Lobachevskii Journal of Mathematics

http://ljm.ksu.ru

Vol. 17, 2005, 47 – 60 \begin{matrix}c \\ ◯ \end{matrix} K. Igudesman

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.

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 → X an mtransformation if 1 ≤ |S(x)| ≤ m for any x ∈ X, where |A| is just a number of elements in A.

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.

Let

S_{k;l}^{−1}(B) ≡ {x ∈ X : |S(x)| = k, |S(x) ∩ B| = l},

where B ⊂ X and k, l ∈ ℕ. Note that sets S_{k;l}^{−1}(B) are pairwise disjoint for the fixed B.

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

Let K : X × B → ℝ^{+} be the function

K(x, B) ≡ \frac{1}{|S(x)|} ∑_{y∈S(x)} χ_{B}(y) .

For each x ∈ X, K(x, ·) : B → ℝ^{+} is a normalized measure and for each B ∈ B, K(·, B) : X → 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 Sμ on (X, B, μ)

Sμ(B) ≡ ∫_{X} K(x, B) dμ = ∑_{k=1}^{m} ∑_{l=1}^{k} \frac{l}{k} μ(S_{k;l}^{−1}(B)).

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

Definition 4. Let the m-transformation S : X → 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 ∈ B

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

for almost all x ∈ X, then S is μ-preserving.

▶ For any measurable B we have

Sμ(B) = ∫_{X} K(x, B) dμ = \frac{1}{k} ∑_{i=1}^{k} ∫_{X} P_{i}(x, B) dμ = μ(B). ◀

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

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

Example 2. Let S : [0, 1] → [0, 1] be defined by

S(x) = \{ \begin{matrix}\{2x, 1 − 2x\}, & x ∈ [0, \frac{1}{2}] \\ \{2x − 1\}, & x ∈ (\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] → [0, 1] be defined by

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

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

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

λ(S_{1}^{−1}(S_{1}(B))) = λ(B ∪ (B − \frac{1}{3})) = 2λ(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] → [0, 1] be defined by

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

Then S isn’t λ-preserving.

▶ For instance,

Sλ([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} ≠ 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 ∈ X : S(x) ∩ B ≠ ∅} denote the full preimage of B.

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

2. Recurrence and ergodic theorems

Let S : X → X be an m-transformation. The n-th iterate of S is denoted by S^{n}. The tree at x_{0} ∈ X is the set {x ∈ X : x ∈ S^{n}(x_{0}) for some n ≥ 0}. Any sequence x_{0}, x_{1}, x_{2}, … with x_{n+1} ∈ S(x_{n}) for all n ≥ 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) ≤ μ(S^{−1}(A)) for any A ∈ B. If μ(B) > 0, then for almost all x ∈ 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 ∈ B : S^{n}(x) ∩ B = ∅ for all n ≥ 1} = B\ ∪_{n=1}^{∞} S^{−n}(B). Consider a collection of sets

A_{1} = A ∪ S^{−1}(A), A_{i} = A ∪ S^{−1}(A_{i−1}), i ≥ 2 .

It is clear that A ∩ S^{−1}(A_{i−1}) = ∅. Hence

μ(A_{i}) = μ(A) + μ(S^{−1}(A_{i−1})) ≥ μ(A) + μ(A_{i−1}) ≥ … ≥ (i + 1)μ(A) .

Therefore, μ(A) = 0. Since μ is nonsingular, μ(S^{−n}(A)) = 0 for any n ≥ 0. This gives μ(B\ ⋃_{n} S^{−n}(A)) = μ(B), and for any x ∈ 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 ∈ B there is an orbit of x that returns infinitely often to B.

▶ Note that Sμ ≪ μ and for any measurable A

μ(A) = Sμ(A) = ∑_{k=1}^{m} ∑_{l=1}^{k} \frac{l}{k} μ(S_{k;l}^{−1}(A)) ≤ 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 ∈ B the orbit {x, 1 − x, 1 − x, …} does not return to B.

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

(Uf)(x) ≡ ∫_{X} f dK(x, ·) = \frac{1}{|S(x)|} ∑_{y∈S(x)} f(y) .

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

∫_{X} f dSμ = ∫_{X} Uf dμ ,

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 Uf is measurable. Given any α ∈ ℝ consider an increasing sequence of rational numbers α_{1} < … < α_{k}, where k ≤ m and ∑_{i=1}^{k} α_{i} < kα. Then the set

B_{α_{1},...,α_{k}} = S^{−1}(f^{−1}(−∞, α_{1}])∩S^{−1}(f^{−1}(α_{1}, α_{2}])∩…∩S^{−1}(f^{−1}(α_{k−1}, α_{k}])

is measurable. Taking the union of B_{α_{1},...,α_{k}} for all possible k ≤ m and α_{1}, …, α_{k}, we conclude that the set {x : (Uf)(x) < α} is measurable.

When f = χ_{B} is the characteristic function of B ∈ B,

∫_{X} χ_{B} dSμ = Sμ(B)

and

∫_{X} Uχ_{B} dμ = ∫_{X} (∫_{X} χ_{B} dK(x, ·)) dμ

= ∫_{X} K(x, B) dμ = Sμ(B) . (2)

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 Sμ-pointwise limit of an increasing sequence of simple functions f_{i}, and the result follows from the fact that Uf is the μ-pointwise limit of the increasing sequence of functions Uf_{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 → X be a measurable m-transformation on (X, B, μ). Then S is μ-preserving if and only if

∫_{X} f dμ = ∫_{X} Uf dμ

for any f ∈ L^{1}.

▶ This follows from the Proposition above and from (2). ◀

Proposition 4. Let S : X → 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 ≤ p ≤ ∞.

▶ It is easily seen that U is a contraction on L^{∞}. By the Jensen inequality |Uf|^{p} ≤ U|f|^{p} for any p ≥ 1 and f ∈ L^{p} (see [5], Chapter 1, Lemma 7.4 for a more general statement). Then

∥Uf∥_{p}^{p} = ∫_{X} |Uf|^{p} dμ ≤ ∫_{X} U|f|^{p} dμ = ∫_{X} |f|^{p} dμ = ∥f∥_{p}^{p} . ◀

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

A_{n}(f) = \frac{1}{n} ∑_{k=0}^{n−1} U^{k}f, n = 1, 2, ….

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) → (X, B, μ) is a measure preserving m-transformation and f ∈ L^{1}. Then there exists a function f^{∗} ∈ L^{1} such that

A_{n}(f) → f^{∗}, μ − a.e.

Furthermore, Uf^{∗} = f^{∗} μ-a.e. and ∫_{X} f^{∗} dμ = ∫_{X} f dμ.

Corollary 3. Let 1 ≤ p < ∞ and let S be a measure preserving mtransformation on (X, B, μ). If f ∈ L^{p}, then there exists f^{∗} ∈ L^{p} such that Uf^{∗} = f^{∗} μ-a.e. and ∥f^{∗} − A_{n}(f)∥_{p} → 0 as n → ∞.

▶ Let us fix 1 ≤ p ≤ ∞ and f ∈ L^{p}. Since ∥A_{n}(f)∥_{p} ≤ ∥f∥_{p}, we have by Fatou’s lemma,

∫_{X} |f^{∗}|^{p} dμ ≤ lim inf_{n→+∞} ∫_{X} |A_{n}(f)|^{p} dμ ≤ ∫_{X} |f|^{p} dμ .

Hence, the operator L : L^{p} → L^{p} defined by L(f) = f^{∗} is a contraction on L^{p}. By Theorem 1 ∥f^{∗} − A_{n}(f)∥_{p} → 0 as n → ∞ for any bounded function f ∈ L^{p}. Let f ∈ L^{p} be a function, not necessarily bounded. For any ε > 0 we can find a bounded function f_{B} ∈ L^{p} such that ∥f −f_{B}∥_{p} < ε. Then, since L is a contraction on L^{p}, we have

∥f^{∗} − A_{n}(f)∥_{p} ≤ ∥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 Uf = 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 ∈ B, such that B\S^{−1}(B) = B^{c}\S^{−1}(B^{c}) = ∅, 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 → X.

(1) S is ergodic

(2) for any B ∈ B, such that μ(B\S^{−1}(B)) = μ(B^{c}\S^{−1}(B^{c})) = 0, μ(B) = 0 or μ(B^{c}) = 0.

(3) for any disjoint sets B_{1}, B_{2} ∈ 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)⇒(1).

(1)⇒(2) Suppose S is ergodic and B ∈ B, such that μ(B\S^{−1}(B)) = μ(B^{c}\S^{−1}(B^{c})) = 0. Let A_{1} = (B ∩ S^{−1}(B)) ∪ (B^{c}\S^{−1}(B^{c})), A_{i} = A_{i−1} ∩ S^{−1}(A_{i−1}) for i ≥ 2, and A = ∩_{i=1}^{∞}A_{i}. We have A_{1} ⊃ A_{2} ⊃ … and

A_{i−1}\A_{i} ⊂ S^{−1}(A_{i−2}\A_{i−1}) ⊂ … ⊂ S^{−i+2}(A_{1}\A_{2}) ⊂ S^{−i+1}(B\S^{−1}(B)) .

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

Let C_{1} = A^{c}, C_{i} = C_{i−1} ∩ S^{−1}(C_{i−1}) for i ≥ 2, and C = ∩_{i=1}^{∞}C_{i}. We have C_{1} ⊃ C_{2} ⊃ … and

C_{i−1}\C_{i} ⊂ … ⊂ S^{−i+2}(C_{1}\C_{2}) ⊂ S^{−i+1}(B^{c}\S^{−1}(B^{c})) ∪ S^{−i+2}(B\A) .

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

C^{c} = A ∪ C_{1}\C ⊂ S^{−1}(A) ∪ S^{−1}(C_{1}\C) ∪ 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)⇒(3) Suppose (2) holds true and let B_{1}, B_{2} ∈ 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} ∩ S^{−1}(C_{i−1}) for i ≥ 2, and C = ∩_{i=1}^{∞}C_{i}. We have C_{1} ⊃ C_{2} ⊃ … and μ(B_{2}\C_{i}) = 0. Therefore μ(C) ≥ μ(B_{2}). Let x ∈ C, then there is at least one point in S(x) that belongs to infinite many of C_{i}. This gives C ⊂ S^{−1}(C). Moreover μ(C^{c}\S^{−1}(C^{c})) = 0 and μ(C^{c}) ≥ μ(B_{1}). By assumption μ(C) = 0 or μ(C^{c}) = 0. This finishes the proof. ◀

Example 5. We will prove the ergodicity of

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

▶ Let

B ⊂ S^{−1}(B) and B^{c} ⊂ S^{−1}(B^{c}) .  (3)

Set A_{1} = {x : {x, 1 − x} ⊂ B}, A_{2} = {x : {x, 1 − x} ⊂ B^{c}} and A_{3} = (A_{1} ∪ A_{2})^{c}.

Let x ∈ A_{1}. By (3)

\frac{1 + x}{2} ∈ B , \frac{2 − x}{2} ∈ B , \frac{1 − x}{2} ∈ B , \frac{x}{2} ∈ B .

Therefore S̄^{−1}(A_{1}) ⊂ A_{1}, where S̄ is the well known ergodic singlevalued transformation S̄(x) = 2x (mod 1), x ∈ [0, 1]. By ergodicity of S̄, 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 ∈ B. By (3) and (4)

\frac{1 + x}{2} ∈ B , \frac{2 − x}{2} ∈ B^{c} a.s., \frac{1 − x}{2} ∈ B^{c} a.s., \frac{x}{2} ∈ B a.s.

Therefore λ(S̄^{−1}(B)\B). By ergodicity of S̄, λ(B) = 0 or λ(B) = 1. ◀

Example 6. The 2-transformation S : [0, 1] → [0, 1]

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

is not ergodic.

▶ For instance, [0, \frac{1}{2}) ⊂ S^{−1}([0, \frac{1}{2})) and [\frac{1}{2}, 1] ⊂ S^{−1}([\frac{1}{2}, 1]) . ◀

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

▶ For each r ∈ ℝ, E_{r} = {x ∈ X : (Uf)(x) = f(x) > r} is measurable. Then E_{r} ⊂ S^{−1}(E_{r}) and E_{r}^{c} ⊂ 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 ∈ 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 ∈ L^{1}, then the limit of the averages f^{∗} = ∫_{X} f dμ is constant a.e. Thus, if μ(B) > 0, then for almost all x ∈ X there is a orbit of x that returns infinitely often to B.

▶ We conclude from Theorem 1 and from Proposition 5, that f^{∗} = ∫_{X} f dμ. 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 ∈ X : |S(x)| ≥ 2} has positive measure. If μ(B) > 0, then for almost all x ∈ 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 ∈ L^{1} such that f(x) ≥ f(y)(f(x) ≤ f(y)), for any y ∈ S(x). Then f is constant a.e.

▶ We have Uf ≤ f, hence the limit of averages f^{∗} ≤ f. By Corollary 4 f = f^{∗} is constant a.e. ◀

4. The Frobenius-Perron operator

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

1. Let f ∈ L^{1} and f ≥ 0. Write

ν(B) = ∫_{X} f(x)K(x, B) dμ .

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

ν(B) = ∫_{B} P f dμ .

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

∫_{B} P f dμ = ∫_{X} f^{+}(x)K(x, B) dμ − ∫_{X} f^{−}(x)K(x, B) dμ

or, more completely,

∫_{B} P f dμ = ∫_{X} f(x)K(x, B) dμ .  (5)

Definition 7. If S : X → X is a nonsingular m-transformation the unique operator P : L^{1} → 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

∫_{X} P f dμ = ∫_{X} f dμ .

Proposition 6. If f ∈ L^{1} and g ∈ L^{∞}, then ⟨P f, g⟩ = ⟨f, Ug⟩, i.e.,

∫_{X} (P f) · g dμ = ∫_{X} f · (Ug) dμ .  (6)

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

∫_{B} P f dμ = ∫_{X} f(x)K(x, B) dμ

and the right hand side is

∫_{X} f · (Uχ_{B}) dμ = ∫_{X} f · (∫_{X} χ_{B} dK(x, ·) ) dμ = ∫_{X} f(x)K(x, B) dμ .

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

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 → X be nonsingular and let f_{∗} ∈ L^{1} be a density function on (X, B, μ). Then P f_{∗} = f_{∗} a.e., if and only if the measure ν = f_{∗} · μ, defined by ν(B) = ∫_{B} f_{∗} dμ, is S-invariant.

▶ Let B ⊂ X be measurable. Then

Sν(B) = ∫_{X} K(x, B) dν = ∫_{X} f_{∗}(x)K(x, B) dμ = ∫_{B} P f_{∗} dμ .

On the other hand

ν(B) = ∫_{B} f_{∗} dμ . ◀

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

supp f ⊂ S^{−1}(supp 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 ∈ S^{−1}(B). Now setting B = (supp f)^{c}, we have P f(x) = 0 for a.a. x ∈ B and, consequently, f(x) = 0 for a.a. x ∈ S^{−1}(B), which means that supp f ⊂ (S^{−1}(B))^{c}. Since (S^{−1}(B))^{c} ⊂ S^{−1}(B^{c}) a.s., this completes the proof. ◀

Proposition 9. Let S : X → 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} = supp g^{+} and B_{2} = supp 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} ⊂ S^{−1}(B_{1}) a.s. and B_{2} ⊂ 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 ∈ [\frac{1}{3}, \frac{2}{3}] and ψ_{1}(x) = αx, ψ_{1}(x) = αx + 1 − α be contracting similarity maps on I = [0, 1] endowed with Lebesgue measure λ. There is a unique compact set C_{α} ⊂ I which satisfies the set equation

C_{α} = ψ_{1}(C_{α}) ∪ ψ_{2}(C_{α}) .

It is easily checked that C_{α} is the middle-β Cantor set for β = 1−2α. Let x ∈ I and f(x) = dim_{H}(C_{α} ∩ (C_{α} + x)) denotes the Hausdorff dimension of the set C_{α} ∩(C_{α} +x). Let B_{ij} = ψ_{i}(C_{α})∩ψ_{j}(C_{α} +x), i, j = 1, 2. From the construction of C_{α} it follows that B_{12} = ∅,

dim_{H}B_{11} = dim_{H}B_{22} = \{ \begin{matrix}f(\frac{x}{α}) , & 0 ≤ x ≤ α \\ 0 , & α < x ≤ 1 \end{matrix}

and

dim_{H}B_{21} = \{ \begin{matrix}0 , & 0 ≤ x < 1 − 2α \\ f(−\frac{x}{α} + \frac{1}{α} − 1) , & 1 − 2α ≤ x < 1 − α \\ f(\frac{x}{α} − \frac{1}{α} + 1) , & 1 − α ≤ x ≤ 1 . \end{matrix}

Since C_{α} ∩ (C_{α} + x) = B_{11} ∪ B_{21} ∪ B_{22}, we have

f(x) = max{dim_{H}B_{ij} : i, j = 1, 2} = max{f(y) : y ∈ S(x)} , (7)

where

S(x) = \{ \begin{matrix}\{\frac{x}{α}\} , & 0 ≤ x < 1 − 2α \\ \{\frac{x}{α}, −\frac{x}{α} + \frac{1}{α} − 1\} , & 1 − 2α ≤ x ≤ α \\ \{−\frac{x}{α} + \frac{1}{α} − 1\} , & α < x ≤ 1 − α \\ \{\frac{x}{α} − \frac{1}{α} + 1\} , & 1 − α < x ≤ 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:

(Pf)(x)=\{ ; \begin{matrix}α(f(1 − α + αx) + f(1 − α − αx) + f(αx)), & 0 ≤ x < \frac{1}{α} − 2 \\ α(f(1 − α + αx) + \frac{1}{2}f(1 − α − αx) + \frac{1}{2}f(αx)), & \frac{1}{α} − 2 ≤ x ≤ 1. \end{matrix}

Assume there exist a stable point f_{∗} of P. Then by Proposition 7 the measure μ = f_{∗} · λ is S-invariant. If in addition S : (I, B, μ) → (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 selfsimilar sets with overlaps (see [2], [7]). Let ψ_{1}, …, ψ_{m} be contracting similarity maps on ℝ^{n}, and let X = ∪_{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∈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 → X are the single-valued measurable transformations such that S(x) = ∪_{i=1}^{m}S_{i}(x) for any x ∈ X, and α_{i} : X → [0, 1] are the measurable functions such that ∑_{i=1}^{m} α_{i}(x) = 1 for any x ∈ X. Let us consider the stochastic kernel

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

and a new measure on X

Sμ(B) ≡ ∫_{X} K(x, B) dμ .

If we choose S_{i} and α_{i} such that Sμ = 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 Sμ 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.

60 K. IGUDESMAN

[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, Kremlevskaya, 18, Kazan, 420008, Russia

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

Received December 8, 2004