Georgian Mathematical Journal Volume 8 (2001), Number 4, 733–752

SOME REMARKS ON FRACTALS GENERATED BY A SEQUENCE OF FINITE SYSTEMS OF CONTRACTIONS

GIORGIO FOLLO

Abstract. We generalize some results shown by J. E. Hutchinson in [7].

Let 𝔉_{n} = {f_{1}^{(n)}, f_{2}^{(n)}, …, f_{m_{n}}^{(n)}} be finite systems of contractions on a complete metric space; then, under some conditions on (𝔉_{n}), there exists a unique non-empty compact set K such that the sequence defined by ((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C)) converges to K in the Hausdorff metric for every non-empty closed and bounded set C.

If the metric space is also separable and for every n, l_{1}^{(n)}, l_{2}^{(n)}, …, l_{m_{n}}^{(n)} there are real numbers strictly between 0 and 1, satisfying the condition ∑_{j=1}^{m_{n}} l_{j}^{(n)} = 1, then there exists a unique probability Radon measure μ_{K} such that the sequence

ν_{n} = ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)})(f_{i_{1}}^{(1)} ∘ f_{i_{2}}^{(2)} ∘ ⋯ ∘ f_{i_{n}}^{(n)})_{♯}ν

weakly converges to μ_{K} for every probability Borel regular measure ν with bounded support (where by f_{♯}ν we denote the image measure of ν under a contraction f). Moreover, K is the support of μ_{K}.

2000 Mathematics Subject Classification: Primary: 28A80; Secondary: 28A33. Key words and phrases: Fractals, construction of measures.

1. Introduction

Let (X, d_{X}) be a complete separable metric space and let f_{1}, f_{2}, …, f_{M} : X → X be contractions. In [7] it is proved that there exists a unique nonempty closed and bounded subset K of X invariant with respect to 𝔉 = {f_{1}, f_{2}, …, f_{M}} i.e., such that

K = 𝔉(K) = ⋃_{j=1}^{M} \overline{f_{j}(K)}.  (1)

Moreover, K is compact and if C_{0} ≠ ∅ is closed and bounded, then the sequence (C_{n}) defined by C_{n} = 𝔉(C_{n−1}) converges to K in the Hausdorff metric.

Let r = {r_{1}, r_{2}, …, r_{M}} be a family of M real numbers in ]0, 1[ with ∑_{j=1}^{M} r_{j} = 1. Then there exists a unique Borel regular (outer) measure μ in

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

X with compact support and of total mass 1 such that μ is invariant with respect to (𝔉, r), i.e.,

μ(A) = ∑_{j=1}^{M} r_{j}μ(f_{j}^{−1}(A)) for every Borel set A ⊆ X.  (2)

Furthermore, the support of μ is the fractal K.

We consider the case in which the system 𝔉 is replaced by a sequence (𝔉_{n}) of finite systems of contractions, i.e., 𝔉_{n} = {f_{1}^{(n)}, f_{2}^{(n)}, …, f_{m_{n}}^{(n)}} with m_{n} ≥ 2. Obviously, we cannot write an expression like (1), but we can still construct a sequence of closed and bounded subsets of X and ask if such a sequence is convergent with respect to the Hausdorff metric. More precisely, if the sequence (𝔉_{n}) satisfies the following two conditions:

• there exists a bounded set Q ⊆ X such that ⋃_{j=1}^{m_{n}} f_{j}^{(n)}(Q) ⊆ Q for any

n ∈ ℕ;

• lim_{n} ∏_{k=1}^{n} ρ^{(k)} = 0; here ρ^{(k)} is the greatest of the Lipschitz constants

of the contractions f_{1}^{(k)}, f_{2}^{(k)}, …, f_{m_{k}}^{(k)};

then there exists a unique non-empty closed and bounded set K ⊆ X such that the sequence ((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C_{0})) converges to K in the Hausdorff metric, for every non empty closed and bounded subset C_{0} of X. Moreover K is compact.

As an interesting example, given d ∈]0, 1[, we construct a d-dimensional compact subset of the real line by considering a sequence (𝔉_{n}) of finite systems of contractive similitudes f_{1}^{(n)}, f_{2}^{(n)}, …, f_{m_{n}}^{(n)} with Lipschitz constants ρ^{(n)} (depending only on n ) such that m_{n}(ρ^{(n)})^{d} = 1. We will study the entropy numbers related to this set.

In Section 4 we consider a generalization of the invariant measure found in [7].

As before, we cannot write an expression like (2). Let X be a complete separable metric space and let for every n, l_{1}^{(n)}, l_{2}^{(n)}, …, l_{m_{n}}^{(n)} ∈]0, 1[ be so that ∑_{j=1}^{m_{n}} l_{j}^{(n)} = 1, then there exists a unique Radon probability measure μ_{K} so that for every Radon probability measure ν on X, with bounded support, the sequence of measures defined by

ν_{n}(A) = ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)})ν((f_{i_{1}}^{(1)} ∘ f_{i_{2}}^{(2)} ∘ ⋯ ∘ f_{i_{n}}^{(n)})^{−1}(A)),

for Borel sets A ⊆ X, weakly converges to μ_{K}.

Moreover, the support of μ_{K} is the fractal K.

2. Notation and preliminary results

In this note (X, d_{X}) will always be a complete metric space. Additional requirements for X will be specified when necessary.

ℕ = {1, 2, … } is the set of all positive integer numbers.

The closed and open balls in X will be indicated by the symbols B_{X}(x_{0}, r) and D_{X}(x_{0}, r):

B_{X}(x_{0}, r)= {x ∈ X | d_{X}(x, x_{0}) ≤ r}, D_{X}(x_{0}, r)= {x ∈ X | d_{X}(x, x_{0}) < r}.

The diameter of a subset A of X is indicated by |A|: |A| = sup_{x,y∈A} d_{X}(x, y)

and its number of elements is indicated by ♯A.

If X is separable and s ≥ 0, H^{s}(A) stands for the s-dimensional Hausdorff measure of A and dimA for its Hausdorff dimension.

If X = ℝ^{N} then we will use the Euclidean metric d_{ℝ^{N}}(x, y) = ∥x − y∥_{2} = \sqrt{∑_{i=1}^{N}(ξ_{i} − η_{i})^{2}}, where x = (ξ_{1}, ξ_{2}, …, ξ_{N}) and y = (η_{1}, η_{2}, …, η_{N}).

Lemma 2.1. Let E ⊆ ℝ^{N} and let d > 0. If f : E → ℝ^{N} is a mapping and c > 0 is a constant such that ∥f(x)−f(y)∥_{2} ≤ c∥x−y∥_{2} for every x, y ∈ E, then H^{d}(f(E)) ≤ c^{d}H^{d}(E).

Proof. See [4], Chapter 2, Proposition 2.2.

2.1. d-sets in ℝ^{N}.

Definition. Let Γ be a closed non-empty subset of ℝ^{N} and let d ∈]0, N]. A positive Borel outer measure μ with support Γ is called a d-measure on Γ if there exist c_{1}, c_{2} ∈]0, +∞[ such that for every x_{0} ∈ Γ and for every r ∈]0, 1]

c_{1}r^{d} ≤ μ(B_{ℝ^{N}}(x_{0}, r)) ≤ c_{2}r^{d}  (1)

holds.

Remark 1. One can replace the condition r ∈]0, 1] in the above definition by the condition r ∈]0, r_{0}]; obviously, the constants c_{1} and c_{2} will be replaced by some constants c_{1}(r_{0}) > 0 and c_{2}(r_{0}) > 0 depending on r_{0}.

Definition. A closed non-empty subset Γ of ℝ^{N} is called a d-set if there exists a d-measure on Γ.

It can be proved that if Γ is a d-set, μ_{1} and μ_{2} are d-measures on Γ. Then there exist constants a, b ∈]0, +∞[ such that aμ_{1}(A) ≤ μ_{2}(A) ≤ bμ_{1}(A) ∀A ⊆ ℝ^{N}. Moreover, the restriction to Γ of the Hausdorff measure H^{d} is a d-measure on Γ; so every d-set has its canonical d-measure and therefore d is unique.

For the proof of these facts see [9], Chapter 2 or [11], Chapter 1.

2.2. Entropy numbers. Let Ω be a bounded subset of X. The n-th entropy number of Ω is defined by

ε_{n}(Ω) = inf {ε > 0 | ∃ x_{1}, x_{2}, …, x_{n} ∈ X such that Ω ⊆ ⋃_{i=1}^{n} B_{X}(x_{i}, ε)}.

The sequence (ε_{n}(Ω)) is monotonically decreasing and tends to zero if and only if Ω is precompact.

See [1] for a complete treatment.

2.3. The Hausdorff metric.

Definition. If x_{0} ∈ X and A ⊆ X, we define the distance between x_{0} and A by

d_{X}(x_{0}, A) = inf_{x∈A} d_{X}(x_{0}, x).

Remark 2. For every x_{0} ∈ X and A ⊆ X,we have d_{X}(x_{0}, A) = d_{X}(x_{0}, \overline{A}). Definition. Let 𝔅 be the class of all non-empty closed bounded subsets

of X.

The Hausdorff metric D on 𝔅 is defined by

D(A, B) = sup {d_{X}(x, B), d_{X}(y, A) | x ∈ A, y ∈ B}.

Remark 3. D is a metric on 𝔅. Moreover, for every A, B ∈ 𝔅

D(A, B) = inf {ε > 0 | A ⊆ ⋃_{y∈B} D_{X}(y, ε) and B ⊆ ⋃_{x∈A} D_{X}(x, ε)}.

Lemma 2.2. Let f : X → X be a Lipschitz function and let ρ = inf {c >

0 | d_{X}(f(x), f(y)) ≤ cd_{X}(x, y) ∀x, y ∈ X} be its Lipschitz constant. Then

D(\overline{f(A)}, \overline{f(B)}) ≤ ρD(A, B) ∀A, B ∈ 𝔅.  (2)

Proof. By remark 2 D(\overline{f(A)}, \overline{f(B)}) = sup \{d_{X}(u, f(B)), d_{X}(v, f(A)) | u ∈

f(A), v ∈ f(B)}.

By the Lipschitz condition on f it follows that d_{X}(f(x), f(B)) ≤ ρd_{X}(x, B) and d_{X}(f(y), f(A)) ≤ ρd_{X}(y, A), ∀x ∈ A ∀y ∈ B and then the (2).

Lemma 2.3. Let {A_{j} | j ∈ J}, {B_{j} | j ∈ J} be two families of elements of 𝔅. Then

D(\overline{⋃_{j∈J} A_{j}}, \overline{⋃_{j∈J} B_{j}}) ≤ sup_{j∈J} D(A_{j}, B_{j})

provided that ⋃_{j∈J} A_{j} and ⋃_{j∈J} B_{j} are bounded.

Proof. Let c > sup_{j∈J} D(A_{j}, B_{j}): for all j ∈ J D(A_{j}, B_{j}) < c and then

A_{j} ⊆ ⋃_{y_{j}∈B_{j}} D_{X}(y_{j}, c) ⊆ ⋃_{i∈J} ⋃_{y_{i}∈B_{i}} D_{X}(y_{i}, c) = _{y∈⋃_{i∈J} B_{i}}^{⋃} D_{X}(y, c).

It follows that

⋃_{j∈J} A_{j} ⊆ _{y∈⋃_{i∈J} B_{i}}^{⋃} D_{X}(y, c).

In the same way we obtain

⋃_{j∈J} B_{j} ⊆ _{x∈⋃_{j∈J} A_{j}}^{⋃} D_{X}(x, c).

Then

D(\overline{⋃_{j∈J} A_{j}}, \overline{⋃_{j∈J} B_{j}}) < c.

2.4. Sequences of indices and product spaces. From now on, (m_{n}) is a fixed sequence of integer numbers, with m_{n} ≥ 2 for all n ∈ ℕ. Moreover, for all n ∈ ℕ we fix l_{1}^{(n)}, l_{2}^{(n)}, …, l_{m_{n}}^{(n)} ∈]0, 1[ so that ∑_{j=1}^{m_{n}} l_{j}^{(n)} = 1.

Definition. For n ∈ ℕ we set I_{n} = {1, 2, …, m_{n}} and I = ∏_{n=1}^{+∞} I_{n}. Each I_{n} is equipped with the discrete topology. On I we consider the function d_{I} : I × I → ℝ,

d_{I}(k, h) = ∑_{j=1}^{+∞} \frac{1}{2^{j}} \frac{|k_{j} − h_{j}|}{1 + |k_{j} − h_{j}|},

where k = (k_{j}), h = (h_{j}), k_{j}, h_{j} ∈ I_{j} for all j ∈ ℕ.

Remark 4. It is well known that d_{I} is a metric on I; moreover d_{I} induces the product topology on I.

It follows that (I, d_{I}) is a compact metric space and then it is complete and separable (see, for example, [2], Chapters 2, 5 and 6).

Definition. Given i_{1}, i_{2}, …, i_{n} ∈ℕ, we define the natural projection π_{i_{1},i_{2},...,i_{n}}: I → ∏_{j=1}^{n} I_{i_{j}} by

π_{i_{1},i_{2},...,i_{n}}(k) = (k_{i_{1}}, k_{i_{2}}, …, k_{i_{n}}).

Remark 5. For every i_{j} ∈ ℕ, π_{i_{j}} is a continuous function by the definition of product topology. Then π_{i_{1},i_{2},...,i_{n}} is continuous.

Definition. Let X and Y be metric spaces, μ an outer measure on X and f : X → Y a function.

The image of μ under f is defined by

f_{♯}μ(A) = μ(f^{−1}(A)) ∀A ⊆ Y.

For the proof of the following two theorems see [10], Chapter 1, Theorems 1.18 and 1.19.

Theorem 2.1. Let X and Y be separable metric spaces. If f : X → Y is continuous and μ is a Radon measure on X with compact support, then f_{♯}μ is a Radon measure. Moreover, if C ⊆ X is the support of μ, then f(C) is the support of f_{♯}μ.

Definition. Let X and Y be separable metric spaces. A mapping f : X → Y is a Borel mapping if f^{−1}(U) is a Borel set for every open set U ⊆ Y.

Let A ⊆ X be a Borel set. A function g : A → [−∞, +∞] is a Borel function if the set {x ∈ A | f(x) < c} is a Borel set for every c ∈ ℝ.

Theorem 2.2. Let X and Y be separable metric spaces and suppose that f : X → Y is a Borel mapping, μ is a Borel measure on X and g is a non-negative Borel function on Y. Then

∫_{Y} gdf_{♯}μ = ∫_{X} (g ∘ f)dμ.

Definition. For every n ∈ ℕ we define a measure τ_{n} on I_{n} by

τ_{n}(A) = ∑_{j∈A} l_{j}^{(n)} ∀A ⊆ I_{n}.

Remark 6. For every n ∈ ℕ, τ_{n} is a Radon measure and τ_{n}(I_{n}) = 1.

Remark 7. From the definition of product measure of two measures it follows that

(τ_{1} × τ_{2} × ⋯ × τ_{n})(A) = ∑_{(k_{1},k_{2},...,k_{n})∈A} ∏_{j=1}^{n} l_{k_{j}}^{(j)} ∀n ∈ ℕ ∀A ⊆ ∏_{j=1}^{n} I_{j}.

In order to define the product measure on I, we need the following theorem.

Theorem 2.3. Let {X_{α} | α ∈ A} be a family of compact Hausdorff spaces and let, for each α ∈ A, μ_{α} be a Radon measure on X_{α}, with μ_{α}(X_{α}) = 1.

Then there exists a unique Radon measure μ on ∏_{α∈A} X_{α} such that μ(∏_{α∈A} X_{α}) = 1 and μ_{α_{1}} × μ_{α_{2}} × ⋯ × μ_{α_{n}} = π_{α_{1},α_{2},...,α_{n}♯}μ for any distinct α_{1}, α_{2}, …, α_{n} ∈ A.

Proof. See [5], Chapter 9, Theorem 9.19.

Remark 8. By the previous theorem there is a unique Radon measure τ on I such that τ(I) = 1 and τ_{i_{1}} × τ_{i_{2}} × ⋯ × τ_{i_{n}} = π_{i_{1},i_{2},...,i_{n}♯}τ for any distinct i_{1}, i_{2}, …, i_{n} ∈ ℕ.

3. Limit sets

3.1. Basic notation. From now on we will use the following notation:

• for any n ∈ ℕ i ∈ I_{n}, f_{i}^{(n)} : X → X is a contraction;

– ρ_{i}^{(n)} = inf{c > 0 | d_{X}(f_{i}^{(n)}(x), f_{i}^{(n)}(y)) ≤ cd_{X}(x, y) ∀x, y ∈ X} (ρ_{i}^{(n)} is the Lipschitz constant of f_{i}^{(n)}),

– ρ^{(n)} = max{ρ_{1}^{(n)}, ρ_{2}^{(n)}, …, ρ_{m_{n}}^{(n)}},

– ρ = sup_{n∈ℕ} ρ^{(n)},

– x_{i}^{(n)} ∈ X is the fixed point of f_{i}^{(n)};

• for every n ∈ ℕ – 𝔉_{n} = {f_{1}^{(n)}, f_{2}^{(n)}, …, f_{m_{n}}^{(n)}},

– for every A ⊆ X 𝔉_{n}(A) = ⋃_{i=1}^{m_{n}} \overline{f_{i}^{(n)}(A)},

– for every A ⊆ X (𝔉_{n} ∘ 𝔉_{n+1})(A) = 𝔉_{n}(𝔉_{n+1}(A));

• for every i ∈ I, n, k ∈ ℕ with k ≤ n:

– f_{i_{1}i_{2}···i_{n}} = f_{i_{1}}^{(1)} ∘ f_{i_{2}}^{(2)} ∘ ⋯ ∘ f_{i_{n}}^{(n)} and x_{i_{1}i_{2}···i_{n}} is its fixed point,

– f_{i_{k}i_{k+1}···i_{n}}^{(k)} = f_{i_{k}}^{(k)} ∘f_{i_{k+1}}^{(k+1)} ∘⋯∘f_{i_{n}}^{(n)} and x_{i_{k}i_{k+1}···i_{n}}^{(k)} is its fixed point;

• F = {x_{i}^{(n)} | n ∈ ℕ i ∈ I_{n}} is the set of all fixed points of the contractions f_{i}^{(n)}.

3.2. Existence and uniqueness.

Lemma 3.1. Let (g_{n}) be a sequence of contraction maps on X, each of them with the Lipschitz constant ρ_{n}. Let us suppose that the following two conditions hold:

(1) there exists a non-empty closed and bounded set Q ⊆ X such that g_{n}(Q) ⊆ Q for every n ∈ ℕ;

(2) lim_{n} ∏_{k=1}^{n} ρ_{k} = 0.

Then there exists a unique x ∈ X so that

lim_{n}(g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x_{0}) = x for every x_{0} ∈ X.

Moreover, x ∈ Q.

Proof. It is easy to prove, by induction, that for every x, y ∈ X and n ∈ ℕ

d_{X}((g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x), (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(y)) ≤ ( ∏_{k=1}^{n} ρ_{k})d_{X}(x, y).

Now, let x_{0} ∈ Q and ε > 0; by the second hypothesis there exists n_{ε} ∈ ℕ so that ∏_{k=1}^{n} ρ_{k} < ε/|Q| ∀n ∈ ℕ with n > n_{ε}.

Then, for all m, n ∈ ℕ with m > n > n_{ε} we have

d_{X}((g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x_{0}), (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{m})(x_{0}))

≤ ( ∏_{k=1}^{n} ρ_{k})d_{X}(x_{0}, (g_{n+1} ∘ g_{n+2} ∘ ⋯ ∘ g_{m})(x_{0})) < ε.

Since X is complete, there exists x ∈ X such that lim_{n}(g_{1}∘g_{2}∘⋯∘g_{n})(x_{0}) = x.

Now we prove that x does not depend on x_{0}.

Let y_{0} ∈ X and n ∈ ℕ:

d_{X}(x, (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(y_{0}))d_{X}(x, (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x_{0}))

+ d_{X}((g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x_{0}), (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(y_{0}))

≤ d_{X}(x, (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x_{0})) + ( ∏_{k=1}^{n} ρ_{k})d_{X}(x_{0}, y_{0})

and by letting n → +∞ we obtain x = lim_{n}(g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(y_{0}).

Definition. Let x_{0} ∈ X be fixed; we define ∀n ∈ ℕ a function p_{n} : ∏_{j=1}^{n} I_{j} → X by

p_{n}(i_{1}, i_{2}, …, i_{n}) = f_{i_{1}i_{2}···i_{n}}(x_{0}).

Remark 1. Obviously, p_{n} depends on x_{0} and it is a continuous function on ∏_{j=1}^{n} I_{j}.

From now on we will suppose that the following two hypotheses are valid:

(1) there exists a non-empty closed bounded set Q ⊆ X such that 𝔉_{n}(Q) ⊆ Q for all n ∈ ℕ; (2) lim_{n} ∏_{k=1}^{n} ρ^{(k)} = 0.

Remark 2. The hypotheses 1 and 2 above are implied by the following:

3. F is bounded;

4. ρ < 1. Indeed, it is obvious that 4 ⇒ 2; moreover, let

Q = ⋂_{n=1}^{+∞} ⋂_{i=1}^{m_{n}} B_{X}(x_{i}^{(n)}, \frac{|F |}{1 − ρ});

we prove that Q satisfies 1.

Q is closed and bounded; moreover F ⊆ Q.

Let n, k ∈ ℕ, i ∈ I_{n} and j ∈ I_{k}; we have ∀x ∈ Q

d_{X}(f_{i}^{(n)}(x), x_{j}^{(k)}) ≤ d_{X}(f_{i}^{(n)}(x), x_{i}^{(n)}) + d_{X}(x_{i}^{(n)}, x_{j}^{(k)})

≤ d_{X}(f_{i}^{(n)}(x), f_{i}^{(n)}(x_{i}^{(n)})) + |F | ≤ ρ_{i}^{(n)}d_{X}(x, x_{i}^{(n)}) + |F |

≤ ρ\frac{|F |}{1 − ρ} + |F | = \frac{|F |}{1 − ρ}

and then

f_{i}^{(n)}(x) ∈ B_{X}(x_{j}^{(k)}, \frac{|F |}{1 − ρ}) ∀k ∈ ℕ ∀j ∈ I_{k}.

Definition. Let x_{0} ∈ X be fixed; we define a function p : I → X by

p(k) = lim_{n} f_{k_{1}k_{2}···k_{n}}(x_{0}).

Remark 3. By Lemma 3.1 the function p is well defined and does not depend on x_{0} ∈ X.

If we take x_{0} ∈ Q, then we would see that p(I) ⊆ Q. We will always suppose x_{0} ∈ Q.

Definition. We denote the set p(I) by K.

Proposition 3.1. The function p is uniformly continuous.

Proof. Let ε > 0 and n_{ε} ∈ ℕ be so that ∏_{i=1}^{n_{ε}} ρ^{(i)} < ε/|Q|. Let δ = 2^{−n_{ε}−1}; for every k, h ∈ I, d_{I}(k, h) < δ implies k_{j} = h_{j} ∀j ∈ ℕ with j ≤ n_{ε} and then, if we suppose x_{0} ∈ Q,

d_{X}(p(k), p(h))

= lim_{j} d_{X}(f_{k_{1}k_{2}···k_{nε}}(f_{k_{nε+1}k_{nε+2}···k_{j}}^{(n_{ε}+1)}(x_{0})), f_{k_{1}k_{2}···k_{nε}}(f_{h_{nε+1}h_{nε+2}···h_{j}}^{(n_{ε}+1)}(x_{0})))

≤ ( ∏_{i=1}^{n_{ε}} ρ^{(i)}) lim_{j} d_{X}(f_{k_{nε+1}k_{nε+2}···k_{j}}^{(n_{ε}+1)}(x_{0}), f_{h_{nε+1}h_{nε+2}···h_{j}}^{(n_{ε}+1)}(x_{0})) < \frac{ε}{|Q|}|Q| = ε.

Corollary 3.2. K is compact and K ⊆ Q.

Proposition 3.3. For every C ∈ 𝔅 lim_{n} D((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C), K) = 0.

Proof. Let x_{0} ∈ Q, ε > 0 and n_{ε} ∈ ℕ be so that ∏_{j=1}^{n_{ε}} ρ^{(j)} < ε/|Q|. For every n ∈ ℕ with n > n_{ε} we have D(p_{n}(π_{1,2,...,n}(I)), K) ≤ ε.

Indeed, for every k ∈ I, we have p(k) ∈ K and

d_{X}(p_{n}(π_{1,2,...,n}(k)), p(k)) ≤ ( ∏_{j=1}^{n} ρ^{(j)})d_{X}(x_{0}, lim_{i} f_{k_{n+1}k_{n+2}···k_{i}}^{(n+1)}(x_{0}))

< ( ∏_{j=1}^{n_{ε}} ρ^{(j)})|Q| < ε

from which d_{X}(p_{n}(π_{1,2,...,n}(k)), K) < ε. On the other hand, ∀x ∈ K there exists k ∈ I so that p(k) = x and then d_{X}(p_{n}(π_{1,2,...,n}(I)), x) < ε.

But

p_{n}(π_{1,2,...,n}(I)) = ⋃_{k_{1}=1}^{m_{1}} ⋃_{k_{2}=1}^{m_{2}} ⋯ ⋃_{k_{n}=1}^{m_{n}}{f_{k_{1}k_{2}···k_{n}}(x_{0})} = (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})({x_{0}}).

Now, let C ∈ 𝔅: by Lemmas 2.3 and 2.2 it follows that ∀n ∈ ℕ

D((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C), (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})({x_{0}})) ≤ ( ∏_{j=1}^{n} ρ^{(j)})D(C, {x_{0}});

then, if n > n_{ε},

D((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C), K)

≤ D((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C), (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})({x_{0}}))

+ D((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(\{x_{0}\}), K) < ε(1 + \frac{D(C, \{x_{0}\})}{|Q|}).

3.3. Some properties of K. We follow the notation of the previous paragraphs.

Definition. Given a finite family of contraction maps G = {g_{1}, g_{2}, …, g_{m}} and a subset A of X, we say that A is invariant with respect to G if G(A) = A.

Remark 4. If 𝔉_{n} = 𝔉 = {f_{1}, f_{2}, …, f_{m}} for every n ∈ ℕ, then K is invariant with respect to 𝔉.

If there exists k ∈ ℕ such that 𝔉_{n+k} = 𝔉_{n} for all n ∈ ℕ, then K is invariant with respect to 𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{k}.

Remark 5. If there exists n ∈ ℕ such that 𝔉_{n+k} = 𝔉_{n} for all k ∈ ℕ, then there is a unique non-empty compact set H ⊆ X which is invariant with respect to 𝔉_{n}. Moreover,

K = ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ⋯ ⋃_{i_{n}=1}^{m_{n}} f_{i_{1}i_{2}···i_{n}}(H).

Lemma 3.2. Let A = ⋃_{j=1}^{N} A_{j} ⊆ X. If A is connected, then |A| ≤ ∑_{j=1}^{N} |A_{j}|.

Proof. We consider the case N = 2, the general statement follows by induction.

Let A = B ⋃ C be a connected subset of X and let x, y ∈ A. If x, y ∈ B or x, y ∈ C; then d_{X}(x, y) ≤ |B| + |C|.

If x ∈ B and y ∈ C, then ∀z ∈ B, ∀w ∈ C,

d_{X}(x, y) ≤ d_{X}(x, z) + d_{X}(z, w) + d_{X}(w, y) ≤ |B| + d_{X}(z, w) + |C|.

It follows that

|A| ≤ |B| + |C| + inf {d_{X}(z, w) | z ∈ B, w ∈ C}.

Let us suppose that inf{d_{X}(z, w) | z ∈ B, w ∈ C} = ε > 0; then B′ = ⋃_{x∈B} D_{X}(x, ε/4) and C′ = ⋃_{y∈C} D_{X}(y, ε/4) are disjoint open sets whose union contains A ⋃ B.

Corollary 3.4. If lim_{n} ∏_{k=1}^{n} ∑_{i=1}^{m_{k}} ρ_{i}^{(k)} = 0, then K is totally disconnected.

Proof. Let x, y ∈ K with x ≠ y and let n_{xy} ∈ ℕ be such that ∏_{k=1}^{n} ∑_{i=1}^{m_{k}} ρ_{i}^{(k)} < d_{X}(x, y)/|Q| ∀n ∈ ℕ with n > n_{xy}. Then, if n > n_{xy}, we have

K = lim_{k}(𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{k})(Q)

= (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})( lim_{k}(𝔉_{n+1} ∘ 𝔉_{n+2} ∘ ⋯ ∘ 𝔉_{k})(Q))

because, by Lemmas 2.2 and 2.3, 𝔉_{1}, 𝔉_{2}, …, 𝔉_{n} are contraction maps with respect to the Hausdorff metric and then they are continuous. It follows that

K ⊆ (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(Q).

Now, let A ⊆ K be connected and such that x, y ∈ A: we have

A ⊆ K ⊆ ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ⋯ ⋃_{i_{n}=1}^{m_{n}} \overline{f_{i_{1}i_{2}···i_{n}}(Q)}

and by Lemma 3.2

|A| ≤ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} |f_{i_{1}i_{2}···i_{n}}(Q)|

≤ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} ρ_{i_{k}}^{(k)})|Q| = |Q| ∏_{k=1}^{n} ∑_{i=1}^{m_{k}} ρ_{i}^{(k)} < d_{X}(x, y) ≤ |A|.

3.4. Examples.

3.4.1. Cantor sets in ℝ. Let X be the set of real numbers with the Euclidean distance; we construct a generalized version of the Cantor set. To do this, we suppose that we are given a real number d ∈]0, 1[ and we construct two sequences (t_{n}) and (d_{n}) in the following way:

• t_{0} = 1 and m_{n}t_{n}^{d} = t_{n−1}^{d};

• d_{n} = \frac{t_{n−1}−m_{n}t_{n}}{m_{n}−1} . Then the set K of the previous paragraph is obtained by setting

f_{i}^{(n)}(x) = \frac{t_{n}x + (i − 1)(t_{n} + d_{n})}{t_{n−1}} = m_{n}^{− \frac{1}{d}} x + \frac{i − 1}{m_{n} − 1}(1 − m_{n}^{− \frac{1}{d}}) ∀n ∈ ℕ ∀i ∈ I_{n}.

Now we show some properties of the fractal set K so obtained. In the definition of the functions p_{n} we assume x_{0} = 0.

Remark 6. The functions p_{n} are given by

p_{n}(i_{1}, i_{2}, …, i_{n}) = ∑_{j=1}^{n}(i_{j} − 1)(t_{j} + d_{j}).  (1)

We prove this by induction: let n ∈ ℕ, we have f_{i_{n}}^{(n)}(0) = \frac{1}{t_{n−1}} (i_{n}−1)(t_{n}+d_{n}) . Let us suppose that for k ∈ ℕ 2 ≤ k ≤ n,

f_{i_{k}i_{k+1}···i_{n}}^{(k)}(0) = \frac{1}{t_{k−1}} ∑_{j=k}^{n}(i_{j} − 1)(t_{j} + d_{j}),  (2)

then

f_{i_{k−1}i_{k}···i_{n}}^{(k−1)}(0) = f_{i_{k−1}}^{(k−1)}(f_{i_{k}i_{k+1}···i_{n}}^{(k)}(0)) = \frac{1}{t_{k−2}} ∑_{j=k−1}^{n}(i_{j} − 1)(t_{j} + d_{j}).

Then (2) holds for any k ≤ n and in particular

p_{n}(i_{1}, i_{2}, …, i_{n}) = f_{i_{1}i_{2}···i_{n}}(0) = ∑_{j=1}^{n}(i_{j} − 1)(t_{j} + d_{j}).

It follows that

p(i) = ∑_{j=1}^{+∞}(i_{j} − 1)(t_{j} + d_{j}) ∀i ∈ I  (3)

and

K = { ∑_{j=1}^{+∞}(i_{j} − 1)(t_{j} + d_{j}) | i_{j} ∈ I_{j} ∀j ∈ ℕ}.  (4)

Example 1. If m_{n} = 2 ∀n ∈ ℕ and d = log_{3} 2, then K is the classical Cantor set.

Indeed, in this case t_{n} = d_{n} = 3^{−n}, f_{1}^{(n)}(x) = x/3 and f_{2}^{(n)}(x) = (x + 2)/3 ∀n ∈ ℕ and by Remark 4 K is invariant with respect to 𝔉 = {f_{1}^{(1)}, f_{2}^{(1)}}.

It may also be noted that (4) becomes

K = \{ ∑_{j=1}^{+∞} \frac{c_{j}}{3^{j}} | c_{j} ∈ \{0, 2\} ∀j ∈ ℕ\}.

Remark 7. It is easy to prove, by induction, that

∀n ∈ ℕ t_{n} = ( ∏_{j=1}^{n} m_{j})^{− \frac{1}{d}} and d_{n} = \frac{1 − m_{n}^{1− \frac{1}{d}}}{m_{n} − 1} ( ∏_{j=1}^{n−1} m_{j})^{− \frac{1}{d}} .

Remark 8. From Remark 6 we have K ⊆ [0, 1] and 0, 1 ∈ K.

Indeed, if we set k, h ∈ I, k_{j} = 1, h_{j} = m_{j} for every j ∈ ℕ, then by (3) , p(k) = 0, p(h) = ∑_{j=1}^{+∞}(t_{j−1} − t_{j}) = t_{0} = 1 and for all i ∈ I 0 ≤ p(i) ≤ p(h) = 1.

It may also be noted that 0 ≤ f_{j}^{(n)}(x) ≤ 1 ∀x ∈ [0, 1] and for every n, j ∈ ℕ, j ≤ m_{n}; then we can set Q = [0, 1].

Remark 9. Given n ∈ ℕ, the ∏_{j=1}^{n} m_{j} intervals of the form [p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n})+t_{n}] are pairwise disjoint and |p_{n}(i_{1}, i_{2}, …, i_{n})− (p_{n}(k_{1}, k_{2}, …, k_{n}) + t_{n})| ≥ d_{n} for any different (i_{1}, i_{2}, …, i_{n}), (k_{1}, k_{2}, …, k_{n}) ∈ ∏_{j=1}^{n} I_{j}.

Moreover, for all (i_{1}, i_{2}, …, i_{n+1}) ∈ ∏_{j=1}^{n+1} I_{j} the interval [p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n})+t_{n}] contains [p_{n}(i_{1}, i_{2}, …, i_{n+1}), p_{n}(i_{1}, i_{2}, …, i_{n+1})+t_{n+1}] and

K = ⋂_{n=1}^{+∞} ( ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ⋯ ⋃_{i_{n}=1}^{m_{n}} [p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n}) + t_{n}]).

Now we are going to prove that K is a d-set if and only if the sequence (m_{n}) is bounded.

Theorem 3.5. We have H^{d}(K) = 1 and so dim K = d.

Proof. See [3] Chapter 1, Theorem 1.15.

Lemma 3.3. For every n ∈ ℕ and (i_{1}, i_{2}, …, i_{n}) ∈ ∏_{j=1}^{n} I_{j}, we have

H^{d}(K ⋂ [p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n}) + t_{n}]) = t_{n}^{d}.

Proof. Let n ∈ ℕ and (i_{1}, i_{2}, …, i_{n}) ∈ ∏_{j=1}^{n} I_{j}; we define

f : K ⋂ [p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n}) + t_{n}] → K ⋂[0, t_{n}],
f(x) = x − p_{n}(i_{1}, i_{2}, …, i_{n}).

It is clear that 0 ≤ f(x) ≤ t_{n}; moreover, f(x) ∈ K by (3) and (1). Then f is well defined.

The function f is one-to-one because it is injective and ∀y ∈ K ⋂[0, t_{n}], y = f(y + p_{n}(i_{1}, i_{2}, …, i_{n})). Moreover, f is an isometry and then, by Lemma 2.1, H^{d}(K ⋂[0, t_{n}]) ≤ H^{d}(K ⋂[p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n}) + t_{n}]). By applying the same arguments to f^{−1} we obtain the opposite inequality.

Finally,

1=H^{d}(K)=H^{d}( ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}}⋯ ⋃_{i_{n}=1}^{m_{n}} (K ⋂ [p_{n}(i_{1}, i_{2},…, i_{n}), p_{n}(i_{1}, i_{2},…, i_{n})+t_{n}]))

= ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} H^{d}(K ⋂ [p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n}) + t_{n}])

= ( ∏_{j=1}^{n} m_{j})H^{d}(K ⋂[0, t_{n}])

and then

H^{d}(K ⋂ [p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n}) + t_{n}]) = H^{d}(K ⋂[0, t_{n}])

= ( ∏_{j=1}^{n} m_{j})^{−1} = t_{n}^{d}.

Proposition 3.6. The set K is a d-set if and only if the sequence (m_{n}) is bounded.

Proof. Let x_{0} ∈ K and r ∈]0, 1]; by Remark 9, for every k ∈ ℕ there exist (i_{1}, i_{2}, …, i_{k}) ∈ ∏_{j=1}^{k} I_{j} such that x_{0} ∈ [p_{k}(i_{1}, i_{2}, …, i_{k}), p_{k}(i_{1}, i_{2}, …, i_{k}) + t_{k}]. Let

n = min {k ∈ ℕ | ∃(i_{1}, i_{2}, …, i_{k}) ∈ ∏_{j=1}^{k} I_{j} so that

x_{0} ∈ [p_{k}(i_{1}, i_{2}, …, i_{k}), p_{k}(i_{1}, i_{2}, …, i_{k}) + t_{k}] ⊆ [x_{0} − r, x_{0} + r]}

(n is well defined because lim_{k} t_{k} = 0 by Remark 7). We prove that

\frac{1}{m_{n}} r^{d} ≤ H^{d}(k ⋂[x_{0} − r, x_{0} + r]) ≤ 2^{1+d}m_{n}r^{d};  (5)

it will follow that K is a d-set if the sequence (m_{n}) is bounded.

By Lemma 3.3 H^{d}(K ⋂[x_{0}−r, x_{0}+r]) ≥ t_{n}^{d} = t_{n−1}^{d}/m_{n}; if t_{n−1} ≤ r, we would have x_{0} ∈ [p_{n−1}(i_{1}, i_{2}, …, i_{n−1}), p_{n−1}(i_{1}, i_{2}, …, i_{n−1}) + t_{n−1}] ⊆ [x_{0} − r, x_{0} + r] and this is absurd; then the first inequality follows.

Let us prove the second inequality: we have [p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n})+ t_{n}] ⊆ [x_{0} − r, x_{0} + r] ⋂[p_{n−1}(i_{1}, i_{2}, …, i_{n−1}), p_{n−1}(i_{1}, i_{2}, …, i_{n−1}) + t_{n−1}] and for every j ∈ ℕ the following implications hold:

• if 1 ≤ j < i_{n−1} − 1 then [x_{0}−r, x_{0}+r] ⋂[p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, j), p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, j)+t_{n−1}] = ∅ because otherwise we would have [p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, i_{n−1}−1), p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, i_{n−1}−1)+t_{n−1}] ⊆ [x_{0}− r, x_{0} + r]; • if i_{n−1} + 1 < j ≤ m_{n−1} then

[x_{0}−r, x_{0}+r] ⋂[p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, j), p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, j)+t_{n−1}] = ∅ because otherwise we would have [p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, i_{n−1}+1), p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, i_{n−1}+1)+t_{n−1}] ⊆ [x_{0}− r, x_{0} + r].

Moreover, at least one of the intervals [p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, i_{n−1} − 1), p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, i_{n−1} − 1) + t_{n−1}] and [p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, i_{n−1} + 1), p_{n−1}(i_{1}, i_{2}, …, i_{n−2}, i_{n−1} + 1) + t_{n−1}] does not intersect [x_{0} − r, x_{0} + r] because otherwise we would have [p_{n−1}(i_{1}, i_{2}, …, i_{n−1}), p_{n−1}(i_{1}, i_{2}, …, i_{n−1}) + t_{n−1}] ⊆ [x_{0} − r, x_{0} + r].

Then H^{d}(K ⋂[x_{0} − r, x_{0} + r]) < 2t_{n−1}^{d} = 2m_{n}t_{n}^{d} ≤ 2^{1+d}m_{n}r^{d}.

Now we suppose that the sequence (m_{n}) is not bounded; by taking x_{0} = 0 and r_{n} = t_{n} + d_{n} for every n ∈ ℕ, we have

\frac{H^{d}(K ⋂[x_{0} − r_{n}, x_{0} + r_{n}])}{r_{n}^{d}} = \frac{H^{d}(K ⋂[0, t_{n}])}{r_{n}^{d}} = \frac{t_{n}^{d}}{(t_{n} + d_{n})^{d}} = (\frac{m_{n} − 1}{mn^{\frac{1}{d}} − 1})^{d}.

Let (m_{n_{k}}) be a subsequence of (m_{n}) such that lim_{k} m_{n_{k}} = +∞; then

lim_{k} \frac{H^{d}(K ⋂[x_{0} − r_{n_{k}}, x_{0} + r_{n_{k}}])}{r_{n_{k}}^{d}} = 0

because d < 1.

Remark 10. If in the above proposition we suppose t_{k} ≤ d_{k} for all k ∈ ℕ, then (5) becomes

\frac{1}{m_{n}} r^{d} ≤ H^{d}(k ⋂[x_{0} − r, x_{0} + r]) ≤ 2^{d}m_{n}r^{d}.  (6)

We recall that by Remark 7 t_{k} ≤ d_{k} if d ≤ log_{(2m_{k}−1)} m_{k}. If d ≤ log_{3} 2, then t_{k} ≤ d_{k} independently of m_{k}.

Example 2. For the classical Cantor set, (6) gives

\frac{1}{2}r^{d} ≤ H^{d}(k ⋂[x_{0} − r, x_{0} + r]) ≤ 2^{1+d}r^{d} ∀x_{0} ∈ K ∀r ∈]0, 1].

Now we estimate the entropy numbers of K under the assumption that t_{k} ≤ d_{k} ∀k ∈ ℕ.

To avoid tedious notation, we set for every k ∈ ℕ,

C_{k} = ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ⋯ ⋃_{i_{n}=1}^{m_{n}} [p_{n}(i_{1}, i_{2}, …, i_{n}), p_{n}(i_{1}, i_{2}, …, i_{n}) + t_{n}].

By Remark 9 K = ⋂_{k=1}^{+∞} C_{k}.

Let k ∈ ℕ and n_{k} = ∏_{j=1}^{k} m_{j}; then ε_{n_{k}}(K) ≤ ε_{n_{k}}(C_{k}) ≤ t_{k}/2.

Since the extreme points of the intervals of C_{k} are in K, it follows that

ε_{n_{k}}(K) = \frac{1}{2}t_{k} = \frac{1}{2}n_{k}^{− \frac{1}{d}} .  (7)

Let h ∈ ℕ be a divisor of m_{k+1}: we compute ε_{hn_{k}}(K).

The set C_{k+1} is a disjoint union of n_{k+1} closed intervals with amplitude t_{k+1} and mutual distance greater than or equal to t_{k+1}. Since hn_{k} divides n_{k+1} = m_{k+1}n_{k}, we can cover all the m_{k+1} closed intervals of C_{k+1} that are included into a single interval of C_{k} with h closed intervals of the form

[p_{k+1}(i_{1}, i_{2}, …, i_{k}, \frac{lm_{k+1}}{h} + 1), p_{k+1}(i_{1}, i_{2}, …, i_{k}, \frac{(l + 1)m_{k+1}}{h} ) + t_{k+1}]

0 ≤ l < h

and, as before, the extreme points of these intervals are in K; so

ε_{hn_{k}}(K) = \frac{1}{2}(\frac{m_{k+1}}{h} t_{k+1} + (\frac{m_{k+1}}{h} − 1)d_{k+1})

and by Remark 7 we have

ε_{hn_{k}}(K) = \frac{(h − 1)m_{k+1}^{1− \frac{1}{d}} + m_{k+1} − h}{2h^{1− \frac{1}{d}} (m_{k+1} − 1)}(hn_{k})^{− \frac{1}{d}} .  (8)

If h is not a divisor of m_{k+1}, we have ε_{hn_{k}}(K) ≤ t_{k}/(2h) and then

ε_{hn_{k}}(K) ≤ \frac{1}{2} h^{\frac{1}{d} −1}(hn_{k})^{− \frac{1}{d}} .  (9)

Let ⌊m_{k+1}/h⌋ = max{n ∈ ℕ | n ≤ m_{k+1}/h}; then we may not cover K by using hn_{k} intervals of length (t_{k+1} + d_{k+1})⌊m_{k+1}/h⌋ because at least one of the intervals of C_{k+1} will not be covered; so it must be

ε_{hn_{k}}(K) > \frac{1}{2} (t_{k+1} + d_{k+1})⌊\frac{m_{k+1}}{h} ⌋,

i.e.,

ε_{hn_{k}}(K) > \frac{h^{\frac{1}{d}} (1 − m_{k+1}^{− \frac{1}{d}})}{2(m_{k+1} − 1)}⌊\frac{m_{k+1}}{h} ⌋(hn_{k})^{− \frac{1}{d}} .  (10)

Finally, if l ∈ ℕ l < n_{k}, then

ε_{hn_{k}+l}(K) = ε_{hn_{k}}(K)  (11)

because the additional l intervals can not be equally distributed between the connected components of C_{k}.

Example 3. Let K be the classical Cantor set; by (7) and (11) we have

ε_{2^{k}+l}(K) = 2^{− \frac{k}{d} −1} = \frac{1}{2}3^{−k}.

It follows that the entropy numbers of the classical Cantor set have the following asymptotic behaviour:

\frac{1}{2}n^{− \frac{1}{d}} ≤ ε_{n}(K) < \frac{3}{2}n^{− \frac{1}{d}} ∀n ∈ ℕ

(see also [6], Example 2.2).

Remark 11. If the sequence (m_{n}) is bounded, then K is a d-set by Proposition 3.6 and we can apply Proposition 3.1 of [6] to see that there exist a, b ∈]0, +∞[ such that

an^{− \frac{1}{d}} ≤ ε_{n}(K) ≤ bn^{− \frac{1}{d}} ∀n ∈ ℕ .

Moreover, by Corollary 2.7 of [6], the box dimension of K is d.

Remark 12. Let us suppose that the sequence (m_{n}) is not bounded and t_{k} ≤ d_{k} ∀k ∈ ℕ. Let (m_{n_{k}}) be a subsequence of (m_{n}) such that lim_{k} m_{n_{k}} = +∞; we set p_{k} = ∏_{j=1}^{n_{k}−1} m_{j} for every k ∈ ℕ; then, by (7) we have

ε_{p_{k}}(K) = \frac{1}{2}p_{k}^{− \frac{1}{d}} .

If m_{n_{k}} is even, then, by setting 2h_{k} = m_{n_{k}}, we have from (8)

ε_{h_{k}p_{k}}(K)(h_{k}p_{k})^{\frac{1}{d}} = \frac{2^{1− \frac{1}{d}} (h_{k} − 1)h_{k} + h_{k}^{1+ \frac{1}{d}}}{2h_{k}(2h_{k} − 1)} .  (12)

If m_{n_{k}} is odd, then, we set 2h_{k} = m_{n_{k}} + 1 and by (10) it follows

ε_{h_{k}p_{k}}(K)(h_{k}p_{k})^{\frac{1}{d}} > \frac{hk^{\frac{1}{d}} (1 − m_{n_{k}}^{− \frac{1}{d}})}{2(m_{n_{k}} − 1)}⌊\frac{m_{n_{k}}}{h_{k}} ⌋,

but ⌊m_{n_{k}}/h_{k}⌋ = ⌊2 − h_{k}^{−1}⌋ = 1; then

ε_{h_{k}p_{k}}(K)(h_{k}p_{k})^{\frac{1}{d}} > \frac{hk^{\frac{1}{d}} (1 − (2h_{k} − 1)^{− \frac{1}{d}} )}{4(h_{k} − 1)}.  (13)

Since lim_{k} h_{k} = +∞, it follows from (12) and (13) that

lim sup_{n} ε_{n}(K)n^{\frac{1}{d}} = +∞.

3.4.2. Sierpiński gaskets. Let X be the plane ℝ^{2} with the Euclidean distance; a generalized version of the Sierpiński gasket may be constructed in the following way.

Let (k_{n}) be a sequence of integer numbers, with k_{n} ≥ 2 for all n ∈ ℕ; the sequence (m_{n}) is given by

m_{n} = ∑_{j=0}^{k_{n}} j = \frac{1}{2} k_{n}(k_{n} + 1) ∀n ∈ ℕ.

For all n ∈ ℕ, i ∈ I_{n} the contraction f_{i}^{(n)} is given by

f_{i}^{(n)}(x, y) = (\frac{x}{k_{n}} , \frac{y}{k_{n}} ) + (a_{i}^{(n)}, b_{i}^{(n)}),

where

a_{i}^{(n)} = \frac{k_{n} − (h_{i}^{(n)} + 1)^{2} + 2(i − 1)}{2k_{n}}, b_{i}^{(n)} = \frac{\sqrt{3}(k_{n} − h_{i}^{(n)} − 1)}{2k_{n}}

and 0 ≤ h_{i}^{(n)} < k_{n} is so that

∑_{j=0}^{h_{i}^{(n)}} j < i ≤ ∑_{j=0}^{h_{i}^{(n)}+1} j.

As before we set K = p(I).

Remark 13. If k_{n} = 2 for all n ∈ ℕ, then K is the Sierpiński gasket. Indeed, in this case, we have m_{n} = 3 for all n ∈ ℕ and

f_{1}^{(n)}(x, y) = (\frac{x}{2}, \frac{y}{2} ) + (\frac{1}{4}, \frac{\sqrt{3}}{4}),

f_{2}^{(n)}(x, y) = (\frac{x}{2}, \frac{y}{2} ),

f_{3}^{(n)}(x, y) = (\frac{x}{2}, \frac{y}{2} ) + (\frac{1}{2}, 0).  (14)

Remark 14. Even for the points of the Sierpiński gasket we can give a representation by means of series.

For the sake of simplicity we only consider the case in which k_{n} = 2 ∀n ∈ ℕ. Let us consider the functions f, g : {1, 2, 3} → ℤ,

f(i) = \{ \begin{matrix}1 & i = 1 \\ 0 & i = 2 \\ 2 & i = 3 \end{matrix} , g(i) = \{ \begin{matrix}1 & i = 1 \\ 0 & i > 1 \end{matrix} .  (15)

Let x_{0} = (0, 0), then for every n ∈ ℕ and i_{1}, i_{2}, …, i_{n} ∈ {1, 2, 3} we have

p_{n}(i_{1}, i_{2}, …, i_{n}) = \frac{1}{2} ∑_{j=1}^{n} \frac{1}{2^{j}} (f(i_{j}), \sqrt{3}g(i_{j})).  (16)

As in Remark 6, even (16) is proven by induction. Let n ∈ ℕ, by (14) and (15) we have

f_{i_{n}}^{(n)}(0, 0) = \frac{1}{4} (f(i_{n}), \sqrt{3}g(i_{n})).

Let us suppose that for k ∈ ℕ k ≤ n,

f_{i_{k+1}i_{k+2}···i_{n}}^{(k+1)}(0, 0) = \frac{1}{2} ∑_{j=k+1}^{n} \frac{1}{2^{j−k}} (f(i_{j}), \sqrt{3}g(i_{j})),

then

f_{i_{k}i_{k+1}···i_{n}}^{(k)}(0, 0) = \frac{1}{4} ∑_{j=k+1}^{n} \frac{1}{2^{j−k}} (f(i_{j}), \sqrt{3}g(i_{j})) + \frac{1}{4}(f(i_{k}), \sqrt{3}g(i_{k}))

= \frac{1}{2} ∑_{j=k}^{n} \frac{1}{2^{j−k+1}} (f(i_{j}), \sqrt{3}g(i_{j})).

In particular, for k = 1 we have (16).

It follows that

p(i) = \frac{1}{2} ∑_{j=1}^{+∞} \frac{1}{2^{j}} (f(i_{j}), \sqrt{3}g(i_{j})) ∀i ∈ I

and

K = \{\frac{1}{2} ∑_{j=1}^{+∞} \frac{1}{2^{j}} (f(i_{j}), \sqrt{3}g(i_{j})) | i_{j} ∈ \{1, 2, 3\} ∀j ∈ ℕ\}.

4. A measure on K

In this section X is a complete separable metric space.

We recall (§ 2.4) that for every n ∈ ℕ, l_{1}^{(n)}, l_{2}^{(n)}, …, l_{m_{n}}^{(n)} are real numbers in ]0, 1[ so that ∑_{j=1}^{m_{n}} l_{j}^{(n)} = 1, τ_{n} is a measure on I_{n} defined by τ_{n}(A) = ∑_{j∈A} l_{j}^{(n)} ∀A ⊆ I_{n} and τ is a unique Radon measure on I such that τ(I) = 1 and τ_{i_{1}} × τ_{i_{2}} × ⋯ × τ_{i_{n}} = π_{i_{1},i_{2},...,i_{n}♯}τ for any distinct i_{1}, i_{2}, …, i_{n} ∈ ℕ.

Definition. We set μ_{K} = p_{♯}τ.

Remark 1. K is the support of μ_{K} and μ_{K}(K) = 1.

Definition. Let (ν_{n}) be a sequence of Radon measures on X. We say that the sequence (ν_{n}) converges weakly to a Radon measure ν if

lim_{n} ∫_{X} fdν_{n} = ∫_{X} fdν ∀f ∈ C_{c}(X).

We denote this fact by writing ν_{n} ⇀ ν.

Remark 2. We recall that if ν is a Borel regular measure on X and for every x ∈ X there is r > 0 such that ν(B_{X}(x, r)) < +∞, then ν is a Radon measure on X. For the proof see [8], chapter 5, theorem V.5.3.

Proposition 4.1. Let ν be a Borel regular measure on X, with bounded support and such that ν(X) = 1. For every n ∈ ℕ we set

ν_{n} = ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)})f_{i_{1}i_{2}···i_{n}♯}ν.

Then ν_{n} ⇀ μ_{K}.

Proof. Let ν be a Borel regular measure on X with bounded support and such that ν(X) = 1; let C be the support of ν and let f ∈ C_{c}(X). We prove that

lim_{n} ∫_{X} fdν_{n} = ∫_{X} fdμ_{K}.

Let ε > 0 and δ > 0 be such that |f(x) − f(y)| < ε ∀x, y ∈ X with d_{X}(x, y) < δ; we consider n′_{ε} ∈ ℕ such that

( ∏_{k=1}^{n′_{ε}} ρ^{(k)}) sup_{x∈C} d_{X}(x, x_{0}) < δ.  (1)

SOME REMARKS ON FRACTALS 751

Moreover, let n′′_{ε} ∈ ℕ be such that |Q| ∏_{j=1}^{n′′_{ε}} ρ^{(j)} < δ; for every n ∈ ℕ with n > n′′_{ε} we have

d_{X}(p_{n}(π_{1,2,...,n}(i)), p(i)) = lim_{k} d_{X}(f_{i_{1}i_{2}···i_{n}}(x_{0}), f_{i_{1}i_{2}···i_{n}}(f_{i_{n+1}i_{n+2}···i_{k}}^{(n+1)}(x_{0})))

≤ |Q| ∏_{j=1}^{n} ρ^{(j)} < δ ∀i ∈ I ∀n ∈ ℕ n > n′′_{ε}

and then

|f(p_{n}(π_{1,2,...,n}(i))) − f(p(i))| < ε ∀i ∈ I ∀n ∈ ℕ n > n′′_{ε}.  (2)

Then, if n > max{n′_{ε}, n′′_{ε}}, we have

| ∫_{X} fdν_{n} − ∫_{X} fdμ_{K} |

= | ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)}) ∫_{X} fdf_{i_{1}i_{2}···i_{n}♯}ν − ∫_{X} fdp_{♯}τ|

≤ | ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)})( ∫_{X} f(f_{i_{1}i_{2}···i_{n}}(x))dν(x) − f(f_{i_{1}i_{2}···i_{n}}(x_{0})))|

+ | ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)})f(f_{i_{1}i_{2}···i_{n}}(x_{0})) − ∫_{I} f(p(i))dτ(i)|

≤ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)})| ∫_{X} f(f_{i_{1}i_{2}···i_{n}}(x))dν(x) − ∫_{X} f(f_{i_{1}i_{2}···i_{n}}(x_{0}))dν(x)|

+ |_{∏_{j=1}^{n} I_{j}}^{∫} f(p_{n}(i_{1}, i_{2}, …, i_{n}))d(τ_{1} × τ_{2} × ⋯ × τ_{n}) − ∫_{I} f(p(i))dτ(i)|

≤ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)}) ∫_{X} |f(f_{i_{1}i_{2}···i_{n}}(x)) − f(f_{i_{1}i_{2}···i_{n}}(x_{0}))|dν(x)

+ | ∫_{I} f(p_{n}(π_{1,2,...,n}(i)))dτ(i) − ∫_{I} f(p(i))dτ(i)|

≤ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)}) ∫_{C} |f(f_{i_{1}i_{2}···i_{n}}(x)) − f(f_{i_{1}i_{2}···i_{n}}(x_{0}))|dν(x)

+ ∫_{I} |f(p_{n}(π_{1,2,...,n}(i))) − f(p(i))|dτ(i).

By (1)

d_{X}(f_{i_{1}i_{2}···i_{n}}(x), f_{i_{1}i_{2}···i_{n}}(x_{0})) < δ ∀x ∈ C

and then

|f(f_{i_{1}i_{2}···i_{n}}(x)) − f(f_{i_{1}i_{2}···i_{n}}(x_{0}))| < ε ∀x ∈ C.  (3)

Then, by (3) and (2) it follows

| ∫_{X} fdν_{n} − ∫_{X} fdμ_{K} | < ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)})εν(C) + ετ(I) = 2ε

because

∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ⋯ ∑_{i_{n}=1}^{m_{n}} ( ∏_{k=1}^{n} l_{i_{k}}^{(k)}) = 1.

Acknowledgement

I wish to thank Prof. H. Triebel who invited me to give a lecture on these topics in Jena and M. Bricchi and Profs. H. Triebel and M. Zähle for the helpful discussion.

References

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3. K. J. Falconer, The geometry of fractal sets. Cambridge University Press, Cambridge, 1985.

4. K. J. Falconer, Fractal geometry. John Wiley & Sons, 1990.

5. G. B. Folland, Real analysis. John Wiley & Sons, 1990.

6. G. Follo, Some remarks on entropy numbers of sets with fractional dimension. Jenaer Schriften zur Mathematik und Informatik, 2000.

7. J. E. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30(1981), 713– 747.

8. K. Jacobs, Measure and integral. Academic Press, New York, 1978.

9. A. Jonsson and H. Wallin, Function spaces on subsets of ℝ^{N}. Harwood Academic Publisher, London, 1984.

10. P. Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge University Press, Cambridge, 1995.

11. H. Triebel, Fractals and spectra: related to Fourier analysis and function spaces. Birkhauser, 1997.

(Received 11.04.2001)

Author’s address: University of Milano Dipartimento di matematica Via Saldini n. 50 Milano 20133 Italy E-mail: follo@socrates.mat.unimi.it