Differentiable Structures of Central Cantor Sets

Rodrigo Bamón ^{∗} Carlos G. Moreira Sergio Plaza ^{†}Jaime Vera ^{‡}

Abstract

Central Cantor sets form a class of symmetric Cantor sets of the real line. Here we give a complete characterization of C^{k+α}–regularity of these Cantor sets. We also give a classification of central Cantor sets up to global and local diffeomorphisms. Examples of central Cantor sets with special dynamical and measure–theoretical properties are also provided. Finally we calculate the fractal dimensions of an arbitrary central Cantor set.

Introduction. Central Cantor sets in the real line form a particular but interesting class of Cantor sets. They are constructed in a simple way, and are useful in providing examples in several mathematical contexts. For instance, in analysis they can be used to prove the sharpness of some recent results concerning the clasical Morse-Sard Theorem [M2]. Even more abundant are their applications in Dynamical Systems. In particular, they were used by R. Bowen [Bo] to construct a horseshoe of class C^{1} with positive Lebesgue measure (see also our example IV.1 ), and by Samanni [Sa] and by Bamón–Plaza–Vera, [B-P-V] to give examples of regular Cantor sets whose self–arithmetic difference is a Cantor set with positive measure. Here we discuss a general class of such examples (see example IV.2). This is related to a conjecture of Palis (see [P]).

∗Partially Supported by Fondecyt Grant # 1930863

†Partially supported by Fondecyt Grant # 1941080 and Dicyt Grant # 9433 P.S.

‡Partially supported by Fondecyt Grant #1941080 and UCANORTE Grant.

Regular or dynamically defined Cantor sets constitutes an important class of subsets of the real line that naturally appear in dynamics (see [P-T], [P-T,1], [P-T,2], [P1], [P2], [M1], [N]). In the present work, we give a complete classifications, for any integer k ≥ 1 and α ∈ [0, 1], of central Cantor set that admit a structure of regular Cantor sets of class C^{k+α} ( Theorem A), as well as a classification of central Cantor sets up to global and local diffeomorphisms, Theorem B. The present work, although self-contained, is a natural continuation of [B-P-V].

We are able to find the explicit and simple conditions for the characterization and classification of C^{k+α} regularity of central Cantot sets, given in Theorems A and B, because of the simple and symmetrical structure of this class of Cantor sets. This contrasts with the conditions for regularity of more general classes of Cantor sets in the line. See [S], [B-F], [P-R], [Pr-T].

I- Basic definitions and Statement of Results

Let \underline{λ} = (λ_{k})_{k∈ℕ^{∗}} be a sequence of real numbers satisfying 0 < λ_{k} < \frac{1}{2} , for all k ∈ ℕ^{∗}, and let I ⊂ ℝ be a closed interval. We define the central Cantor set K_{\underline{λ},I} associated to the sequence \underline{λ} with support interval I as follows: Let P_{\underline{λ},I}^{0} = I. For each i ∈ ℕ^{∗}, P_{\underline{λ},I}^{i} is the set consisting of 2^{i} closed disjoint intervals, obtained from P_{\underline{λ},I}^{i−1} by removing from each of its intervals the central open interval of proportion 1−2λ_{i}. We call these removed intervals the gaps of the i–th step of the construction of K_{\underline{λ},I}, and the intervals of P_{\underline{λ},I}^{i} the fundamental intervals of the i–th step of the construction of K_{\underline{λ},I}. Finally define K_{\underline{λ},I} = ∩_{i=0}^{∞}P_{\underline{λ},I}^{i}.

Let a = |I| be the lenght of the interval I. Then the lenght of any n–th step fundamental interval is a · π_{n}, where π_{n} = π_{n}(\underline{λ}) = Π_{i=1}^{n}λ_{i}, and the lenght of any n–th step gap is a · g_{n}, where g_{n} = g_{n}(\underline{λ}) = (1 − 2λ_{n})π_{n−1}.

Case we do not mention the interval I we will assume I = [0, 1], and we will use the notations K\underline{λ} and P i ; \underline{λ} instead of K_{\underline{λ},I} and P_{\underline{λ},I}^{i}.

Now let us define an important class of Cantor sets, which naturally appear in Dynamical Systems. We say that a Cantor set K ⊂ ℝ is dynamically defined or regular if:

i) there exist compact disjoint intervals \overline{K}_{1}, \overline{K}_{2}, …, \overline{K}_{r} such that K ⊂ \overline{K}_{1} ∪ \overline{K}_{2} ∪ ⋯ ∪ \overline{K}_{r} and the boundary of each \overline{K}_{i} is contained in K; ii) there is an expanding map ψ : \overline{K}_{1} ∪ \overline{K}_{2} ∪ ⋯ ∪ \overline{K}_{r} → ℝ such that:

a) for each i = 1, …, r, ψ(\overline{K}_{i}) is the convex hull of the union of some intervals \overline{K}_{j}’s;

b) for each i = 1, …, r, and n large enough, ψ^{n}(K ∩ \overline{K}_{i}) = K;

c) K = ∩_{n=0}^{∞} ψ^{−n} (\overline{K}_{1} ∪ \overline{K}_{2} ∪ ⋯ ∪ \overline{K}_{r}).

We call \{ \overline{K}_{1}, \overline{K}_{2}, …, \overline{K}_{r}\} a Markov partition for K and D = \overline{K}_{1} ∪ \overline{K}_{2} ∪ ⋯ ∪ \overline{K}_{r} a Markov domain of K.

We say that a Cantor set is a regular of class C^{k} (resp. C^{k+α}) if ψ is a C^{k} (resp. C^{k+α}) function. In many applications we restrict our attention to regular Cantor sets which are at least C^{1+ε}, for some ε > 0.

The main goal of this work is to characterize the central Cantor sets which are regular of a given class of differentiability. We also classify central Cantor sets up to global and local diffeomorphisms. We have the following results.

Theorem A. Let K be a central Cantor set defined by the sequence \underline{λ} = (λ_{i})_{i∈ℕ^{∗}}. Then K admits a structure of regular Cantor set (of class at least C^{1}) if and only if there is r ∈ ℕ^{∗} such that the following conditions hold:

a) lim_{n→∞} \frac{λ_{n+r}}{λ_{n}} = 1 ;

b) lim_{n→∞} \frac{1 − 2λ_{n+r}}{1 − 2λ_{n}} = 1 ;

c) the sequence (λ_{n+1} · λ_{n+2} ⋯ λ_{n+r})_{n∈ℕ} converges to some λ ≠ 0.

Let r_{0} = min{r ∈ ℕ^{∗} : such that conditions a), b) and c) above hold}, and let k ∈ ℕ^{∗}. Then K admits a structure of regular Cantor set of class C^{k} if, and only if, the following equivalent conditions hold:

i) lim_{n→∞} \frac{|λ_{n+1} · λ_{n+2} ⋯ λ_{n+r_{0}} − λ|}{π_{n}^{k−1}} = 0 , where λ = lim_{n→∞} λ_{n+1} ⋯ λ_{n+r_{0}};

ii) lim_{n→∞} \frac{|λ_{n+1} · λ_{n+2} ⋯ λ_{n+r_{0}} − λ|}{λ^{(k−1)n/r_{0}}} = 0 .

If k > 1 then these conditions are also equivalent to the following:

iii) For each j = 1, 2, …, r_{0} the sequence (λ_{r_{0}n+j})_{n∈ℕ} converges to some μ_{j} ∈ (0, \frac{1}{2})

such that lim_{n→∞} \frac{|λ_{r_{0}n+j} − μ_{j}|}{λ^{(k−1)n}} = 0, j = 1, 2, …, r_{0}. We have λ = μ_{1} · μ_{2} ⋯ μ_{r_{0}}.

Let k ∈ ℕ^{∗} and α ∈ (0, 1]. Then K admits a structure of regular Cantor set of class C^{k+α} if and only if the following equivalent conditions hold:

i’) lim sup_{n→∞} \frac{|λ_{n+1} ⋯ λ_{n+r_{0}} − λ|}{π_{n}^{k−1+α}} < ∞;

ii’) lim sup_{n→∞} \frac{|λ_{n+1} ⋯ λ_{n+r_{0}} − λ|}{λ^{(k−1+α)n/r_{0}}} < ∞;

iii’) lim sup_{n→∞} \frac{|λ_{r_{0}n+j} − μ_{j}|}{λ^{(k−1+α)n}} < ∞, j = 1, 2, …, r_{0} .

Remarks. • Condition iii) above does not necessarily hold when k = 1. In fact, we will show that if 0 < a ≤ b ≤ \frac{1}{2} then there is a sequence \underline{λ} = (λ_{n})_{n∈ℕ} dense in [a, b] such that K\underline{λ} is C^{1}. (see example IV.3).

• It follows from above that μ_{j} ≠ 0, j = 1, 2, …, r_{0}, since μ_{1} · μ_{2} ⋯ μ_{r_{0}} = λ. Moreover, if K is regular of class C^{q}, q = k or q = k + α, with q > 1, then μ_{j} < \frac{1}{2}, j = 1, 2, …, r_{0}.

• The condition in order for K to admit a structure of regular Cantor set of class C^{∞} is that the equivalent conditions i), ii) and iii) must be satisfied for each k ∈ ℕ^{∗}.

• We will use Theorem A to provide examples, for each q ≥ 1 , q = k or q = k + α and c ∈ [0, 2), of central Cantor sets K\underline{λ} which are of class C^{q} but not of class C^{q+ε}, for all ε > 0, such that K\underline{λ} − K\underline{λ} = \{ x − y : x, y ∈ K\underline{λ} \} is a Cantor set of c Lebesgue measure.

• To endow K with a regular structure we need a Markov partition with at least 2^{r_{0}} intervals. For example the central Cantor set {1/3, 1/4, 1/3, 1/4, 1/3, ...} is C^{r} for every r ∈ ℕ ∪ \{ω\} with a Markov partition of four intervals: P 2 ; \underline{λ}.

Theorem B. Let K_{\underline{λ},I} and K_{˜\underline{λ},J} be two Central Cantor sets. Then there is a C^{1} diffeomorphism φ : I → J such that φ(K_{\underline{λ},I}) = K_{˜\underline{λ},J} if and only if the following conditions hold:

i) lim_{n→∞} \frac{1 − 2λ̃_{n}}{1 − 2λ_{n}} = 1;

ii) lim_{n→∞} \frac{π_{n}(˜\underline{λ})}{π_{n}(\underline{λ})} = a ≠ 0 .

In this case there is such a diffeomorphism satisfying φ′(x) = a \frac{|J|}{|I|} , for all x ∈ K_{\underline{λ},I} , and φ (P_{\underline{λ},I}^{r}) = P_{˜\underline{λ},J}^{r} , for all r ∈ ℕ , i.e., φ preserves the structure of central Cantor sets.

Let k ∈ ℕ^{∗}. There is such a diffeomorphism of class C^{k} if and only if moreover

iii) lim_{n→∞} \frac{|g_{n}(˜\underline{λ}) − a g_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k}} = 0 .

If α ∈ (0, 1] then there is such a diffeomorphism of class C^{k+α} if and only if moreover

iii’) lim sup_{n→∞} \frac{|g_{n}(˜\underline{λ}) − a g_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k+α}} < ∞.

Necessary and sufficient conditions for the existence of a local diffeomorphism (resp. a local diffeomorphism of class C^{q}, q = k or q = k + α , k ∈ ℕ^{∗} , α ∈ (0, 1]), φ : U → V such that φ(U ∩ K_{\underline{λ},I}) = V ∩ K_{\underline{λ̃},J}, where U and V are open intervals and U ∩ K_{\underline{λ},I} ≠ ∅ are that the corresponding conditions above should be verified for some \underline{λ̃}^{(p)}, p ∈ ℤ, instead of \underline{λ̃} where \underline{λ̃}^{(p)} = λ̃_{n+p}, if n + p ≥ 1, and λ̃_{n}^{(p)} = 1/3, otherwise.

As a consequence of Theorem B, we conclude that if K\underline{λ} has a structure of C^{q}-regular Cantor set, q = k or q = k + α , k ∈ ℕ^{∗}, α ∈ (0, 1], and if q > 1 then there is a C^{q} diffeomorphism φ : [0, 1] → [0, 1] such that φ(K\underline{λ}) = K\underline{λ̃}, where \underline{λ̃} is the periodic sequence defined by λ̃_{n} = μ\overline{n}, where \overline{n} ∈ \{1, 2, …, r_{0}\} satisfy n ≡ \overline{n} (mod r_{0}), μ\overline{n} being as in Theorem A. Note that K\underline{λ̃} is an affine Cantor set in the sense of [P-T].

Finally we compute the fractal dimensions of a central Cantor set: for each interval I and sequence \underline{λ} = (λ_{n})_{n∈ℕ},

• τ(K_{\underline{λ},I}) = τ(K\underline{λ}) = inf_{n∈ℕ} \frac{λ_{n}}{1 − 2λ_{n}} ;

• τ_{loc}(K_{\underline{λ},I}) = τ_{loc}(K\underline{λ}) = lim inf_{n→∞} \frac{λ_{n}}{1 − 2λ_{n}} ;

• d(K_{\underline{λ},I}) = d(K\underline{λ}) = lim sup_{n→∞} \frac{−n log 2}{log π_{n}(\underline{λ})} ;

• HD(K_{\underline{λ},I}) = HD(K\underline{λ}) = lim inf_{n→∞} \frac{−n log 2}{log π_{n}(\underline{λ})} .

II - Structure of diffeomorphisms of central Cantor sets

In this section we shall prove results that characterize combinatorially diffeomorphisms and local diffeomorphisms of central Cantor sets.

Proposition 1 Let K_{1} = K_{\underline{λ},J_{1}} and K_{2} = K_{˜\underline{λ},J_{2}} be two central Cantor sets, and let U, V be two intervals such that ∂U ∩ K_{1} = ∅ and ∂V ∩ K_{2} = ∅. Let φ : U → V be a C^{1} diffeomorphism such that φ(U ∩ K_{1}) = V ∩ K_{2}. Then for any ε > 0 there exist m, n_{0} ∈ ℕ such that

• U ∩ K_{1} = (∪_{r=1}^{m}I_{r}) ∩ K_{1}, where I_{r} is a fundamental interval of the n_{0}–th step of the construction of K_{1}, r = 1, 2, …, m;

• |I_{1}| = |I_{2}| = ⋯ = |I_{m}| < ε;

• φ(I_{r}) is a fundamental interval of the construction of K_{2}, r = 1, 2, …, m.

Remark. We can substitute the hypotheseses ∂U ∩ K_{1} = ∅ and ∂V ∩ K_{2} = ∅ by U ∩ K_{1} and V ∩ K_{2} are open and closed in K_{1} and K_{2}, respectively.

Proof. First we give some definitions. Let K be a Cantor set. A gap of K is a connected component of its complement, perhaps unbounded. We say that an interval J between two gaps U_{1}, U_{2} of K is a bridge if |U_{1}| ≥ \frac{6}{5} |U_{2}| or |U_{2}| ≥ \frac{6}{5} |U_{1}|, and there are no gaps U contained in J with |U| ≥ \frac{5}{6} min (|U_{1}|, |U_{2}|). Note that a bridge of a central Cantor set must necessarily be a fundamental interval of some step of its construction. In fact, a bridge must lie between two gaps of different steps, U_{1} , U_{2} where U_{1}, say, correspond to a more advanced step than U_{2}, and if the interval between U_{1} and U_{2} is not a fundamental interval then it must contain a gap of the same step of U_{1}, and hence with the same lenght of U_{1}, thus it cannot be a bridge.

We say that the n–th step of the construction of a central Cantor set K_{\underline{λ},I} is a good step if g_{n}(\underline{λ}) ≥ \frac{3}{2} g_{m}(\underline{λ}) , for all m > n. In this case, if g_{k}(\underline{λ}) ≥ \frac{6}{5} g_{n}(\underline{λ}) then any fundamental interval bounded by a gap of the k–th step and a gap of the n–th step of the construction of K_{\underline{λ},I} must be a bridge. Also, note that there are arbitrarily advanced good steps for any K_{\underline{λ},I}. In fact, if it is not the case, then there exist sufficiently large n_{0} such that for n ≥ n_{0} there is m > n with g_{m}(\underline{λ}) > \frac{2}{3} g_{n}(\underline{λ}), but this imply that for a positive constant (fixed) C and infinitely many n ∈ ℕ, g_{n}(\underline{λ}) > C · (\frac{2}{3})^{n} , which contradicts the

fact g_{n}(\underline{λ}) < π_{n−1}(\underline{λ}) < (\frac{1}{2})^{n−1} .

Now we can decompose U = Ũ_{1} ∪ Ĩ_{1} ∪ Ũ_{2} ∪ ⋯ ∪ Ĩ_{ℓ} ∪ Ũ_{ℓ+1}, for some ℓ ∈ ℕ in such a way that for each i, Ũ_{i} is a gap of K_{1}, in a more precise way Ũ_{1} and Ũ_{ℓ+1} are intersections of gaps of K_{1} with U, and Ĩ_{j} are fundamental intervals of K_{1} such that

sup_{x,y∈S_{I_{j}}} \frac{|φ′(y)|}{|φ′(x)|} < \frac{10}{9} .

Here, if Ĩ_{j} = [a, a + δ] , SĨ_{j} denote the interval [ a − 2δ , a + 3δ].

Let n_{0} ∈ ℕ such that the n_{0}–th step of the construction of K_{1} is a good step, the fundamental intervals of this n_{0}–th step have lenght less than ε and each of the gaps of this step have lenght less than \frac{1}{2} min_{1≤i≤ℓ+1} (|Ũ_{1}|). We will prove that the fundamental intervals of this n_{0}–th step contained in U, say I_{1}, I_{2}, …, I_{r} satisfies our statements.

We must prove that φ(I_{j}) is a fundamental interval of K_{2} , 1 ≤ j ≤ r. Let us consider the decomposition of one of these intervals Ĩ_{r}, corresponding to the n_{0}–th step of the construction of K_{1}, say Ĩ_{r} = I_{m} ∪ U_{1} ∪ I_{m+1} ∪ ⋯ ∪ U_{2^{s}−1} ∪ I_{m+2^{s}−1}, where U_{i} are gaps, and the decomposition is given in increasing order of the intervals. Then, by the above construction, I_{m} lies between Ũ_{r} and U_{1}, where |Ũ_{r}| > 2 |U_{1}|, and any gap contained in I_{m} have length less than or equal to \frac{2}{3} |U_{1}|, so, since sup_{x,y∈S_{I_{r}}} \frac{|φ′(y)|}{|φ′(x)|} < \frac{10}{9} (distortion estimated of φ in SĨ_{r}), φ(I_{m}) lies between φ(Ũ_{r}) and φ(U_{1}), where |φ(Ũ_{r})| > \frac{9}{5} |φ(U_{1})| and any gap contained in φ(I_{m}) have lenght less than or equal to \frac{20}{27} |φ(U_{1})| < \frac{5}{6} |φ(U_{1})|, hence φ(I_{m}) is a bridge and therefore a fundamental interval.

Now let us consider the decomposition of Ĩ_{r} corresponding to a good step of the construction of K_{1} such that the gaps of this step have length less than \frac{1}{2} min_{1≤i≤2^{s}} (|U_{i}|), say Ĩ_{r} = I′_{1} ∪ U′_{1} ∪ I′_{2} ∪ ⋯ ∪ U′_{2^{p}−1} ∪ I′_{2^{p}}. Note that each interval I_{ℓ} is decomposed in a union of gaps and 2^{p−s} of the intervals I′_{j} , m ≤ ℓ ≤ m + 2^{s} − 1. More precisely, I_{m+r} is the convex hull of ∪_{i=1}^{2^{p−s}} I′_{2^{p−s}r+i}. We know that φ(I_{m}) is a fundamental interval of K_{2} and, by the same reason, φ(I′_{1}) is also a fundamental interval of K_{2}, say of the n_{1}–th step.

Now we will prove that φ send the gaps U′_{1}, U′_{3}, …, U′_{2^{p}−1} of the same step of the fundamental intervals I′_{j} in the gaps of the n_{1}–step contained in φ(Ĩ_{r}). First of all, φ(U′_{1}) is the first gap after φ(I′_{1}), which is the first gap of the n_{1}–th step contained in φ(Ĩ_{r}). Suppose by induction, that φ(U′_{2k−1}) is the k–th gap of the n_{1}–th step contained in φ(Ĩ_{r}).

We have |U′_{2k+1}| = |U′_{2k−1}|, and the left end of U′_{2k+1} is at least 2 · |I′_{2k}| after the right end of U′_{2k−1}. Then, because of the distortion estimate of φ , |φ(U′_{2k+1})| > \frac{5}{6} |φ(U′_{2k−1})|, and the left end of φ(U′_{2k+1}) is at least \frac{5}{3} |φ(I′_{2k})| ≥ \frac{3}{2} |φ(I′_{1})| after the right end of φ(U′_{2k−1}). Thus, either φ(U′_{2k+1}) = V_{k+1}, the (k +1)–th gap of the n_{1}–th step contained in φ(Ĩ_{r}) or either φ(U′_{2k+1}) is after V_{k+1}. The last case can not occur, because |V_{k+1}| = |V_{k}| = |φ(U′_{2k−1})|, and the left end of V_{k+1} is at least 2 |φ(I′_{1})| after the right end of φ(U′_{2k−1}), and so, by the same argument applied with φ^{−1} instead of φ , φ^{−1}(V_{k+1}) would be a gap of K_{1} between U′_{2k−1} and U′_{2k+1}, which lies at least \frac{3}{2} |I′_{1}| = \frac{3}{2} |I′_{2k}| after the right end of U′_{2k−1}, satisfying |φ^{−1}(V_{k+1})| ≥ \frac{3}{4} |U′_{2k−1}|, which is absurd, then our claim is proved.

Since, U′_{2^{p−s}r} = U_{r} , 1 ≤ r ≤ 2^{s} − 1, we have, by the construction of I′_{j} that U′_{2^{p−s}r} is a gap that lies between U′_{2^{p−s}r−1} and U′_{2^{p−s}r+1}, and |U′_{2^{p−s}r}| ≥ 2 |U_{2^{p−s}r−1}| , so φ(U′_{2^{p−s}r}) is a gap that lies between V_{2^{p−s−1}r} and V_{2^{p−s−1}r+1} and |φ(U′_{2^{p−s}r})| ≥ \frac{3}{2} |V_{2^{p−s−1}r}|, so φ(U′_{2^{p−s}r}) must be the unique gap of older step between V_{2^{p−s−1}r} and V_{2^{p−s−1}r+1}. This proves that φ(I_{m+r}) is the convex hull of the union of 2^{p−s} consecutive fundamental intervals of the n_{1}–th step of the construction of K_{2}, 0 ≤ r ≤ 2^{s} − 1, and since φ(I_{m}) is itself a fundamental interval of the construction of K_{2}, all the φ(I_{m+r}) , 0 ≤ r ≤ 2^{s} − 1 must also be fundamental intervals of the same step of the construction of K_{2} .

Corollary 1 Let K_{\underline{λ},I} and K_{\underline{_{λ}},J} be two central Cantor sets, and let φ : I → J be a C^{1} diffeomorphism such that sup_{x,y∈I} \frac{|φ′(y)|}{|φ′(x)|} ≤ 4 and φ(K_{\underline{λ},I}) = K_{\underline{λ̃},J} . Then φ map the decomposition of the m–th step of K_{\underline{λ},I} into the decomposition of the m–th step of K_{\underline{λ̃},J} , i.e., φ(P_{\underline{λ},I}^{m}) = P_{\underline{λ̃},J}^{m} , for all m ∈ ℕ.

Proof. Let m ∈ ℕ, and consider the decomposition I = I_{1} ∪ U_{1} ∪ ⋯ ∪ U_{2^{m}−1} ∪ I_{2^{m}} of the m–th step of the construction of K_{\underline{λ},I}. By Proposition 1, there exist n_{0} ∈ ℕ such that if I = I′_{1} ∪ U′_{1} ∪ … ∪ U′_{2^{n_{0}}−1} ∪ I′_{2^{n_{0}}} is the n_{0}–th step of the construction of K_{\underline{λ},I} then |I′_{1}| = |I′_{2}| = ⋯ = |I′_{2^{n_{0}}}| < |I_{1}|, and φ(I′_{r}) is a fundamental interval of the construction of K_{2}, r = 1, 2, …, 2^{n_{0}}, but then all φ(I′_{r}) must be fundamental intervals of the n_{0}–th step of the construction of K_{2}, otherwise there will be fundamental intervals φ(I′_{r_{1}}) of a step before the n_{0}–th and φ(I′_{r_{2}}) of a step after the n_{0}–th, and we will have \frac{|φ(I′_{r_{1}})|}{|φ(I′_{r_{2}})|} > 4,

which is a contradiction, since |I′_{r_{1}} | = |I′_{r_{2}} | and sup_{x,y∈I} \frac{|φ′(y)|}{|φ′(x)|} ≤ 4. This proves our result for the n_{0}–th step of the construction of K_{1} and a fortiori for the m-th step, which is less advanced than the n_{0}-th.

Corollary 2 Under conditions of Proposition 1, φ map decompositions of I_{r} in fundamental intervals and gaps in the corresponding ones of φ(I_{r}), 1 ≤ r ≤ m, provided the I_{r} are conveniently chosen. In particular, φ maps sufficiently small fundamental intervals of K_{1} in fundamental intervals of K_{2}.

Proof. This is an immediate consequence of Corollary 1, since if K is a central Cantor set and I is a fundamental interval of K then I ∩ K is itself a central Cantor set, and we have sup_{x,y∈I_{r}} \frac{|φ′(y)|}{|φ′(x)|} < \frac{10}{9} < 4 , 1 ≤ r ≤ m, for I_{r} obtained as in the proof of Proposition 1.

Example II.1 Let λ ∈ (0, \frac{1}{2}), and K_{λ} = K\underline{λ}, where λ_{n} = λ , for all n ∈ ℕ. Then there are C^{∞}- diffeomorphisms φ : [0, 1] → [0, 1] such that φ(K_{λ}) = K_{λ}, whose distortion Dφ = sup_{x,y∈[0,1]} \frac{|φ′(y)|}{|φ′(x)|} are arbitrarily close to 1/λ^{2}. In fact, define

φ(x) = \{ \begin{matrix}\frac{x}{λ} , & x ∈ [0, λ^{2}] \\ x + 1 − 2λ + λ^{2} , & x ∈ [λ(1 − λ), λ] \\ λx + 1 − λ , & x ∈ [1 − λ, 1] . \end{matrix}

Note that [0, λ] is a fundamental interval of K_{λ} but φ([0, λ]) = [0, 1 − λ + λ^{2}] is not. Letting λ tend to \frac{1}{2}, we see that the constant 4 in Corollary 1 is the best possible.

Proposition 2 Let K_{1} = K_{\underline{λ},I} and K_{2} = K_{\underline{λ̃},J} be two central Cantor, and let φ : U → V be a C^{1} diffeomorphism, where U and V are intervals such that int(U) ∩ K_{1} ≠ ∅ and φ(U ∩ K_{1}) = V ∩ K_{2}. Then there exist p ∈ ℤ such that lim_{n→∞} \frac{1−2λ̃_{n+p}}{1−2λ_{n}} = 1 and lim_{n→∞} \frac{π_{n+p}(\underline{λ̃})}{π_{n}(\underline{λ})} = a, for some a ≠ 0. We also have that φ′|_{K_{1}∩U} is locally constant. Let k ∈ ℕ^{∗}. If φ is C^{k}, then there exist p ∈ ℤ as before such that

lim_{n→∞} \frac{|g_{n+p}(\underline{λ̃}) − a g_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k}} = 0

and

lim_{n→∞} \frac{|π_{n+p}(\underline{λ̃}) − a π_{n}(\underline{λ})|}{(π_{n}(\underline{λ}))^{k}} = 0 .

If α ∈ (0, 1] and φ ∈ C^{k+α} then there exist p ∈ ℤ as before such that

lim sup_{n→∞} \frac{|g_{n+p}(\underline{λ̃}) − a g_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k+α}} < ∞

and

lim sup_{n→∞} \frac{|π_{n+p}(\underline{λ̃}) − a π_{n}(\underline{λ})|}{(π_{n}(\underline{λ}))^{k+α}} < ∞ .

Proof. We can, if necessary, reduce the interval U to satisfy hypothesis of Proposition 1. We will restrict our attention to φ|_{I_{j}}, where I_{j} is one of the fundamental intervals obtained in Proposition 1, Corollary 2. I_{j} is a fundamental interval of the n_{0}–th step of the construction of K_{1}, for some n_{0} ∈ ℕ, and φ(I_{j}) is a fundamental interval of the (n_{0} + p)–th step of the construction of K_{2}, for some p ∈ ℤ, such that, for r ∈ ℕ , φ map fundamental intervals and gaps of the (n_{0} +r)–th step of the construction of K_{1} contained in I_{j} in fundamental intervals and gaps of the (n_{0} + p + r)–th step of the construction of K_{1}. In particular, if x ∈ K_{1} ∩ I_{j}, and I_{n}(x) is the fundamental interval of the n–th step containing x, then φ′(x) = lim_{n→∞} \frac{|φ(I_{n}(x))|}{|I_{n}(x)|} = \frac{|J|}{|I|} · lim_{n→∞} \frac{π_{n+p}(\underline{λ̃})}{π_{n}(\underline{λ})}, so lim_{n→∞} \frac{π_{n+p}(\underline{λ̃})}{π_{n}(\underline{λ})} exists and it is equal to some a ≠ 0 such that φ′(x) = a \frac{|J|}{|I|} , for all x ∈ K_{1} ∩ I_{j}. If U_{n}(x) is the gap of the n–th step of the construction of K_{1} contained in I_{n−1}(x) then lim_{n→∞} \frac{1−2λ̃_{n+p}}{1−2λ_{n}} = lim_{n→∞} \frac{|φ(U_{n}(x))|/|φ(I_{n−1}(x))|}{|U_{n}(x)|/|I_{n−1}(x))|} = \frac{lim_{n→∞} |φ(U_{n}(x))|/|U_{n}(x)|}{lim_{n→∞} |φ(I_{n−1}(x))|/|I_{n−1}(x)|} = \frac{φ′(x)}{φ′(x)} = 1. Let k ∈ ℕ^{∗} and suppose φ is C^{k}. If k = 1 the results are proved. If k > 1, since φ′|_{K_{1}∩I_{j}} is constant, φ^{(r)}|_{K_{1}∩I_{j}} identically 0, for r = 2, 3, …, k. Then |\frac{π_{n+p}(\underline{λ̃})}{π_{n}(\underline{λ})} − a | = \frac{|I|}{|J|} ·

|\frac{|φ(I_{n}(x))|}{|I_{n}(x)|} − φ′(x)| = \frac{|I|}{|J|} |φ′(c(n, x)) − φ′(x)|, for some c(n, x) ∈ I_{n}(x). Since, φ^{(r)}(x) = 0 , 2 ≤ r ≤ k, we have lim_{y→x} \frac{|φ′(y)−φ′(x)|}{|y−x|^{k−1}} = 0, and since |c(n, x) − x| ≤ |I_{n}(x)| = |I| · π_{n}(\underline{λ}),

we have lim_{n→∞} \frac{|φ′(c(n,x))−φ′(x)|}{(π_{n}(\underline{λ}))^{k−1}} = 0, and then lim_{n→∞} \frac{|π_{n+p}(\underline{_{λ}})−a π_{n}(\underline{λ})|}{(π_{n}(\underline{λ}))^{k}} = 0. Let (U_{n}) be

a sequence of gaps of the n–th step of the construction of K_{1} contained in I_{j}, and x_{n} be

the left end of U_{n}. Then |\frac{g_{n+p}(\underline{_{λ}})}{g_{n}(\underline{λ})} − a| = \frac{|I|}{|J|} |\frac{|φ(U_{n})|}{|U_{n}|} − φ′(x_{n})| = \frac{|I|}{|J|} |φ′(c_{n}) − φ′(x_{n})|, for

some c_{n} ∈ U_{n}, and our result about the g′_{n}s follows from lim_{n→∞} \frac{|φ′(c_{n})−φ′(x_{n})|}{|c_{n}−x_{n}|^{k−1}} = 0, which

follow from φ′(c_{n}) = φ′(x_{n}) + \frac{φ^{(k)}(ξ_{n})}{(k−1)!} (c_{n} − x_{n})^{k−1}, for some ξ_{n} ∈ (x_{n}, c_{n}) ⊂ U_{n}. We have lim_{n→∞} φ^{(k)}(ξ_{n}) = 0, since φ is C^{k}. The claims in case φ is C^{k+α} follows exactly as before from |φ′(y) − φ′(x)| ≤ M |x − y|^{k+α−1} , for all y ∈ I_{j} , and all x ∈ K_{1} ∩ I_{j}, where M is a constant. This follow from φ′(y) − φ′(x) = \frac{φ^{(k)}(c)}{(k−1)!} (y − x)^{k−1}, where c is between x and y, and |φ^{(k)}(c)| = |φ^{(k)}(c) − φ^{(k)}(x)| ≤ M |c − x|^{α} ≤ M |y − x|^{α} , for k ≥ 2, since φ is C^{k+α}, and, for φ ∈ C^{1+α}, |φ′(y) − φ′(x)| ≤ M |y − x|^{α}.

Remark. The conclusions of Proposition 2 hold for any p, such that there is a fundamental interval I_{j} of the n_{0}–th step of the construction of K_{1} as in Proposition 1, Corollary 2 such that φ(I_{j}) is a fundamental interval of the (n_{0} + p)–th step of the construction of K_{2}. In this case we have φ′(x) = a \frac{|J|}{|I|} , for all x ∈ K_{1} ∩ I_{j}, where a = lim_{n→∞} \frac{π_{n+p}(\underline{λ̃})}{π_{n}(\underline{λ})}.

Corollary If K\underline{λ} admits a structure of regular Cantor (at least of class C^{1}), then there is r ∈ ℕ^{∗} such that lim_{n→∞} \frac{λ_{n+r}}{λ_{n}} = 1 , and lim_{n→∞} \frac{1−2λ_{n+r}}{1−2λ_{n}} = 1, and the sequence (λ_{n+1} · λ_{n+2} ⋯ λ_{n+r})_{n∈ℕ} converges to some λ ≠ 0. If K\underline{λ} admits a structure of regular Cantor C^{k} (resp. C^{k+α}) then there exist r as before such that lim_{n→∞} \frac{|λ_{n+1}·λ_{n+2}···λ_{n+r}−λ|}{π_{n}^{k−1}} = 0 (resp. lim sup_{n→∞} \frac{|λ_{n+1}·λ_{n+2}···λ_{n+r}−λ|}{π_{n}^{k+α−1}} < ∞.

Proof. If K\underline{λ} is a regular Cantor defined by ψ as defined in section I., this is a direct consequence of Proposition 2 (and the above remark) applied to K_{1} = K_{2} = K\underline{λ}, and V being some connected component of the domain of ψ, U = ψ(V) and φ = (ψ|_{V})^{−1}.

III - Construction of diffeomorphisms between central Cantor sets

Proposition 3 Let K_{1} = K_{\underline{λ},I} and K_{2} = K_{\underline{λ̃},J} be two central Cantor sets such that lim_{n→∞} \frac{1−2λ̃_{n}}{1−2λ_{n}} = 1 and lim_{n→∞} \frac{π_{n}(\underline{λ̃})}{π_{n}(\underline{λ})} = α ≠ 0. Then there is a C^{1} diffeomorphism φ : I → J such that φ(K_{\underline{λ},I}) = K_{\underline{λ̃},J}, and φ(P_{\underline{λ},I}^{r}) = P_{\underline{λ̃},J}^{r} for all r ∈ ℕ, and thus φ′(x) = α \frac{|J|}{|I|} , for all x ∈ K_{\underline{λ},I}.

Let k ∈ ℕ^{∗}. If lim_{n→∞} \frac{|g_{n}(\underline{λ̃})−α g_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k}} = 0 then we can obtain φ of class C^{k}.

Let β ∈ (0, 1]. If lim sup_{n→∞} \frac{|g_{n}(\underline{λ̃})−α g_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k+β}} < ∞, then we obtain φ of class C^{k+β}.

Proof. We will define φ in the gaps of K_{\underline{λ},I}, the definition of φ in K_{\underline{λ},I} being done by continuity. Let n ∈ ℕ, I = I_{1}^{(n)} ∪ U_{1}^{(n)} ∪ ⋯ ∪ U_{2^{n}−1}^{(n)} ∪ I_{2^{n}}^{(n)}, and J = J_{1}^{(n)} ∪ V_{1}^{(n)} ∪ ⋯ ∪ V_{2^{n}−1}^{(n)} ∪ J_{2^{n}}^{(n)}, be the corresponding decompositions of the n–th step of the constructions of K_{1} and K_{2}, respectively. We will have φ(I_{r}^{(n)}) = J_{r}^{(n)}and φ(U_{r}^{(n)}) = V_{r}^{(n)}, for each r, n. In

particular, if x ∈ K_{1}, φ′(x) = lim_{n→∞} ^{\frac{|J_{r}^{(n)}|}{|I_{r}^{(n)}|}} = lim_{n→∞} \frac{|J|·π_{n}(\underline{λ̃})}{|I|·π_{n}(\underline{λ})}, since |I_{r}^{(n)}| = |I| · π_{n}(\underline{λ})

and |J_{r}^{(n)}| = |J| · π_{n}(\underline{λ̃}) , for all r, so φ′(x) = α \frac{|J|}{|I|} , for all x ∈ K_{\underline{λ},I}.

Let ψ : ℝ → ℝ be a C^{∞} function such that ψ(x) ≥ 0 , for all x ∈ ℝ , and ψ(x) = 0 , x ∉ [0, 1], and ∫_{0}^{1} ψ(x) dx = 1. Say ψ(x) = \frac{ψ̃(x)}{∫_{0}^{1} ψ̃(t) dt}, where

ψ̃(x) = \{ \begin{matrix}e^{\frac{1}{x(x−1)}} , & x ∈ [0, 1] \\ 0, & x ∉ [0, 1] , \end{matrix}

Let U = (a, b) be a gap of the n–th step of K_{\underline{λ},I} and let V = (c, d) be the corresponding gap of the n–th step of K_{\underline{λ̃},J}. We have |U| = |I| · g_{n}(\underline{λ}) and |V | = |J| · g_{n}(\underline{λ̃}). We define, for n large enough φ|_{U} as follows: φ(a) = c and for x ∈ [a, b], φ′(x) =

\frac{|J|}{|I|} (α + (\frac{g_{n}(\underline{_{λ}})}{g_{n}(\underline{λ})} − α) ψ (\frac{x−a}{b−a})). Since lim_{n→∞} \frac{g_{n}(\underline{λ̃})}{g_{n}(\underline{λ})} = α, this expression for φ′(x) is

strictly positive for n ≥ n_{0}, where n_{0} is a sufficiently large positive integer. For the gaps (a, b) of the steps before the n_{0}–th, we define φ in [a, b] as follows: where (c, d) is the gap of K_{2} corresponding to (a, b), set φ(a) = c, and for x ∈ [a, b] , set φ′(x) = α \frac{|J|}{|I|} · exp (ρ ψ (\frac{x−a}{b−a})), where ρ ∈ ℝ is choosen so that ∫_{a}^{b} φ′(x) dx = d − c.

The function φ just constructed satisfies φ(K_{\underline{λ},I}) = K_{\underline{_{λ}},J}, and is a C^{1}–diffeomorphism, since φ′|_{I−K_{\underline{λ},I}} is strictly positive and tends to α \frac{|J|}{|I|} in the gaps of the n–th step,

uniformly when n → ∞ and lim_{n→∞} ^{\frac{|φ(I_{r}^{(n)})|}{|I_{r}^{(n)}|}} = α \frac{|J|}{|I|}, uniformly, so φ′|_{K_{\underline{λ},I}} ≡ α \frac{|J|}{|I|}, and φ′

is continuous in I. Suppose we have lim_{n→∞} \frac{|g_{n}(\underline{λ̃})−α g_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k}} = 0. Then φ is of class C^{k}, the derivative of order k of φ in a gap (a, b) of the n–th step are given, for k ≥ 2, by φ^{(k)}(x) = \frac{|J|}{|I|^{k}} \frac{(g_{n}(\underline{λ̃})−α g_{n}(\underline{λ}))}{(g_{n}(\underline{λ}))^{k}} · ψ^{(k)} (\frac{x−a}{b−a}), which tends to 0 uniformly when n → ∞, and φ^{(k)}|_{K_{\underline{λ},I}} ≡ 0. So φ^{(k)} a continuous function in I.

Now suppose β ∈ (0, 1] and lim sup_{n→∞} \frac{|g_{n}(\underline{λ̃})−α g_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k+β}} < ∞. Let x, y ∈ I. Since φ^{(k)} is constant in K_{\underline{λ},I} , for all k ∈ ℕ^{∗}, we have |φ^{(k)}(x) − φ^{(k)}(y)| ≤ 2 max (|φ^{(k)}(x) − φ^{(k)}(ã)| , |φ^{(k)}(y) − φ^{(k)}(b̃)|), where ã = x (resp. b̃ = y) if x ∈ K_{\underline{λ},I} (resp. y ∈ K_{\underline{λ̃},J}) and ã (resp. b̃) is an end of the gap containing x (resp. y). Thus we can suppose that x and y are in the closure of the same gap (a, b), and so |φ^{(k)}(x) − φ^{(k)}(y)| = \frac{|J|}{|I|^{k}} \frac{|g_{n}(\underline{λ̃})−αg_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k}}·|ψ^{k+1}(ξ)|·\frac{|x−y|}{b−a}, for some ξ ∈ [0, 1], where (a, b) is a gap of the n-th step of K_{\underline{λ},I}, and so

\frac{|φ^{(k)}(x) − φ^{(k)}(y)|}{|x − y|^{β}} = \frac{|J|}{|I|^{k+β}} · \frac{|g_{n}(\underline{λ̃}) − αg_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k+β}} · |φ^{(k+1)}(ξ)| (\frac{|x − y|}{b − a})^{1−β} ,

which is bounded (since φ^{(k+1)} is bounded and |x − y| ≤ b − a) by

\frac{|J|}{|I|^{k+β}} · sup_{n∈ℕ} \frac{|g_{n}(\underline{λ̃}) − α g_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k+β}} · sup_{x∈[0,1]} |ψ^{(k+1)}(x)| .

So φ is of class C^{k+β}.

Corollary Let K_{1} = K_{\underline{λ},I} and K_{2} = K_{\underline{λ̃},J} be two central Cantor sets, and p ∈ ℤ such that lim_{n→∞} \frac{1−2λ̃_{n+p}}{1−2λ_{n}} = 1 and lim_{n→∞} \frac{π_{n+p}(\underline{λ̃})}{π_{n}(\underline{λ})} = α ≠ 0. Then, if m ∈ ℕ^{∗} , m + p ∈ ℕ^{∗} , Ĩ is a fundamental interval of the m–th step of K_{1} and J̃ is a fundamental interval of the (m + p)–th step of K_{2}, then there exist a C^{1} diffeomorphism φ : Ĩ → J̃ such that φ(K_{1} ∩ Ĩ) = K_{2} ∩ J̃, φ(P_{\underline{λ},I}^{r} ∩ Ĩ) = P_{\underline{λ̃},J}^{r+p} ∩ J̃, for r ≥ m, and thus φ′(x) = α \frac{|J|}{|I|} , for

all x ∈ K_{1} ∩ Ĩ. Let k ∈ ℕ and β ∈ (0, 1]. Then, if lim_{n→∞} \frac{|g_{n+p}(\underline{λ̃})−αg_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k}} = 0 (resp.

lim sup_{n→∞} \frac{|g_{n+p}(\underline{λ̃})−αg_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k+β}} < ∞) then we can obtain φ of class C^{k} (resp. of class C^{k+β}).

Remark. If α = 0 the result still valid, except for φ be not a diffeomorphism, but is a homeomorphism with φ′(x) = 0 ⇔ x ∈ K_{1}. This is used in [M2] to give examples related to critical situations of Morse-Sard’s theorem.

IV - Proof of Theorems

Proof of Theorem A. Let us first prove the stated equivalences. If k = 1 then the conditions i) and ii) are the same. Let P_{n} = λ_{n+1} · λ_{n+2} ⋯ λ_{n+r_{0}}. If q > 1 , ( q = k or q = k+α) then each of the conditions i), ii), i’), ii’) implies that P_{n} converges exponentially to λ, i.e., there are constants M > 0 and C ∈ (0, 1) such that |P_{n} − λ| ≤ M · C^{n} , for all n ∈ ℕ. So Π_{n=1}^{∞} (P_{r_{0} n}/λ) converges exponentially, so π_{n}^{q−1} = O (λ^{(q−1)n/r_{0}}) and λ^{(q−1)n/r_{0}} = O (π_{n}^{q−1}). (We are using the notations a_{n} = O(b_{n}) ⇔ lim sup_{n→∞} \frac{a_{n}}{b_{n}} < ∞, and a_{n} = o(b_{n}) ⇔ lim_{n→∞} \frac{a_{n}}{b_{n}} = 0 .)

So i) ⇔ ii) and i’) ⇔ ii’). If k > 1 (resp. k + α > 1)

\frac{λ_{r_{0}(n+1)+j}}{λ_{r_{0}n+j}} = \frac{P_{r_{0}n+j}}{P_{r_{0}n+j−1}} = \frac{λ+o(λ^{(k−1)n})}{λ+o(λ^{(k−1)n})} , ( resp. \frac{λ+O(λ^{(k+α−1)n})}{λ+O(λ^{(k+α−1)n})} = 1 + o(λ^{(k−1)n})) ( resp. 1 + O(λ^{(k+α−1)n}) ).

So λ_{r_{0}(n+1)+j} = λ_{r_{0}n+j} + o(λ^{(k−1)n}) (resp. λ_{r_{0}n+j} + O(λ^{(k+α−1)n})), and thus (λ_{r_{0}n+j}) converges exponentially to some μ_{j}, and |λ_{r_{0}n+j} − μ_{j}| = ∑_{m=n}^{∞} |λ_{r_{0}m+j} − λ_{r_{0}(m+1)+j}| = ∑_{m=n}^{∞} o (λ^{(k−1)m}) (resp. ∑_{m=n}^{∞} O (λ^{(k+α−1)m}) = o (λ^{(k−1)n}) ) (resp. O (λ^{(k+α−1)n})). If some of the μ_{j} is equal to 1/2 then (λ_{r_{0}n+j}) converges exponentially to 1/2, contradicting lim_{n→∞} \frac{1−2λ_{n+r_{0}}}{1−2λ_{n}} = 1. Moreover, μ_{1} · μ_{2} · μ_{3} ⋯ μ_{r_{0}} = lim_{n→∞} P_{r_{0}n} = λ ≠ 0, so μ_{j} ≠ 0 , for all j. Thus we proved μ_{j} ∈ (0, 1/2) , for all j. Reciprocally, if λ_{r_{0}n+j} = μ_{j} + o(λ^{(k−1)n}) (resp. λ_{r_{0}n+j} = μ_{j} + O(λ^{(k+α−1)n}) then P_{n} = λ + o(λ^{(k−1)n/r_{0}}) (resp. P_{n} = λ + λ^{(k+α−1)n/r_{0}}.) Then i′) ⇔ ii′) ⇔ iii′) and, if k > 1, i) ⇔ ii) ⇔ iii). Proposition 2, Corollary shows that if K is C^{k} (resp. K is C^{k+α}) then conditions a), b), c) and condition i) (resp. i’)) hold for some r instead of r_{0}. We will show that this holds also for r = r_{0}. In the case C^{1} there is nothing to prove. If k > 1 (resp. k + α > 1), lim_{n→∞} \frac{λ_{n+r_{0}}}{λ_{n}} = 1 implies μ_{i} = μ_{j} if i ≡ j

(mod r_{0}), so lim_{n→∞} \frac{|λ_{rn+j}−μ_{j}|}{λ^{(k−1)n}} = 0, 1 ≤ j ≤ r, if and only if, lim_{n→∞} ^{\frac{|λ_{n}−μ_{\overline{n}_{(r)}} |}{π_{n}^{k−1}}} = 0

(resp. lim sup_{n→∞} \frac{|λ_{rn+j}−μ_{j}|}{λ^{(k−1)n}} < ∞, 1 ≤ j ≤ r, if and only if lim sup_{n→∞} ^{\frac{|λ_{n}−μ_{\overline{n}_{(r)}} |}{π_{n}^{k+α−1}}} < ∞), where \overline{n}_{(p)} ∈ \{1, 2, …, p\} satisfies n ≡ \overline{n}_{(p)} (mod p), but, since μ_{\overline{n}_{(r)}} = μ_{\overline{n}_{(r_{0})}} (because r_{0} = min\{p ∈ G / p > 0\}, where G = \{p ∈ ℤ / lim_{n→∞} \frac{λ_{n+p}}{λ_{n}} = lim_{n→∞} \frac{1−2λ_{n+p}}{1−2λ_{n}} = 1 \} is a subgroup of ℤ, so G = r_{0} ℤ and r_{0}|r, thus \overline{n}_{(r)} ≡ \overline{n}_{(r_{0})}(mod r_{0})), we see that the validity of i) (resp. of i’)) for r and for r_{0} are equivalent. This shows that necessity ot the conditions.

To prove the sufficiency of the conditions, note that lim_{n→∞} \frac{|g_{n+r_{0}}(\underline{λ})−λg_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k}} = 0 (resp.

lim sup_{n→∞} \frac{|g_{n+n_{0}}(\underline{λ})−λg_{n}(\underline{λ})|}{(g_{n}(\underline{λ}))^{k+α}} < ∞), since \frac{g_{n+r_{0}}(\underline{λ})}{g_{n}(\underline{λ})} = \frac{1−2λ_{n+r_{0}}}{1−2λ_{n}} · \frac{π_{n+r_{0}−1}(\underline{λ})}{π_{n−1}(\underline{λ})} = (1 + o(πk−1_{n}))(1 +

o(π_{n}^{k−1}) (resp. (1+O(πk+α−1_{n}))(λ+O(πk+α−1_{n}))) = λ+O(πk−1_{n}) (resp. λ+O(πk+α−1_{n})), and

π_{n}(\underline{λ}) = O(g_{n}(\underline{λ})), provided k > 1 (resp. k + α > 1), since lim sup_{n→∞} λ_{n} < 1/2. The case of class C^{1} is trivial. Now we can use Proposition 3, Corollary, in the following way: If I = I_{1}∪U_{1}∪I_{2}∪⋯∪U_{2^{r_{0}}−1}∪I_{2^{r_{0}}} is the decomposition of the r_{0}–th step of the construction of K then there are diffeomorphisms φ_{j} : I → I_{j} of class C^{k} (resp. C^{k+α}) such that φ_{j}(K) = K ∩ I_{j} and φ_{j} map decompositions of the construction of K into corresponding decompositions of the construction of K ∩I_{j} (so φ′_{j}|_{K} ≡ lim_{n→∞} λ_{n+1} ·λ_{n+2} ⋯ λ_{n+r_{0}} = λ). Let ψ : P_{\underline{λ},I}^{r_{0}} → I be defined by ψ|_{I_{j}} = φ_{j}^{−1}. Then ψ′|_{K} ≡ λ^{−1} > 1. Then there is k_{0} ∈ ℕ and σ > 1 such that ψ′(x) > σ , for all x ∈ P k_{0}r_{0}_{\underline{λ},I}. By the construction we

have ψ(P (k+1)r_{0}_{\underline{λ},I}) = P kr_{0}_{\underline{λ},I}. So, if δ = inf\{(ψ^{k_{0}})′(x) | x ∈ P k_{0}r_{0}_{\underline{λ},I}\} and N ∈ ℕ satisfy

δ · σ^{N} > 1, then ψ^{N+k_{0}} : P (N+k_{0})r_{0}_{\underline{λ},I} → I is an expansive function of class C^{k} (resp. C^{k+α}) that makes K be a regular Cantor of class C^{k} (resp. C^{k+α}), because (ψ^{N+k_{0}})′(x) =

(ψ^{k_{0}})′(ψ^{N}(x)) · Π_{k=0}^{N−1}ψ′(ψ^{k}(x)) ≥ δ · σ^{N}, since ψ^{k}(x) ∈ P k_{0}r_{0}_{\underline{λ},I} , for all i ≤ N.

Example IV.1 . The Lebesgue measure of a central Cantor K\underline{λ} is lim_{n→∞} 2^{n}π_{n}(\underline{λ}). Given a ∈ (0, 1) there is a C^{1} Cantor set K\underline{λ} with Lebesgue measure equal to a. For this we take

λ_{n} = \frac{1}{2} · \frac{1 − \frac{1−a}{n}}{1 − \frac{1−a}{n+1}} .

If K is C^{1+ε} Cantor set then its Lebesgue measure is 0.

Example IV.2 . If λ_{n} < 1/3 for all n ∈ ℕ then K\underline{λ} − K\underline{λ} = \{ x − y / x, y ∈ K\underline{λ} \} is a Cantor set with measure 2 · Π_{n=1}^{∞} (3λ_{n}) (see [Sa] or [B-P-V]). Given q ≥ 1 (q = k or q = k + α) and a ∈ (0, 2) we can give example of a central Cantor K\underline{λ} of class C^{q} but not of class C^{q+ε}, for all ε > 0 such that m(K\underline{λ} − K\underline{λ}) = a, where m denote the Lebesgue measure in ℝ. For this we take

λ_{n} = \frac{1}{3} \frac{1 − \frac{3^{q}(2−a)}{2·3^{\sqrt{n}+(q−1)n}}}{1 − \frac{3^{q}(2−a)}{2·3^{\sqrt{n+1}+(q−1)(n+1)}}}

if q < ∞, and

λ_{n} = \frac{1}{3} · \frac{1 − \frac{3(2−a)}{2·3^{n^{2}}}}{1 − \frac{3(2−a)}{2·3^{(n+1)^{2}}}} ,

if q = ∞.

We remark that if λ_{n} < 1/3 , for all n, lim_{n→∞} λ_{n} = 1/3 and K\underline{λ} is C^{q} , q > 1, then K\underline{λ} −K\underline{λ} is a Cantor with positive Lebesgue measure, because \underline{λ} = (λ_{n})_{n∈ℕ} converges exponentially to 1/3 and so Π_{n=1}^{∞}(3λ_{n}) > 0.

Example IV.3 . Let 0 ≤ a < b ≤ 1/2. The K_{λ} is of class C^{1}, where

\begin{matrix}λ_{2n−1} = \frac{a + b}{2} + (\frac{b − a}{2}) (1 − \frac{1}{\sqrt{n}}) sin(log n) \\ λ_{2n} = \frac{ab}{λ_{2n−1}} \end{matrix}

is dense in [a, b].

Proof of Theorem B. The necessity and sufficiency of the conditions for local diffeomorphisms follow from Proposition 2 and Proposition 3, Corollary, respectively. The suficiency of the conditions for diffeomorphisms (i) and (ii)) follow from Proposition 3. We must only prove that these conditions are also necessary. By Proposition 1, Corollary 2, we can decompose I = I_{1} ∪ U_{1} ∪ I_{2} ∪ ⋯ ∪ U_{2^{n_{0}}−1} ∪ I_{2^{n_{0}}}, where I_{j} are the fundamental intervals of the n_{0}–th step of the construction of K\underline{λ}, I, such that φ(I_{j}) is a fundamental interval of the construction of K_{\underline{λ̃},J}, say of the n(j)–th step, for j = 1, 2, …, 2^{n_{0}}. Moreover, φ map decompositions of I_{j} in fundamental intervals and gaps in the corresponding decompositions of φ(I_{j}). A diffeomorphism between two central Cantor sets with this last property is called a good diffeomorphism. Let A = {p ∈ ℤ / for sufficiently large n ∈ ℕ, given fundamental intervals Ĩ of the n–th step of K_{\underline{λ},I} and J̃ of the (n + p)–th step of K_{\underline{λ̃},J} there is a good diffeomorphism ψ : Ĩ → J̃ at least as differentiable as φ such that ψ(Ĩ∩K_{\underline{λ},I}) = J̃∩K_{\underline{λ̃},J}\}, and B = \{p ∈ ℤ / for sufficiently large n ∈ ℕ, given fundamental intervals Ĩ of the n–th step of K_{\underline{λ̃},J} and J̃ of the (n + p)–th step of K_{\underline{λ̃},J} there is a good diffeomorphism ψ : Ĩ → J̃ at least as diferentiable as φ such that ψ(Ĩ∩K_{\underline{λ̃},J}) = J̃∩K_{\underline{λ̃},J}\}. Then:

• n(j) − n_{0} ∈ A, j = 1, 2, …, 2^{n_{0}} (take restrictions of φ|_{I_{j}})

• p, q ∈ A ⇒ p − q ∈ B; and p, q ∈ B ⇒ p − q ∈ B (be diffeomorphic by a good diffeomorphism is an equivalence relation). In particular, n(i) − n(j) ∈ B, for all i, j.

This implies that B = d · ℤ for some d ∈ ℤ, and A = p + dℤ, for all p ∈ A. If 0 ∉ A then d > 1 and X ≡ p (mod d), for all X ∈ A, where d|/p, so we can take p ∈ {1, 2, …, d − 1} such that n(j) ≡ n_{0} + p (mod d), for all j. Let N = max(n(j) + p). Then J contains 2^{N} fundamental intervals of the N–th step of K_{\underline{λ̃},J} and φ(I_{j}) contains 2^{N−n(j)} of these intervals, so ∑_{j=1}^{2^{n_{0}}} 2^{N−n(j)} = 2^{N} but n(j) ≡ n_{0} + p (mod d), so 2^{N−n(j)} ≡ 2^{N−n_{0}−p} (mod 2^{d} − 1), and thus 2^{N} = ∑_{j=1}^{2^{n_{0}}} 2^{N−n(j)} ≡ 2^{n_{0}}2^{N−n_{0}−p} = 2^{N−p} (mod 2^{d} − 1), so (2^{d} − 1)|2^{N−p}(2^{p} − 1), contradiction. So 0 ∈ A, and the result follows from Proposition 2.

Example IV.4. The Central Cantor sets defined by the sequences (\frac{1}{3}, \frac{1}{4}, \frac{1}{3}, \frac{1}{4}, …) and (\frac{1}{4}, \frac{1}{3}, \frac{1}{4}, \frac{1}{3}, …) are not diffeomorphics but are locally diffeomorphics.

V - Calculation of numerical invariants of central Cantor sets

i) Thickness: Given a Cantor set K and a gap U of K, let I_{U} be the interval bounded by U and the closest gap V that is larger than U. We define τ(U) = \frac{|I_{U}|}{|U|} and

τ(K) = inf{τ(U) / U is a bounded gap of K}, the thickness of K.

If K = K\underline{λ} is a central Cantor set, then the interval I_{U} in the above definition must be a fundamental interval of either U or V so, if I_{U} is a fundamental interval of the n–th

step, we have τ(U) = \frac{|I_{U}|}{|U|} = \frac{|I_{U}|}{min (|U| , |V |)} ≥ \frac{π_{n}(\underline{λ})}{g_{n}(\underline{λ})} = \frac{λ_{n}}{1 − 2λ_{n}}. On the other hand, if

U is the first gap of the n–th, τ(U) = \frac{π_{n}(\underline{λ})}{g_{n}(\underline{λ})} = \frac{λ_{n}}{1 − 2λ_{n}}. Therefore, τ(K) = inf_{n∈ℕ} \frac{λ_{n}}{1 − 2λ_{n}}.

Given p ∈ K we define τ_{loc}(k, p) = lim sup_{ε_{1},ε_{2}→0} τ (\overline{K ∩ (p − ε_{2} , p + ε_{2})}). By the above

arguments we can conclude that τ_{loc} (K\underline{λ} , p) = lim inf_{n→∞} \frac{λ_{n}}{1 − 2λ_{n}}, for all p ∈ K\underline{λ}.

ii) Limit Capacity : Let K be a Cantor set and ε > 0. We define N(K, ε) as the minimum number of intervals of length ε needed to cover K, and the limit capacity d(K)

of K is defined by d(K) = lim sup_{ε→0} −\frac{log N(K, ε)}{log ε}.

If K = K\underline{λ} is a central Cantor set then we can cover K\underline{λ} by 2^{n+1} intervals of lenght ε for

each ε > 0 satisfying π_{n+1}(K\underline{λ}) ≤ ε < π_{n}(\underline{λ}), and so d (K\underline{λ}) ≤ lim sup_{n→∞} −(n+1) \frac{log 2}{log π_{n}(\underline{λ})} =

lim sup_{n→∞} −n \frac{log 2}{log π_{n}(\underline{λ})}.

On the other hand, if a_{1} < a_{2} < ⋯ < a_{2^{n}} are the left ends of the fundamental intervals of the n–th step of K\underline{λ}, we have a_{i+1} − a_{i} > π_{n}(\underline{λ}), for each i, so, if π_{n+1}(\underline{λ}) ≤ ε < π_{n}(\underline{λ}) then and interval of lenght ε contains at most one of the a′_{i}s and so we must at least 2^{n} intervals of lenght ε to cover K\underline{λ}, since a_{i} ∈ K\underline{λ}, for all i. Thus N(K\underline{λ}, ε) ≥ 2^{n}

and, − \frac{log N(K\underline{λ}, ε)}{log ε} ≥ − \frac{n log 2}{log π_{n}(\underline{λ})} and so d (K\underline{λ}) = lim sup_{n→∞} − \frac{n log 2}{log π_{n}(\underline{λ})}.

iii) Hausdorff dimension. Given a Cantor set K ⊂ ℝ a finite cover of K by open intervals U = {U_{i}}_{i∈I} we define the diameter of diam U = max{|U_{i}| , i ∈ I}. Given α ∈ ℝ, we define H_{α}(U) = ∑_{i∈I} |U_{i}|^{α}. The Hausdorff α–measure of K is m_{α}(K) = lim_{ε→0}(inf{H_{α}(U) / U is a covering of K with diam(U) < ε}). The Hausdorff dimension of K is defined as being the unique real number HD(K) such that

\{ \begin{matrix}m_{α}(K) = +∞ , & α < HD(K) \\ m_{α}(K) = 0 , & α > HD(K) . \end{matrix}
If K = K\underline{λ}, K can be covered by 2^{n} intervals of lenght π_{n}(\underline{λ}), so, if α > lim inf_{n→∞}

− \frac{n log 2}{log π_{n}(\underline{λ})}, there are infinitely many n ∈ ℕ such that 2^{n} (π_{n}(\underline{λ}))^{α} < 1, and thus HD(K\underline{λ}) ≤ α.

Let U be a covering of K\underline{λ} by m interval of small lenght and let α ∈ ℝ. Let n ∈ ℕ such that m < 2^{n−1}. Then the set C of fundamental intervals of the n–th step which are strictly contained in some interval of U has at least 2^{n−1} elements. Now, if U_{i} ∈ U, let n(U_{i}) = #\{I ⊂ C / I ⊂ U_{i}\}, and choose a U_{i_{0}} that minimizes \frac{|U_{i}|^{α}}{n(U_{i})}. Since ∑ n(U_{i}) ≥

2^{n−1} and ∑ |U_{i}|^{α} = H_{α}(U), we have \frac{|U_{i_{0}}|}{n(U_{i_{0}})} ≤ \frac{H_{α}(U)}{2^{n−1}}. If 2^{k} ≤ n(U_{i_{0}}) < 2^{k+1}, then U_{i_{0}}

constains 2^{k} consecutive fundamental intervals of the n–th step, and so it contains a whole fundalmental interval of the (n − k + 1)–th step. Thus |U_{i_{0}}| ≥ π_{n−k+1}(\underline{λ}) and \frac{H_{α}(U)}{2^{n−1}} ≥

\frac{|U_{i_{0}}|^{α}}{n(U_{i_{0}})} > \frac{(π_{n−k+1}(\underline{λ}))^{α}}{2^{n+1}} thus H_{α}(U) > 2^{n−k−2} (π_{n−k+1}(\underline{λ}))^{α}. This proves that for each

covering U of K\underline{λ} there is N ∈ ℕ such that (∗) π_{N}(\underline{λ}) ≤ diam(U) and 2^{N} · (π_{N}(\underline{λ}))^{α} < 8·H_{α}(U), in particular, if α < lim inf_{n→∞} − \frac{n log 2}{log π_{n}(\underline{λ})}, we have lim_{n→∞} 2^{n} (π_{n}(\underline{λ}))^{α} = +∞ implies H_{α}(U) = +∞, by (∗). So we have HD(K\underline{λ}) ≥ α, which proves that HD(K\underline{λ}) = lim inf_{n→∞} − \frac{n log 2}{log π_{n}(\underline{λ})}.

Using the same arguments we can conclude that for each α ∈ ℝ, α > 0, \frac{1}{8} lim inf_{n→∞} 2^{n}· (π_{n}(\underline{λ}))^{α} ≤ m_{α}(K\underline{λ}) ≤ lim inf_{n→∞} 2^{n} · (π_{n}(\underline{λ}))^{α}.

In resume, the above results show that for each interval I and sequence \underline{λ} = (λ_{n})_{n∈ℕ},

• τ(K_{\underline{λ},I}) = τ(K\underline{λ}) = inf_{n∈ℕ} \frac{λ_{n}}{1 − 2λ_{n}} ;

• τ_{loc}(K_{\underline{λ},I}) = τ_{loc}(K\underline{λ}) = lim inf_{n→∞} \frac{λ_{n}}{1 − 2λ_{n}} ;

• d(K_{\underline{λ},I}) = d(K\underline{λ}) = lim sup_{n→∞} \frac{−n log 2}{log π_{n}(\underline{λ})} ;

• HD(K_{\underline{λ},I}) = HD(K\underline{λ}) = lim inf_{n→∞} \frac{−n log 2}{log π_{n}(\underline{λ})} .

Moreover, for each α > 0, m_{α}(K_{\underline{λ},I}) = |I|^{α}·m_{α}(K\underline{λ}) and m_{α}(K\underline{λ}) satisfies \frac{1}{8} lim inf_{n→∞} 2^{n}·

(π_{n}(\underline{λ}))^{α} ≤ m_{α}(K\underline{λ}) ≤ lim inf_{n→∞} 2^{n} · (π_{n}(\underline{λ}))^{α}.

Acknowledgements. This paper is an outgrowth of research initiated during the visit of one of the autors, C. G. Moreira to Universidad de Santiago de Chile (June, 1994) [ the author is grateful to this institution for financial support (grants DICYT # 9433P.S.and # 9433L.B.)] and completed during the visit of S. Plaza to IMPA and C. G. Moreira to Université de Paris–Sud, France. The authors are grateful to IMPA/CNPq for financial support and stimulating working conditions while we prepared the final version of this work. Finally we thank Jacob Palis for valuable suggestions for the final version. Also thanks to Juan Rivera for his valuable suggestion during the preparation of the final version.

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Depto. de Matemática, Universidad de Chile, Casilla 653 Santiago - CHILE. e-mail: rbamon@abello.dic.uchile.cl

IMPA. Estrada Dona Castorina 110, Jardim Botánico 22460-320, Rio de Janeiro - BRASIL. e-mail: gugu@impa.br

Depto. de Matemática, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago - CHILE. e-mail: splaza@fermat.usach.cl

Depto. de Matemática, Facultad de Ciencias, Universidad Católica del Norte, Avda. Angamos 0610, Antofagasta - CHILE. e-mail: jvera@socompa.cecun.ucn.cl