)
<
1
3
,
23
then there exists some δ > 0, such that for all yi −T
′
i
(x) < δ,
∑
i
pi
yi
<
1
3
.
Since T is C
2
, then select ε, such that x−y < ε =⇒ T
′
i
(x)−T
′
i
(y) < δ. If we select a partition
with max
k
Lk
 < ε, then T
′
i
(x)− inf
x∈Lk
T
′
i
(x) < δ. It follows that γ
1 <
1
3
.
Then we have proved that there exists some N, such that VI(PT
N f) ≤ αVI
f + K∥ f ∥1, where
0 ≤ α < 1 and K > 0 are constants that independent to f . Again, We apply the same approach
as demonstrated in the proof in Chapter 3 with the generalization of YosidaKakutani theorem [6],
then it follows that T has an ACIM.
Note that the conditions given above are only sufficient to guarantee the existence of an ACIM.
References [2] and [4] provide examples of transformations that do not meet these conditions but
still admit an ACIM.
24
Chapter 5
Further research directions
This thesis primarily focuses on one dimensional piecewise transformations. Generalizing the
research to multidimensional systems seems to be a good project. This could provide deeper
insights into complex behaviors that are more representative of realworld phenomena.
For example, we may investigate the existence and properties of ACIMs in twodimensional
piecewise expanding systems. For automorphisms on the twodimensional torus, which conditions
guarantee the existence of ACIMs? We notice that some work has explored the scenario of higherdimensional piecewise expanding systems, see [7] [8]. This line of inquiry could significantly
expand our understanding of dynamical systems in multidimensional contexts, offering insights
into more complex structures.
25
References
1. Lasota, A. & Yorke, J. A. On the Existence of Invariant Measures for Piecewise Monotonic
Transformations. Transactions of the American Mathematical Society 186, 481–488. doi:10.
1090/s00029947197303357581 (1973).
2. Boyarsky, A. & Gora, P. Laws of Chaos: Invariant Measures and Dynamical Systems in One
Dimensions 1st ed (Birkhauser Boston, Boston, 1997). ¨
3. Keller, G. Stochastic Stability in Some Chaotic Dynamical Systems. Monatshefte Fur Mathe ¨
matik 94, 313–333. doi:10.1007/BF01667385 (1982).
4. Pelikan, S. Invariant Densities for Random Maps of the Interval. Transactions of the American
Mathematical Society 282, 813–825. doi:10.1090/s00029947198407227761 (1984).
5. Dunford, N. & Schwartz, J. T. Linear Operators Wiley Classics Library ed (Interscience Publishers, New York, 1988).
6. Marchi, E. & Zo, F. A Generalization of the YosidaKakutani Ergodic Theorem. Studia Mathematica 71, 107–111 (1981).
7. Eslami, P. Inducing schemes for multidimensional piecewise expanding maps. Discrete and
Continuous Dynamical Systems 42, 353–368. doi:10.3934/dcds.2021120 (2022).
8. Buzzi, J., Paccaut, F. & Schmitt, B. Conformal Measures for Multidimensional Piecewise
Invertible Maps. Ergodic Theory and Dynamical Systems 21, 1035–1049 (2001).
26
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Jia, Bining
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Core Title
The existence of absolutely continuous invariant measures for piecewise expanding operators and random maps
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Master of Science
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Applied Mathematics
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202408
Publication Date
08/30/2024
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