Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 17, 2005, 47 – 60

Konstantin B. Igudesman
DYNAMICS OF FINITE-MULTIVALUED TRANSFORMATIONS
(submitted by M. Malakhaltsev)

Abstract. We consider a transformation of a normalized measure space such that the image of any point is a ﬁnite set. We call such a transformation an $m$ -transformation. In this case the orbit of any point looks like a tree. In the study of $m$ -transformations we are interested in the properties of the trees. An $m$ -transformation generates a stochastic kernel and a new measure. Using these objects, we introduce analogies of some main concept of ergodic theory: ergodicity, Koopman and Frobenius-Perron operators etc. We prove ergodic theorems and consider examples. We also indicate possible applications to fractal geometry and give a generalization of our construction.

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2000 Mathematical Subject Classiﬁcation. 37A05, 28D05, 28A80.

Key words and phrases. Ergodic theory, dynamic system, self-similar set.

The author was supported in part by RF Education Ministry and DAAD grant 331 4 00 088.

1. Main deﬁnitions and examples

Throughout the paper $(X, ℬ, μ)$ denotes a normalized measure space. Let $m$ be a positive integer.

Deﬁnition 1. We call a multivalued transformation $S : X \to X$ an $m$ -transformation if $1 \leq ∣ S (x) ∣ \leq m$ for any $x \in X$ , where $∣ A ∣$ is just a number of elements in $A$ .

Let

S_{k; l}^{- 1} (B) \equiv {x \in X : ∣ S (x) ∣ = k, ∣ S (x) \cap B ∣ = l},

where $B \subset X$ and $k, l \in ℕ$ . Note that sets $S_{k; l}^{- 1} (B)$ are pairwise disjoint for the ﬁxed $B$ .

Deﬁnition 2. The $m$ -transformation $S : X \to X$ is measurable if $S_{k; l}^{- 1} (B) \in ℬ$ for all $B \in ℬ$ and $k, l \in ℕ$ .

Let $K : X \times ℬ \to ℝ^{+}$ be the function

K (x, B) \equiv \frac{1}{∣ S (x) ∣} \sum_{y \in S (x)} χ_{B} (y) .

For each $x \in X$ , $K (x, \cdot) : ℬ \to ℝ^{+}$ is a normalized measure and for each $B \in ℬ$ , $K (\cdot, B) : X \to ℝ^{+}$ is measurable by the Deﬁnition 2. Therefore $K$ is a stochastic kernel that describes the $m$ -transformation $S$ . We will use $K$ as a tool for proving some results. Fore a more complete study of stochastic kernels the reader is referred to [5].

For any measurable $m$ -transformation $S$ we deﬁne a new measure $S μ$ on $(X, ℬ, μ)$

S μ (B) \equiv \int_{X} K (x, B) d μ = \sum_{k = 1}^{m} \sum_{l = 1}^{k} \frac{l}{k} μ (S_{k; l}^{- 1} (B)) .

Deﬁnition 3. We say the measurable $m$ -transformation $S : X \to X$ preserves measure $μ$ or that $μ$ is $S$ -invariant if $S μ = μ$ .

Deﬁnition 4. Let the $m$ -transformation $S : X \to X$ preserve measure $μ$ . The quadruple $(X, ℬ, μ, S)$ is called an m-dynamical system.

The next proposition gives a number of examples of $m$ -dynamical systems.

Proposition 1. Let ${S_{i}}_{1}^{k}$ be a ﬁnite collection of the $μ$ -preserving $m_{i}$ -transformations of $(X, ℬ, μ)$ and let $S (x) = ⋃_{i = 1}^{k} S_{i} (x)$ be measurable. Let $K, K_{i}$ be the stochastic kernels that generates $S$ , $S_{i}$ , respectively. If for any $B \in ℬ$

K (x, B) = \frac{1}{k} \sum_{i = 1}^{k} K_{i} (x, B)

(1)

for almost all $x \in X$ , then $S$ is $μ$ -preserving.

$▸$ For any measurable $B$ we have

S μ (B) = \int_{X} K (x, B) d μ = \frac{1}{k} \sum_{i = 1}^{k} \int_{X} P_{i} (x, B) d μ = μ (B) . ◂

In the following examples $λ$ denotes the Lebesgue measure on $[0, 1]$ .

Example 1. Let $S : [0, 1] \to [0, 1]$ be deﬁned by $S (x) = {x, 1 - x}$ . Then $S$ is $λ$ -preserving.

Example 2. Let $S : [0, 1] \to [0, 1]$ be deﬁned by

S (x) = \{\begin{matrix} {2 x, 1 - 2 x}, & x \in [0, \frac{1}{2}] \\ {2 x - 1}, & x \in (\frac{1}{2}, 1] . \end{matrix}

Then $S$ is $λ$ -preserving.

The following example show that not every $λ$ -preserving $m$ -transformation is union of $λ$ -preserving transformations.

Example 3. Let $S : [0, 1] \to [0, 1]$ be deﬁned by

S (x) = \{\begin{matrix} {\frac{3}{2} x}, & x \in [0, \frac{1}{3}) \\ {\frac{3}{2} x, \frac{3}{2} x - \frac{1}{2}}, & x \in [\frac{1}{3}, \frac{2}{3}] \\ {\frac{3}{2} x - \frac{1}{2}}, & x \in (\frac{2}{3}, 1] . \end{matrix}

Then $S$ is $λ$ -preserving, but $S$ can not be represented as union of $λ$ -preserving transformations.

$▸$ Assume $S (x) = \cup_{i = 1}^{k} S_{i} (x)$ , where $S_{i}$ are the $λ$ -preserving transformations. Then there are a measurable set $B \subset [\frac{1}{3}, \frac{2}{3}]$ of positive measure and transformation $S_{i}$ (for instance $S_{1}$ ), such that $S_{1} (B) \subset [0, \frac{1}{2}]$ . We have

λ (S_{1}^{- 1} (S_{1} (B))) = λ (B \cup (B - \frac{1}{3})) = 2 λ (B) and λ (S_{1} (B)) = \frac{3}{2} λ (B) .

Since $S_{1}$ is the $λ$ -preserving transformation, $λ (S_{1} (B)) = λ (B) = 0$ . $◂$

Example 4. Let $S : [0, 1] \to [0, 1]$ be deﬁned by

S (x) = \{\begin{matrix} {2 x, 1 - 2 x, x}, & x \in [0, \frac{1}{2}] \\ {2 x - 1, x}, & x \in (\frac{1}{2}, 1] . \end{matrix}

Then $S$ isn’t $λ$ -preserving.

$▸$ For instance,

S λ ([0, \frac{1}{2}]) = \frac{2}{3} λ ([0, \frac{1}{2}]) + \frac{1}{2} λ ([\frac{1}{2}, \frac{3}{4}]) = \frac{11}{24} \neq λ ([0, \frac{1}{2}]) .

Nevertheless, we can represent $S$ as the union of the $λ$ -preserving transformations $S_{1} (x) = x$ and $S_{2}$ from Example 2. Of course, (1) does not hold true. $◂$

Let $S^{- 1} (B) = {x \in X : S (x) \cap B \neq \emptyset}$ denote the full preimage of $B$ .

Deﬁnition 5. A measurable $m$ -transformation $S : X \to X$ is said to be nonsingular if for any $B \in ℬ$ such that $μ (B) = 0$ , we have $μ (S^{- 1} (B)) = 0$ , i.e., $S μ ≪ μ$ .

2. Recurrence and ergodic theorems

Let $S : X \to X$ be an $m$ -transformation. The $n$ -th iterate of $S$ is denoted by $S^{n}$ . The tree at $x_{0} \in X$ is the set ${x \in X : x \in S^{n} (x_{0}) for some n \geq 0}$ . Any sequence $x_{0}, x_{1}, x_{2}, \dots$ with $x_{n + 1} \in S (x_{n})$ for all $n \geq 0$ is called the orbit of $x_{0}$ .

In the study of $m$ -dynamical systems, we are interested in properties of the trees. For example, in the recurrence of trees of $S$ , i.e., the property that if the tree in $x$ starts in a speciﬁed set, some orbits of $x$ return to that set inﬁnitely many times.

Proposition 2. Let $S$ be a nonsingular $m$ -transformation on $(X, ℬ, μ)$ and let $μ (A) \leq μ (S^{- 1} (A))$ for any $A \in ℬ$ . If $μ (B) > 0$ , then for almost all $x \in B$ there is an orbit of $x$ that returns inﬁnitely often to $B$ .

$▸$ Let $B$ be a measurable set with $μ (B) > 0$ , and let us deﬁne the set $A$ of points that never return to $B$ , i.e., $A = {x \in B : S^{n} (x) \cap B = \emptyset for all n \geq 1} = B ∖ \cup_{n = 1}^{\infty} S^{- n} (B)$ . Consider a collection of sets

A_{1} = A \cup S^{- 1} (A), A_{i} = A \cup S^{- 1} (A_{i - 1}), i \geq 2 .

It is clear that $A \cap S^{- 1} (A_{i - 1}) = \emptyset$ . Hence

μ (A_{i}) = μ (A) + μ (S^{- 1} (A_{i - 1})) \geq μ (A) + μ (A_{i - 1}) \geq \dots \geq (i + 1) μ (A) .

Therefore, $μ (A) = 0$ . Since $μ$ is nonsingular, $μ (S^{- n} (A)) = 0$ for any $n \geq 0$ . This gives $μ (B ∖ ⋃_{n} S^{- n} (A)) = μ (B)$ , and for any $x \in B ∖ ⋃_{n} S^{- n} (A)$ there exists an orbit of $x$ that returns inﬁnitely often to $B$ . $◂$

If $S$ is measure preserving, then we have an analogue of Poincare’s Recurrence Theorem.

Corollary 1. Let $S$ be a measure-preserving $m$ -transformation on $(X, ℬ, μ)$ . If $μ (B) > 0$ , then for almost all $x \in B$ there is an orbit of $x$ that returns inﬁnitely often to $B$ .

$▸$ Note that $S μ ≪ μ$ and for any measurable $A$

μ (A) = S μ (A) = \sum_{k = 1}^{m} \sum_{l = 1}^{k} \frac{l}{k} μ (S_{k; l}^{- 1} (A)) \leq μ (S^{- 1} (A)) . ◂

Example 1 shows there are orbits that do not return to $B$ . If $B = [0, \frac{1}{2})$ , then for any $x \in B$ the orbit ${x, 1 - x, 1 - x, \dots}$ does not return to $B$ .

For any nonsingular $m$ -transformation $S$ and function $f$ on $X$ we deﬁne a new function $U f$ on $X$ by the equality

(U f) (x) \equiv \int_{X} f d K (x, \cdot) = \frac{1}{∣ S (x) ∣} \sum_{y \in S (x)} f (y) .

Proposition 3. If $S$ is a nonsingular $m$ -transformation and $f$ is a real-valued measurable function on $X$ , then

\int_{X} f d S μ = \int_{X} U f d μ,

in the sense that if one of these integrals exists then so does the other integral and the two integrals are equal.

$▸$ We ﬁrst show that $U f$ is measurable. Given any $α \in ℝ$ consider an increasing sequence of rational numbers $α_{1} < \dots < α_{k}$ , where $k \leq m$ and $\sum_{i = 1}^{k} α_{i} < k α$ . Then the set

B_{α_{1}, \dots, α_{k}} = S^{- 1} (f^{- 1} (- \infty, α_{1}]) \cap S^{- 1} (f^{- 1} (α_{1}, α_{2}]) \cap \dots \cap S^{- 1} (f^{- 1} (α_{k - 1}, α_{k}])

is measurable. Taking the union of $B_{α_{1}, \dots, α_{k}}$ for all possible $k \leq m$ and $α_{1}, \dots, α_{k}$ , we conclude that the set ${x : (U f) (x) < α}$ is measurable.

When $f = χ_{B}$ is the characteristic function of $B \in ℬ$ ,

\int_{X} χ_{B} d S μ = S μ (B)

and

\begin{matrix} \int_{X} U χ_{B} d μ = \int_{X} (\int_{X} χ_{B} d K (x, \cdot)) d μ \\ = \int_{X} K (x, B) d μ = S μ (B) . & (2) \end{matrix}

Since $U$ is a linear operator, the formula is also true for simple functions. If $f$ is a nonnegative measurable function, then $f$ is the $S μ$ -pointwise limit of an increasing sequence of simple functions $f_{i}$ , and the result follows from the fact that $U f$ is the $μ$ -pointwise limit of the increasing sequence of functions $U f_{i}$ and the monotone convergence theorem. Finally, any measurable function $f$ can be written as the difference $f = f^{+} - f^{-}$ of two nonnegative measurable functions, so the formula is true in general. $◂$

Corollary 2. Let $S : X \to X$ be a measurable $m$ -transformation on $(X, ℬ, μ)$ . Then $S$ is $μ$ -preserving if and only if

\int_{X} f d μ = \int_{X} U f d μ

for any $f \in ℒ^{1}$ .

$▸$ This follows from the Proposition above and from (2). $◂$

Proposition 4. Let $S : X \to X$ be a $μ$ -preserving $m$ -transformation on $(X, ℬ, μ)$ . Then the positive linear operator $U$ is a contraction on $ℒ^{p}$ for any $1 \leq p \leq \infty$ .

$▸$ It is easily seen that $U$ is a contraction on $ℒ^{\infty}$ . By the Jensen inequality $∣ U f ∣^{p} \leq U ∣ f ∣^{p}$ for any $p \geq 1$ and $f \in ℒ^{p}$ (see [5], Chapter 1, Lemma 7.4 for a more general statement). Then

∥ U f ∥_{p}^{p} = \int_{X} ∣ U f ∣^{p} d μ \leq \int_{X} U ∣ f ∣^{p} d μ = \int_{X} ∣ f ∣^{p} d μ = ∥ f ∥_{p}^{p} . ◂

For a function $f$ on $X$ and an $m$ -transformation $S : X \to X$ , we deﬁne the averages

A_{n} (f) = \frac{1}{n} \sum_{k = 0}^{n - 1} U^{k} f, n = 1, 2, \dots .

From the Birkhoff Ergodic Theorem for Markov operators (see [4] for the details) and from the Proposition above we get the following theorem.

Theorem 1. Suppose $S : (X, ℬ, μ) \to (X, ℬ, μ)$ is a measure preserving $m$ -transformation and $f \in ℒ^{1}$ . Then there exists a function $f^{*} \in ℒ^{1}$ such that

A_{n} (f) \to f^{*}, μ - a . e .

Furthermore, $U f^{*} = f^{*}$ $μ$ -a.e. and $\int_{X} f^{*} d μ = \int_{X} f d μ$ .

Corollary 3. Let $1 \leq p < \infty$ and let $S$ be a measure preserving $m$ -transformation on $(X, ℬ, μ)$ . If $f \in ℒ^{p}$ , then there exists $f^{*} \in ℒ^{p}$ such that $U f^{*} = f^{*}$ $μ$ -a.e. and $∥ f^{*} - A_{n} (f) ∥_{p} \to 0$ as $n \to \infty$ .

$▸$ Let us ﬁx $1 \leq p \leq \infty$ and $f \in ℒ^{p}$ . Since $∥ A_{n} (f) ∥_{p} \leq ∥ f ∥_{p}$ , we have by Fatou’s lemma,

\int_{X} ∣ f^{*} ∣^{p} d μ \leq {liminf}_{n \to + \infty} \int_{X} ∣ A_{n} (f) ∣^{p} d μ \leq \int_{X} ∣ f ∣^{p} d μ .

Hence, the operator $L : ℒ^{p} \to ℒ^{p}$ deﬁned by $L (f) = f^{*}$ is a contraction on $ℒ^{p}$ . By Theorem 1 $∥ f^{*} - A_{n} (f) ∥_{p} \to 0$ as $n \to \infty$ for any bounded function $f \in ℒ^{p}$ . Let $f \in ℒ^{p}$ be a function, not necessarily bounded. For any $ɛ > 0$ we can ﬁnd a bounded function $f_{B} \in ℒ^{p}$ such that $∥ f - f_{B} ∥_{p} < ɛ$ . Then, since $L$ is a contraction on $ℒ^{p}$ , we have

∥ f^{*} - A_{n} (f) ∥_{p} \leq ∥ f_{B}^{*} - A_{n} (f_{B}) ∥_{p} + ∥ A_{n} (f - f_{B}) ∥_{p} + ∥ {(f - f_{B})}^{*} ∥_{p},

which can be made arbitrarily small. $◂$

3. Ergodicity

Assume $U f = f$ for some measurable function $f$ . It is very important to know condition on $S$ under which $f$ is constant.

Deﬁnition 6. We call a nonsingular $m$ -transformation $S$ ergodic if for any $B \in ℬ$ , such that $B ∖ S^{- 1} (B) = B^{c} ∖ S^{- 1} (B^{c}) = \emptyset$ , $μ (B) = 0$ or $μ (B^{c}) = 0$ .

It is obvious that if $S$ is the union of $μ$ -preserving $m$ -transformations (see Proposition 1) one of which is not ergodic, then $S$ is not ergodic.

Theorem 2. The following three statements are equivalent for any nonsingular $m$ -transformation $S : X \to X$ .

$S$ is ergodic
for any $B \in ℬ$ , such that $μ (B ∖ S^{- 1} (B)) = μ (B^{c} ∖ S^{- 1} (B^{c})) = 0$ , $μ (B) = 0$ or $μ (B^{c}) = 0$ .
for any disjoint sets $B_{1}, B_{2} \in ℬ$ , such that $μ (B_{1} ∖ S^{- 1} (B_{1})) = μ (B_{2} ∖ S^{- 1} (B_{2})) = 0$ , $μ (B_{1}) = 0$ or $μ (B_{2}) = 0$ .

$▸$ It is evident that $(3) \Rightarrow (1)$ .

$(1) \Rightarrow (2)$ Suppose $S$ is ergodic and $B \in ℬ$ , such that $μ (B ∖ S^{- 1} (B)) = μ (B^{c} ∖ S^{- 1} (B^{c})) = 0$ . Let $A_{1} = (B \cap S^{- 1} (B)) \cup (B^{c} ∖ S^{- 1} (B^{c})), A_{i} = A_{i - 1} \cap S^{- 1} (A_{i - 1})$ for $i \geq 2$ , and $A = \cap_{i = 1}^{\infty} A_{i}$ . We have $A_{1} \supset A_{2} \supset \dots$ and

A_{i - 1} ∖ A_{i} \subset S^{- 1} (A_{i - 2} ∖ A_{i - 1}) \subset \dots \subset S^{- i + 2} (A_{1} ∖ A_{2}) \subset S^{- i + 1} (B ∖ S^{- 1} (B)) .

Therefore, $μ (A △ B) = 0$ . Let $x \in A$ , then there is at least one point in $S (x)$ that belongs to inﬁnite many of $A_{i}$ . This gives $A \subset S^{- 1} (A)$ .

Let $C_{1} = A^{c}, C_{i} = C_{i - 1} \cap S^{- 1} (C_{i - 1})$ for $i \geq 2$ , and $C = \cap_{i = 1}^{\infty} C_{i}$ . We have $C_{1} \supset C_{2} \supset \dots$ and

C_{i - 1} ∖ C_{i} \subset \dots \subset S^{- i + 2} (C_{1} ∖ C_{2}) \subset S^{- i + 1} (B^{c} ∖ S^{- 1} (B^{c})) \cup S^{- i + 2} (B ∖ A) .

Therefore, $μ (C △ B^{c}) = 0$ . Let $x \in C$ , then there is at least one point in $S (x)$ that belongs to inﬁnite many of $C_{i}$ . This gives $C \subset S^{- 1} (C)$ . Moreover,

C^{c} = A \cup C_{1} ∖ C \subset S^{- 1} (A) \cup S^{- 1} (C_{1} ∖ C) \cup S^{- 1} (A) = S^{- 1} (C^{c}) .

We conclude from the ergodicity of $S$ that $μ (B^{c}) = μ (C) = 0$ or $μ (B) = μ (C^{c}) = 0$ .

$(2) \Rightarrow (3)$ Suppose $(2)$ holds true and let $B_{1}, B_{2} \in ℬ$ be the disjoint sets, such that $μ (B_{1} ∖ S^{- 1} (B_{1})) = μ (B_{2} ∖ S^{- 1} (B_{2})) = 0$ . Let $C_{1} = B_{1}^{c}, C_{i} = C_{i - 1} \cap S^{- 1} (C_{i - 1})$ for $i \geq 2$ , and $C = \cap_{i = 1}^{\infty} C_{i}$ . We have $C_{1} \supset C_{2} \supset \dots$ and $μ (B_{2} ∖ C_{i}) = 0$ . Therefore $μ (C) \geq μ (B_{2})$ . Let $x \in C$ , then there is at least one point in $S (x)$ that belongs to inﬁnite many of $C_{i}$ . This gives $C \subset S^{- 1} (C)$ . Moreover $μ (C^{c} ∖ S^{- 1} (C^{c})) = 0$ and $μ (C^{c}) \geq μ (B_{1})$ . By assumption $μ (C) = 0$ or $μ (C^{c}) = 0$ . This ﬁnishes the proof. $◂$

Example 5. We will prove the ergodisity of

S (x) = \{\begin{matrix} {2 x, 1 - 2 x}, & x \in [0, \frac{1}{2}] \\ {2 x - 1}, & x \in (\frac{1}{2}, 1] . \end{matrix}

$▸$ Let

B \subset S^{- 1} (B) and B^{c} \subset S^{- 1} (B^{c}) .

(3)

Set $A_{1} = {x : {x, 1 - x} \subset B}$ , $A_{2} = {x : {x, 1 - x} \subset B^{c}}$ and $A_{3} = {(A_{1} \cup A_{2})}^{c}$ .

Let $x \in A_{1}$ . By (3)

\frac{1 + x}{2} \in B, \frac{2 - x}{2} \in B, \frac{1 - x}{2} \in B, \frac{x}{2} \in B .

Therefore ${\bar{S}}^{- 1} (A_{1}) \subset A_{1}$ , where $\bar{S}$ is the well known ergodic single-valued transformation $\bar{S} (x) = 2 x (mod 1)$ , $x \in [0, 1]$ . By ergodicity of $\bar{S}$ , $λ (A_{1}) = 0$ or $λ (A_{1}) = 1$ . Similarly, $λ (A_{2}) = 0$ or $λ (A_{2}) = 1$ .

Since $λ (A_{1}) = 1$ leads to $λ (B^{c}) = 0$ and $λ (A_{2}) = 1$ leads to $λ (B) = 0$ , we need only consider

λ (A_{3}) = 1 .

(4)

Let $x \in B$ . By (3) and (4)

\frac{1 + x}{2} \in B, \frac{2 - x}{2} \in B^{c} a.s., \frac{1 - x}{2} \in B^{c} a.s., \frac{x}{2} \in B a.s.

Therefore $λ ({\bar{S}}^{- 1} (B) ∖ B)$ . By ergodicity of $\bar{S}$ , $λ (B) = 0$ or $λ (B) = 1$ . $◂$

Example 6. The 2-transformation $S : [0, 1] \to [0, 1]$

S (x) = \{\begin{matrix} {\frac{3}{2} x}, & x \in [0, \frac{1}{3}) \\ {\frac{3}{2} x, \frac{3}{2} x - \frac{1}{2}}, & x \in [\frac{1}{3}, \frac{2}{3}] \\ {\frac{3}{2} x - \frac{1}{2}}, & x \in (\frac{2}{3}, 1] . \end{matrix}

is not ergodic.

$▸$ For instance, $[0, \frac{1}{2}) \subset S^{- 1} ([0, \frac{1}{2})) and [\frac{1}{2}, 1] \subset S^{- 1} ([\frac{1}{2}, 1]) . ◂$

Proposition 5. Let $S$ be ergodic. If $f$ is measurable and $(U f) (x) = f (x)$ a.e., then $f$ is constant a.e.

$▸$ For each $r \in ℝ$ , $E_{r} = {x \in X : (U f) (x) = f (x) > r}$ is measurable. Then $E_{r} \subset S^{- 1} (E_{r})$ and $E_{r}^{c} \subset S^{- 1} (E_{r}^{c})$ , hence $E_{r}$ has measure 0 or 1. But if $f$ is not constant a.e., there exists an $r \in ℝ$ such that $0 < μ (E_{r}) < 1$ . Therefore $f$ must be constant a.e. $◂$

Corollary 4. If a measure preserving $m$ -transformation $S$ is ergodic and $f \in ℒ^{1}$ , then the limit of the averages $f^{*} = \int_{X} f d μ$ is constant a.e. Thus, if $μ (B) > 0$ , then for almost all $x \in X$ there is a orbit of $x$ that returns inﬁnitely often to $B$ .

$▸$ We conclude from Theorem 1 and from Proposition 5, that $f^{*} = \int_{X} f d μ$ . To prove the second statement we consider $f = χ_{B}$ and apply Corollary 1. $◂$

Corollary 5. Let measure preserving $m$ -transformation $S$ be ergodic and $μ (S_{11}^{- 1} (X)) < 1$ , i.e., the set ${x \in X : ∣ S (x) ∣ \geq 2}$ has positive measure. If $μ (B) > 0$ , then for almost all $x \in X$ there are uncountable many orbits of $x$ that return inﬁnitely often to $B$ .

$▸$ We just apply the Corollary above to the sets $B$ and $S_{11}^{- 1} {(X)}^{c}$ . $◂$

Corollary 6. Let $S$ be a measure preserving ergodic $m$ -transformation and $f \in ℒ^{1}$ such that $f (x) \geq f (y) (f (x) \leq f (y))$ , for any $y \in S (x)$ . Then $f$ is constant a.e.

$▸$ We have $U f \leq f$ , hence the limit of averages $f^{*} \leq f$ . By Corollary 4 $f = f^{*}$ is constant a.e. $◂$

4. The Frobenius-Perron operator

Assume that a nonsingular $m$ -transformation $S : X \to X$ on a normalized measure space is given. We deﬁne an operator $P : ℒ^{1} \to ℒ^{1}$ in two steps.

1. Let $f \in ℒ^{1}$ and $f \geq 0$ . Write

ν (B) = \int_{X} f (x) K (x, B) d μ .

Then, by the Radon-Nikodym Theorem, there exists a unique element in $ℒ^{1}$ , which we denoted by $P f$ , such that

ν (B) = \int_{B} P f d μ .

2. Now let $f \in ℒ^{1}$ be arbitrary, not necessarily nonnegative. Write $f = f^{+} - f^{-}$ and deﬁne $P f = P f^{+} - P f^{-}$ . From this deﬁnition we have

\int_{B} P f d μ = \int_{X} f^{+} (x) K (x, B) d μ - \int_{X} f^{-} (x) K (x, B) d μ

or, more completely,

\int_{B} P f d μ = \int_{X} f (x) K (x, B) d μ .

(5)

Deﬁnition 7. If $S : X \to X$ is a nonsingular $m$ -transformation the unique operator $P : ℒ^{1} \to ℒ^{1}$ deﬁned by equation (5) is called the Frobenius-Perron operator corresponding to $S$ .

It is straightforward to show that $P$ is a positive linear operator and

\int_{X} P f d μ = \int_{X} f d μ .

Proposition 6. If $f \in ℒ^{1}$ and $g \in ℒ^{\infty}$ , then $〈 P f, g 〉 = 〈 f, U g 〉$ , i.e.,

\int_{X} (P f) \cdot g d μ = \int_{X} f \cdot (U g) d μ .

(6)

$▸$ Let $B$ be a measurable subset of $X$ and $g = χ_{B}$ . Then the left hand side of (6) is

\int_{B} P f d μ = \int_{X} f (x) K (x, B) d μ

and the right hand side is

\int_{X} f \cdot (U χ_{B}) d μ = \int_{X} f \cdot (\int_{X} χ_{B} d K (x, \cdot)) d μ = \int_{X} f (x) K (x, B) d μ .

Hence (6) is veriﬁed for characteristic functions. Since the linear combinations of characteristic functions are dense in $ℒ^{\infty}$ , (6) holds for all $f \in ℒ^{1}$ and $g \in ℒ^{\infty}$ . $◂$

The following proposition says that a density $f_{*}$ is a ﬁxed point of $P$ if and only if it is a density of an $S$ -invariant measure $ν$ , absolutely continuous with respect to a measure $μ$ .

Proposition 7. Let $S : X \to X$ be nonsingular and let $f_{*} \in ℒ^{1}$ be a density function on $(X, ℬ, μ)$ . Then $P f_{*} = f_{*}$ a.e., if and only if the measure $ν = f_{*} \cdot μ$ , deﬁned by $ν (B) = \int_{B} f_{*} d μ$ , is $S$ -invariant.

$▸$ Let $B \subset X$ be measurable. Then

S ν (B) = \int_{X} K (x, B) d ν = \int_{X} f_{*} (x) K (x, B) d μ = \int_{B} P f_{*} d μ .

On the other hand

ν (B) = \int_{B} f_{*} d μ . ◂

Proposition 8. Let $S : X \to X$ be a nonsingular $m$ -transformation and $P$ the associated Frobenius-Perron operator. Assume that an $f \geq 0, f \in ℒ^{1}$ is given. Then

s u p p f \subset S^{- 1} (s u p p P f) a.s.

$▸$ By the deﬁnition of the Frobenius-Perron operator, we have $P f (x) = 0$ a.e. on $B$ implies that $f (x) = 0$ for a.a. $x \in S^{- 1} (B)$ . Now setting $B = {(s u p p f)}^{c}$ , we have $P f (x) = 0$ for a.a. $x \in B$ and, consequently, $f (x) = 0$ for a.a. $x \in S^{- 1} (B)$ , which means that $s u p p f \subset {(S^{- 1} (B))}^{c}$ . Since ${(S^{- 1} (B))}^{c} \subset S^{- 1} (B^{c})$ a.s., this completes the proof. $◂$

Proposition 9. Let $S : X \to X$ be a nonsingular $m$ -transformation and $P$ the associated Frobenius-Perron operator. If $S$ is ergodic, then there is at most one stationary density $f_{*}$ of $P$ .

$▸$ Assume that $S$ is ergodic and that $f_{1}$ and $f_{2}$ are different stationary densities of $P$ . Set $g = f_{1} - f_{2}$ , so that $P g = g$ . Since $P$ is a Markov operator, $g^{+}$ and $g^{-}$ are both stationary densities of $P$ . By assumption, $f_{1}$ and $f_{2}$ are not only different but are also densities we have $g^{+} ≢ 0$ and $g^{-} ≢ 0$ . Set

B_{1} = s u p p g^{+} and B_{2} = s u p p g^{-} .

It is evident that $B_{1}$ and $B_{2}$ are disjoint sets and both have positive measure. By Proposition 8, we have

B_{1} \subset S^{- 1} (B_{1}) a.s. and B_{2} \subset S^{- 1} (B_{2}) a.s.

But, from Theorem 2 it follows that $μ (B_{1}) = 0$ or $μ (B_{2}) = 0$ . $◂$

5. Applications and generalization

We now apply the method of $m$ -transformation to the intersection of two middle- $β$ Cantor sets (see [8] and the references given there).

Let $α \in [\frac{1}{3}, \frac{2}{3}]$ and $ψ_{1} (x) = α x$ , $ψ_{1} (x) = α x + 1 - α$ be contracting similarity maps on $I = [0, 1]$ endowed with Lebesgue measure $λ$ . There is a unique compact set $C_{α} \subset I$ which satisﬁes the set equation

C_{α} = ψ_{1} (C_{α}) \cup ψ_{2} (C_{α}) .

It is easily checked that $C_{α}$ is the middle- $β$ Cantor set for $β = 1 - 2 α$ . Let $x \in I$ and $f (x) = d i m_{H} (C_{α} \cap (C_{α} + x))$ denotes the Hausdorff dimension of the set $C_{α} \cap (C_{α} + x)$ . Let $B_{i j} = ψ_{i} (C_{α}) \cap ψ_{j} (C_{α} + x), i, j = 1, 2$ . From the construction of $C_{α}$ it follows that $B_{12} = \emptyset$ ,

d i m_{H} B_{11} = d i m_{H} B_{22} = \{\begin{matrix} f (\frac{x}{α}), & 0 \leq x \leq α \\ 0, & α < x \leq 1 \end{matrix}

and

d i m_{H} B_{21} = \{\begin{matrix} 0, & 0 \leq x < 1 - 2 α \\ f (- \frac{x}{α} + \frac{1}{α} - 1), & 1 - 2 α \leq x < 1 - α \\ f (\frac{x}{α} - \frac{1}{α} + 1), & 1 - α \leq x \leq 1 . \end{matrix}

Since $C_{α} \cap (C_{α} + x) = B_{11} \cup B_{21} \cup B_{22}$ , we have

f (x) = max {d i m_{H} B_{i j} : i, j = 1, 2} = max {f (y) : y \in S (x)},

(7)

where

S (x) = \{\begin{matrix} {\frac{x}{α}}, & 0 \leq x < 1 - 2 α \\ {\frac{x}{α}, - \frac{x}{α} + \frac{1}{α} - 1}, & 1 - 2 α \leq x \leq α \\ {- \frac{x}{α} + \frac{1}{α} - 1}, & α < x \leq 1 - α \\ {\frac{x}{α} - \frac{1}{α} + 1}, & 1 - α < x \leq 1 \end{matrix}

(compare with Examples 2 and 5 under $α = \frac{1}{2}$ ).

Using Leibniz’s rule, we ﬁnd the Frobenius-Perron operator corresponding to $S$ :

(P f) (x) = \{\begin{matrix} α (f (1 - α + α x) + f (1 - α - α x) + f (α x)), & 0 \leq x < \frac{1}{α} - 2 \\ α (f (1 - α + α x) + \frac{1}{2} f (1 - α - α x) + \frac{1}{2} f (α x)), & \frac{1}{α} - 2 \leq x \leq 1 . \end{matrix}

Assume there exist a stable point $f_{*}$ of $P$ . Then by Proposition 7 the measure $μ = f_{*} \cdot λ$ is $S$ -invariant. If in addition $S : (I, ℬ, μ) \to (I, ℬ, μ)$ is ergodic, then by (7) and Corollary 6 $f$ is constant $μ$ -a.e. The same method works in case of the intersection of two arbitrary self-similar sets.

Using $m$ -transformations, we can develop a new approach to the self-similar sets with overlaps (see [2], [7]). Let $ψ_{1}, \dots, ψ_{m}$ be contracting similarity maps on $ℝ^{n}$ , and let $X = \cup_{i = 1}^{m} ψ_{i} (X)$ be an attractor of the iterated function system. Given normalized measure $μ$ on $X$ we consider $m$ -transformation of $X$

S (x) = ⋃_{{i : x \in ψ_{i} (X)}} ψ_{i}^{- 1} (x) .

Assume, using the Frobenius-Perron operator corresponding $S$ , we have found $S$ -invariant ergodic measure on $X$ . This measure gives us an interesting information about $X$ . For instance, if the conditions of Corollary 5 hold true, we see that a.a. points of $X$ have uncountable many of addresses (see [3] for details).

From these examples we see that the main problem of the investigation is to ﬁnd an $S$ -invariant ergodic measure. To decide this problem we propose a following generalization of an $m$ -transformation.

Given $m$ -transformation $S$ on a normalized measure space $(X, ℬ, μ)$ we consider a collection of pairs ${S_{i}, α_{i}}_{i = 1}^{m}$ , where $S_{i} : X \to X$ are the single-valued measurable transformations such that $S (x) = \cup_{i = 1}^{m} S_{i} (x)$ for any $x \in X$ , and $α_{i} : X \to [0, 1]$ are the measurable functions such that $\sum_{i = 1}^{m} α_{i} (x) = 1$ for any $x \in X$ . Let us consider the stochastic kernel

K (x, B) = \sum_{i = 1}^{m} α_{i} (x) χ_{B} (S_{i} (x))

and a new measure on $X$

S μ (B) \equiv \int_{X} K (x, B) d μ .

If we choose $S_{i}$ and $α_{i}$ such that $S μ = μ$ , we can employ the results of this paper to the measure preserving transformation $S$ .
Acknowledgement
We are very grateful to Christoph Bandt for useful discussions on properties of the measure $S μ$ generated by the stochastic kernel.

References

[1] Boyarsky, A. and Gora, P. (1997), Laws of Chaos. Invariant Measures and Dynamical Systems in one Dimension, Birkhauser, Boston.

[2] Broomhead, D., Montaldi, J. and Sidorov, N. (2004), Golden gaskets: variations on the Sierpinski sieve, Nonlinearity, 17, 1455 – 1480.

[3] Falconer, K.J. (1990), Fractal Geometry, Wiley.

[4] Foguel, S. (1980), Selected Topics in the Study of Markov Operators, Carolina Lecture Series, Department of Mathematics, University of North Carolina.

[5] Krengel, V. (1985), Ergodic Theorems, Walter de Gruyter, N. York.

[6] Lasota, A. and Mackey, M. (1994), Chaos, Fractals, and Noise, Appl. Math. Sci. 97, Springer-Verlag, New York.

[7] Ngai, S.M. and Wang, Y. (2001), Hausdorff dimension of self-similar sets with overlaps, J. Lond. Math. Soc. 63, 655 – 672.

[8] Peres, Y. and Solomyak, B. (1998), Self-similar measures and intersections of Cantor sets, Trans. Amer. Math. Soc. 350, 4065 – 4087.

Kazan State University, Kazan, 420008, Russia

E-mail address: Konstantin.Igudesman@ksu.ru

Received December 8, 2004