Hausdorff dimension estimates for invariant sets of k -1-maps

A. Mirle, Department of Mathematics, University of Technology Dresden, Mommsenstr. 13, 01062 Dresden

Introduction

Many chaotic systems are characterized by a very complicated dynamics on invariant sets and a complex geometric structure of these sets, given by a non-integer Hausdorff dimension. Section 1 of this paper will briefly recall the definition of this dimension.

The Douady-Oesterlé theorem (see [1]) gives an upper bound of the Hausdorff dimension of negatively invariant sets of smooth maps defined on open subsets of ℝ^{n}. In [4] a generalization of this theorem for maps on Riemannian manifolds is given.

For special classes of maps it may be possible to estimate the dimension of an invariant set although the generalized Douady-Oesterlé theorem is not applicable. One of such classes consists of k -1-maps, i. e. maps which assign k different points to the same image. Section 2 of this paper gives an exact definition of k -1-maps. Resulting from this special property of the maps some conditions of the Douady-Oesterlé theorem, such as the smoothness of the considered map and an upper bound condition for a singular value function, can be weakened in order to get estimates of the Hausdorff dimension of invariant sets.

1 Hausdorff dimension

Let us recall the definition of the Hausdorff dimension for compact subsets of a metric space. Let (X, ρ) be a metric space and K ⊂ X a compact subset. For arbitrary real numbers ε > 0 and d ≥ 0 we consider the Hausdorff premeasure at level ε and of order d of K given by

μ(K, d, ε) = inf ∑_{i} r_{i}^{d},

where the infimum is taken over all finite covers of K by balls of radius r_{i} ≤ ε. For fixed d and K the function μ(K, d, ·) is monotone decreasing. Hence the limit

μ(K, d) = lim_{ε→0+0} μ(K, d, ε)

exists and is called outer Hausdorff measure of order d of K. For every compact set K there exists a critical number d^{∗} with

μ(K, d) = \{ \begin{matrix}∞ & for all & 0 ≤ d < d^{∗} \\ 0 & for all & d > d^{∗}. \end{matrix}

This critical number can be characterized as

d^{∗} = sup{d ≥ 0 | μ(K, d) = ∞}

as well. It is called Hausdorff dimension of K and denoted by dim_{H}(K).

2 The class of k -1-maps

Let M be a smooth n-dimensional Riemannian manifold, U an open subset of M and K ⊂ U a compact set.

For a map φ : U → M we define the following property:

(E1) There are a finite number of compact subsets K_{i} ⊂ K, i = 1, …, k, with K = ⋃_{i=1}^{k} K_{i} and φ(K_{i}) = K for all i = 1, …, k and a natural number m < n, such that for the set

F := {u ∈ K | ∃i, j : i ≠ j, u ∈ K_{i} ∩ K_{j}}.

the inequality dim_{H}(φ^{p}(F)) ≤ m is satisfied for all p ∈ ℤ.

Property (E1) guarantees, that every point of the set K has at least k different preimages under the map φ, if non of the preimages is in the set F. The condition concerning the dimension of the set F ensures that the subsets K_{i} do not overlap each other, i. e. the intersection of two different sets K_{i} and K_{j} must have a smaller dimension than K. Since the dimension of K is not known, dim_{H}(F) can not be compared with dim_{H}(K). Therefore the integer m is introduced, and then we look for estimates of dim_{H}(K) greater than m. Often the set F is contained in a mdimensional submanifold of M, and if φ has a few special properties then the condition dim_{H}(φ^{p}(F)) ≤ m is automatically satisfied for arbitrary p ∈ ℤ.

The sets K_{i} from property (E1) define a cover of K. The property (E1) also implies φ(K) = K. This means that K is an invariant set for the map φ.

Definition Let M be a smooth n-dimensional Riemannian manifold and U ⊂ M be an open subset. A map φ : U → M is called k -1-map if there exist a compact set K ⊂ U and a cover {K_{i}}_{i=1}^{k} of K such that (E1) is satisfied.

The following two examples of maps on subsets of ℝ and ℝ^{2}, respectively, are to illustrate the property (E1).

Example 1 Let ε > 0 be arbitrary and φ : U = (−ε, 1 + ε) → ℝ be defined by

φ(u) = αu(1 − u),

where α is a parameter. This map is the well-known logistic map and describes for parameters 0 < α ≤ 4 the evolution of a population in a bounded area.

Consider the parameter α = 4 and the sets K = [0, 1], K_{1} = [0, \frac{1}{2}] and K_{2} = [\frac{1}{2}, 1]. It can easily be seen that φ(K_{1}) = K and φ(K_{2}) = K is satisfied. The sets φ^{p}(F), p ∈ ℤ, consist of a countable number of points only, so their dimension is zero, i. e. m = 0. Thus the logistic map with parameter α = 4 is a 2-1-map with respect to K = [0, 1] and the cover {K_{1}, K_{2}} in the sense of the definition mentioned above.

Example 2 Let ε > 0 be arbitrary and φ : U = (−ε, 1 + ε) × (−ε, 1 + ε) → ℝ^{2} be defined by

φ(u, v) = (α (\frac{1}{2} − |u − \frac{1}{2}|) , βv) ,

where α > 0 and β > 0 are parameters. The first component of φ is the well-known tent map.

For β ≠ 1 any invariant set of the map φ is contained in (−ε, 1 + ε) × {0}. For the parameter α = 2 we can choose K = [0, 1] × \{0\}, K_{1} = [0, \frac{1}{2}] × \{0\} and K_{2} = [\frac{1}{2}, 1] × \{0\}, and then we have φ(K_{1}) = K and φ(K_{2}) = K. The set F = \{(\frac{1}{2}, 0)\} implies m = 0. Thus the considered map with α = 2 and arbitrary β ≠ 1 is a 2-1-map with respect to K and the cover {K_{1}, K_{2}}.

An estimation of the Hausdorff dimension of invariant sets of a k -1-map analogously to [1] and [4] will be found if the considered map φ satisfies some additional properties. Suppose that φ : U → M is a k -1-map with respect to a compact set K ⊂ U and a cover {K_{i}}_{i=1}^{k} of K. We say that such a map has the property (E2), if the following conditions are satisfied:

(E2a) The map φ is continuous on K.
(E2b) The image of every nonempty open subset of U under φ has interior points.

(E2c) For any i ∈ {1, …, k} the map φ_{i} := φ|_{K_{i}} can be extended to a C^{1}-map on an open neighbourhood U_{i} ⊂ U of K_{i}.

In the following we will refer the maps φ_{i} in property (E2c) as partial maps.

Now we analyse the iteration of k -1-maps. It will be shown that the p-th power φ^{p} of a k -1-map φ is a k^{p}-1-map for any natural number p.

Notation For any natural number p we will denote the subsets of K related to the map φ^{p} and the corresponding partial maps by

\begin{matrix}K_{i_{1},... ,i_{p}} := φ^{−(p−1)}(K_{i_{1}}) ∩ … ∩ φ^{−1}(K_{i_{p−1}}) ∩ K_{i_{p}}, \\ φ_{i_{1},... ,i_{p}} := φ^{p}|_{K_{i_{1},... ,ip}}. \end{matrix}

Proposition 1 Let φ : U → M be a k -1-map with respect to a compact set K ⊂ U and a cover {K_{i}}_{i=1}^{k} of K such that the property (E2) is satisfied, and let p be an arbitrary natural number. Then the sets K_{i_{1},... ,i_{p}}, (i_{1}, …, i_{p}) ∈ {1, …, k}^{p}, define a cover of K, so that φ^{p} is a k^{p}-1-map with respect to this cover and satisfies (E2).

Proof Property (E1) contains dim_{H}((φ^{p})^{q}(F_{p})) ≤ m, q ∈ ℤ, with

\begin{aligned}F_{p} = \{u ∈ K | ∃(i_{1}, …, i_{p}), (j_{1}, …, j_{p}) : & (i_{1}, …, i_{p}) ≠ (j_{1}, …, j_{p}), \\ & u ∈ K_{i_{1},... ,i_{p}} ∩ K_{j_{1},... ,j_{p}}\}. \end{aligned}

If we can show that F_{p} ⊂ φ^{−p+1}(F) for arbitrary p ∈ ℕ, then for arbitrary q ∈ ℤ there is dim_{H}((φ^{p})^{q}(F_{p})) ≤ dim_{H}((φ^{p(q−1)}(F)) ≤ m.

The proof of the proposition with this special form of property (E1) can be done by induction.

It is clear that for p = 1 the assertion is true.

Suppose that for an arbitrary p ∈ ℕ the sets K_{i_{1},... ,i_{p}}, (i_{1}, …, i_{p}) ∈ {i, …, k}^{p}, are such that φ^{p} has the properties (E1) and (E2) with respect to this cover, where dim_{H}(F_{p}) ≤ m is satisfied in the special form F_{p} ⊂ φ^{−p+1}(F).

Now we are going to prove the assertion for p + 1. The preimage of a compact set under a continuous map with compact domain is always compact. Considering the map φ|_{K} restricted to the domain K it follows with (E2a) that for every (i_{1}, …, i_{p}) the set φ^{−1}(K_{i_{1},... ,i_{p}}) is compact. Furthermore the relation

φ^{−1}(K_{i_{1},... ,i_{p}}) = φ^{−1}(φ^{−p+1}(K_{i_{1}}) ∩ … ∩ K_{i_{p}}) = φ^{−p}(K_{i_{1}}) ∩ … ∩ φ^{−1}(K_{i_{p}})

holds. Therefore any set K_{i_{1},... ,i_{p+1}} is compact as the intersection of the two compact sets K_{i_{p+1}} and φ^{−1}(K_{i_{1},... ,i_{p}}).

From the equality

\begin{aligned}φ(K_{i_{1},... ,i_{p+1}}) & = φ (φ^{−1}(K_{i_{1},... ,i_{p}}) ∩ K_{i_{p+1}}) = K_{i_{1},... ,i_{p}} ∩ φ(K_{i_{p+1}}) \\ & = K_{i_{1},... ,i_{p}} ∩ K = K_{i_{1},... ,i_{p}} \end{aligned}

together with the assumption φ^{p}(K_{i_{1},... ,i_{p}}) = K the relation φ^{p+1}(K_{i_{1},... ,i_{p+1}}) = K follows.

Because of

φ(K_{i_{1},... ,i_{p+1}} ∩ K_{j_{1},... ,j_{p+1}}) ⊂ φ(K_{i_{1},... ,i_{p+1}}) ∩ φ(K_{i_{1},... ,i_{p+1}}) = K_{i_{1},... ,i_{p}} ∩ K_{j_{1},... ,j_{p}}

we have φ(F_{p+1}) ⊂ F_{p} and, therefore, F_{p+1} ⊂ φ^{−1}(F_{p}) ⊂ φ^{−p}(F).

Therefore φ^{p+1} is a k^{p+1}-1-map, the condition dim_{H}(F_{p+1}) ≤ m has the special form F_{p+1} ⊂ φ^{−p}(F).

The properties (E2a) and (E2b) are obviously satisfied for φ^{p+1}, because they are assumed for φ.

Finally (E2c) has to be shown. For every partial map φ_{i_{1},... ,i_{p+1}} we have the equality φ_{i_{1},... ,i_{p+1}} = φ_{i_{1},... ,i_{p}} ∘ φ_{i_{p+1}}. Because of (E2c) the map φ_{i_{p+1}} can be extended to a C^{1}-map on a neighbourhood U_{i_{p+1}} of K_{i_{p}+1}. The same is true for φ_{i_{1},... ,i_{p}} and some neighbourhood U_{i_{1},... ,i_{p}} of K_{i_{1},... ,i_{p}}. Therefore the set U_{i_{1},... ,i_{p+1}} := U_{i_{p+1}} ∩ φ^{−1}(U_{i_{1},... ,i_{p}}) is a neighbourhood of K_{i_{1},... ,i_{p+1}}, on which the map φ_{i_{1},... ,i_{p},i_{p+1}} can be extended as C^{1}-map.

Consequently the validity of the result for p + 1 is shown and therefore Proposition 1 is proved. ■

3 The suspension of the original map

Now we consider a map φ : U → M defined on an open subset U of a n-dimensional smooth manifold M and satisfying the properties (E1) and (E2) with a compact set K ⊂ U and a cover {K_{i}}_{i=1}^{k} of K.

To use the k -1-property of a map to obtain Hausdorff dimension estimates of invariant sets it is suitable to consider a suspension φ̃ of the map φ on a (n + 1)-dimensional product space given by

φ̃: K × [0, 1] → K × [0, 1] with

φ̃(u, v) = (φ(u), \frac{v + i(u) − 1}{k}) ,where i(u) = min\{j ∈ \{1, …, k\} | u ∈ K_{j}\}.

For the new map φ̃ a set K′ with (K \ ⋃_{p=−∞}^{∞} φ^{p}(F)) × [0, 1] ⊂ K′ ⊂ K × [0, 1] is invariant. If an upper bound of the Hausdorff dimension of K̃ = K × [0, 1] can be found, then we have an estimate of the dimension of K as well, since (see [7]) the inequality

dim_{H}(K̃) ≥ dim_{H}(K) + dim_{H}([0, 1])

holds and therefore one has

dim_{H}(K) ≤ dim_{H}(K̃) − 1.

In the next section it is shown how to find an upper bound for the Hausdorff dimension of the set K̃.

By the construction of φ̃ it becomes clear that this map is not differentiable at the boundaries between two different sets of {K̃_{i} := K_{i} × [0, 1] | i = 1, …, k}. Therefore we consider the map φ̃ separately on the sets (K_{i} \F)×[0, 1] ⊂ K̃_{i}, where F is defined as in Section 2, i. e. we decompose the map φ̃ into partial maps analogously to the decomposition of φ. For the partial maps of φ̃^{p}, p ∈ ℕ, the following proposition holds:

Proposition 2 Let M be a smooth n-dimensional Riemannian manifold, U ⊂ M an open set and φ : U → M a map satisfying the properties (E1) and (E2) with respect to a compact set K ⊂ U and a cover {K_{i}}_{i=1}^{k} of K. Then for any p ∈ ℕ and any p-tupel (i_{1}, …, i_{p}) ∈ {1, …, k}^{p} the map

\begin{matrix}φ̃_{i_{1},... ,i_{p}} : (K_{i_{1},... ,i_{p}} \ ⋃_{j=0}^{p−1} φ^{−j}(F)) × [0, 1] → K × [0, 1], given by \\ φ̃_{i_{1},... ,i_{p}}(u, v) = (φ^{p}(u), \frac{v}{k^{p}} + ∑_{j=1}^{p} \frac{i_{j} − 1}{k^{j}}) , \end{matrix}

can be extended to a C^{1}-map on a neighbourhood of K_{i_{1},... ,i_{p}} × [0, 1]. This extension, denoted by φ̃_{i_{1},... ,i_{p}} as well, coincides with φ̃^{p} on

(K_{i_{1},... ,i_{p}} \ ⋃_{j=0}^{p−1} φ^{−j}(F)) × [0, 1]

and satisfies

φ̃_{i_{1},... ,i_{p}}(K_{i_{1},... ,i_{p}} × [0, 1]) = K × [∑_{j=1}^{p} \frac{i_{j} − 1}{k^{j}}, ∑_{j=1}^{p−1} \frac{i_{j} − 1}{k^{j}} + \frac{i_{p}}{k^{p}} ] .

Proof We prove this proposition again by induction.

For p = 1 and any i ∈ {1, …, k} the map

φ̃_{i}(u, v) = (φ(u), \frac{v + i − 1}{k}), u ∈ K_{i} \ F, v ∈ [0, 1],

is identical with φ̃ on (K_{i} \ F) × [0, 1]. Every partial map φ̃_{i} can be extended to a C^{1}-map on an open set containing K̃_{i} because of property (E2c) and the affine-linear second component of the map φ̃. For simplicity we denote these extensions by φ̃_{i} as well. From (E2c) and m ≤ n − 1 with m defined in property (E1) it follows that φ̃_{i}(K_{i} × [0, 1]) = K × [\frac{i−1}{k}, \frac{i}{k}], i = 1, …, k. So the assertion for p = 1 is shown.

Now let p be a natural number such that for any p-tupel (i_{1}, …, i_{p}) ∈ {1, …, k}^{p} the map φ̃_{i_{1},... ,i_{p}} can be extended to a C^{1}-map on a neighbourhood of K_{i_{1},... ,i_{p}} ×[0, 1]. This extension, denoted by φ̃_{i_{1},... ,i_{p}} as well, coincides on (K_{i_{1},... ,i_{p}} \ ⋃_{j=0}^{p−1} φ^{−j}(F)) × [0, 1] with φ̃^{p}, and satisfies

φ̃_{i_{1},... ,i_{p}}(K_{i_{1},... ,i_{p}} × [0, 1]) = K × [∑_{j=1}^{p} \frac{i_{j} − 1}{k^{j}}, ∑_{j=1}^{p−1} \frac{i_{j} − 1}{k^{j}} + \frac{i_{p}}{k^{p}} ] .

Because of

\begin{aligned}K_{i_{1},... ,i_{p+1}} \ ⋃_{j=0}^{p} φ^{−j}(F) & = & (φ^{−1}(K_{i_{1},... ,i_{p}}) ∩ K_{i_{p+1}}) \ (⋃_{j=1}^{p} φ^{−j}(F) ∪ F ) \\ & = & φ^{−1} (K_{i_{1},... ,i_{p}} \ ⋃_{j=0}^{p−1} φ^{−j}(F)) ∩ (K_{i_{p+1}} \ F) \end{aligned}

the equation

φ̃_{i_{1},... ,i_{p}} ∘ φ̃_{i_{p+1}}(u, v) = φ̃_{i_{1},... ,i_{p}} (φ(u), \frac{v + i_{p+1} − 1}{k})
= (φ^{p+1}(u), \frac{v + i_{p+1} − 1}{k^{p+1}} + ∑_{j=1}^{p} \frac{i_{j} − 1}{k^{j}})

= (φ^{p+1}(u), \frac{v}{k^{p}} + ∑_{j=1}^{p} \frac{i_{j} − 1}{k^{j}})

is valid for any (p+1)-tupel (i_{1}, …, i_{p+1}) ∈ {1, …, k}^{p+1} and any (u, v) ∈ K_{i_{1},... ,i_{p+1}}×

[0, 1]. This means that φ̃_{i_{1},... ,i_{p}} ∘φ̃_{i+1} on (K_{i_{1},... ,i_{p+1}} \ ⋃_{j=0}^{p} φ^{−j}(F))×[0, 1] is identical with φ̃^{p+1}. The maps φ̃_{i_{1},... ,i_{p}} and φ̃_{i_{p+1}} can be extended to a C^{1}-map on a neighbourhood U_{i_{1},... ,i_{p}} of K_{i_{1},... ,i_{p}} ×[0, 1] and U_{i_{p+1}} of K_{i_{p+1}} ×[0, 1], respectively. Moreover, there holds the relation

φ_{i_{p+1}}(K_{i_{p+1}} × [0, 1]) = K × [\frac{i_{p+1} − 1}{k}, \frac{i_{p+1}}{k} ] .

The set U_{i_{1},... ,i_{p+1}} := U_{i_{p+1}} ∩ φ_{i_{1},... ,i_{p}}^{−1}(U_{i_{1},... ,i_{p}}) is a neighbourhood of K_{i_{1},... ,i_{p+1}} × [0, 1]. Note that on this set the map φ̃_{i_{1},... ,i_{p}} ∘ φ̃_{i+1} is C^{1}, and

\begin{aligned}φ̃_{i_{1},... ,i_{p}} ∘ φ̃_{i+1} (K_{i_{1},... ,i_{p+1}} × [0, 1]) & = & φ̃_{i_{1},... ,i_{p}} (K_{i_{1},... ,i_{p}} × [\frac{i_{p+1} − 1}{k^{p}}, \frac{i_{p+1}}{k^{p}} ]) \\ & = & K × [∑_{j=1}^{p+1} \frac{i_{j} − 1}{k^{j}}, ∑_{j=1}^{p} \frac{i_{j} − 1}{k^{j}} + \frac{i_{p+1}}{k^{p+1}} ] \end{aligned}

holds. So the result is shown for p + 1 and Proposition 2 is proved. ■

Since we now have defined the maps φ̃_{i}, i = 1, …, k, on a neighbourhood of K_{i}×[0, 1], the underlying manifold for the suspended map φ̃ is the product manifold M × ℝ.

4 The dimension estimate

Using the suspension procedure of the previous section the following theorem yields an upper estimate for the Hausdorff dimension of an invariant set of a k -1-map, which can be considered as an extension of the generalized Douady-Oesterlé theorem (see [4]) to this special class of maps. Note that the assumptions of the original Douady-Oesterlé theorem ([1]) and its generalization for manifolds ([4]) are not satisfied:

• The C^{1}-property of φ on U is replaced by the property (E2c).

• The singular value condition ω_{d,K}(φ) < 1 is weakened.

Let us briefly recall the definition of the singular value function ω mentioned above. Consider a linear operator L : E → E′, where E and E′ are p-dimensional Euclidean spaces. Denote by L^{∗} the adjoint operator of L, i. e. the unique linear operator L^{∗} : E′ → E given by ⟨Lx, y⟩_{E′} = ⟨x, L^{∗}y⟩_{E} for all x ∈ E and y ∈ E′. The singular values of L are the eigenvalues of the positive semi-definite operator \sqrt{L^{∗}L}, ordered with respect to its size and multiplicity and denoted by α_{1}(L) ≥ α_{2}(L) ≥ … ≥ α_{p}(L). For an arbitrary j ∈ {0, 1, …, p} we introduce for L the notation

ω_{j}(L) = \{ \begin{matrix}α_{1}(L) · . . . · α_{j}(L) & , for j > 0, \\ 1 & , for j = 0. \end{matrix}

For an arbitrary number d ∈ (0, p], written in the form d = d_{0} + s with d_{0} ∈ {0, 1, …, p − 1} and s ∈ (0, 1], we define

ω_{d}(L) = ω_{d_{0}}(L)^{1−s} · ω_{d_{0}+1}(L)^{s}.

Now let N be a smooth p-dimensional Riemannian manifold, U ⊂ N open, K ⊂ U compact, f : U → N a C^{1}-map and d_{u}f : T_{u}N → T_{f(u)}N the tangent map of f at the point u ∈ U. For an arbitrary number d ∈ (0, p] we define the value

ω_{d,K}(f) = sup_{u∈K} ω_{d}(d_{u}f).

Now we give an estimation of the Hausdorff dimension for invariant sets of k -1-maps with the property (E2). If max_{i=1,... ,k} ω_{d,K_{i}}(φ_{i}) < 1 is satisfied for a number d ∈ [0, n]

then we can use the techniques of Douady and Oesterlé nearly in the original form. But if this condition for the singular value function is not satisfied for a k -1-map, then we can use the suspension of the map described in the previous section to get dimension estimates in the interval (n − 1, n].

Theorem 1 Suppose that M is a smooth n-dimensional Riemannian manifold, U ⊂ M is an open set, K ⊂ U is compact and φ : U → M is a k -1-map with respect to the cover {K_{i}}_{i=1}^{k} of K and satisfies (E2). Suppose also that there exists a number s ∈ (0, 1] such that

max_{i=1,... ,k} ω_{n,K_{i}}(φ_{i}) < k^{s}

is satisfied for the partial maps φ_{i} defined in (E2c), that α_{j}(d_{u}φ_{i}) ≥ \frac{1}{k} for any u ∈ K, 1 ≤ j ≤ n and 1 ≤ i ≤ k, and that for any p ∈ ℕ there exists a number ε_{p} > 0 such that

μ(K × [0, 1], d, ε) = ∑_{(i_{1},... ,i_{k})∈{1,... ,k}^{p}} μ(K_{i_{1},... ,i_{p}} × [0, 1], d, ε)  (∗)

is satisfied for d = n + s and any ε ∈ (0, ε_{p}), where {K_{i_{1},... ,i_{p}} | (i_{1}, …, i_{p}) ∈ {1, …, k}^{p}} is the cover of K generated by {K_{i}}_{i=1}^{k} on the base of Proposition 1, and μ(·, d, ε) denotes the Hausdorff premeasure at level ε and of order d in the product manifold M × ℝ. Then the Hausdorff dimension of K can be estimated by

dim_{H}(K) ≤ n − 1 + s.

Proof From the assumption of the theorem it follows, that there exists a number

ν ∈ (0, k) with max_{i=1,... ,k} ω_{n,K_{i}}(φ_{i}) = ν^{s}. With the notations φ̃, K̃, φ̃_{i}, K̃_{i} (i = 1, …, k)

introduced in the previous section we have that all the singular values of φ_{i} are singular values of φ̃_{i} as well, but φ̃_{i} has the additional singular value \frac{1}{k}. By assumption \frac{1}{k} is the smallest singular value of φ̃_{i}. Therefore the inequality

ω_{n+s,K̃_{i}}(φ̃_{i}) ≤ (\frac{ν}{k} )^{s} < 1

is true. So for every l > 0 there exists a p_{0} with

(\frac{ν}{k} )^{sp} < l

for any p > p_{0}. It follows from Proposition 1 that the set K can be covered by the compact sets

K_{i_{1},... ,i_{p}} = φ^{−p+1}(K_{i_{1}}) ∩ … ∩ φ^{−1}(K_{i_{p−1}}) ∩ K_{i_{p}}

where φ^{p}(K_{i_{1},... ,i_{p}}) = K.

Due to Proposition 2 on each of the sets (K_{i_{1},... ,i_{p}} \ ⋃_{j=0}^{p−1} φ^{−j}(F)) × [0, 1] the map φ̃^{p} can be considered as the product φ̃_{i_{1}} ∘ … ∘ φ̃_{i_{p}}. Any such map φ̃_{i_{1}} ∘ … ∘ φ̃_{i_{p}} can be extended to a C^{1}-map on a neighbourhood of K̃_{i_{1},... ,i_{p}}. This extension is denoted by φ̃_{i_{1},... ,i_{p}}. For d = n + s it follows that

ω_{d,K̃_{i_{1},... ,ip}}(φ̃_{i_{1},... ,i_{p}}) = sup_{u∈K̃_{i_{1},... ,ip}} ω_{d}(d_{u}φ̃_{i_{1},... ,i_{p}})

= sup_{u∈K_{i_{1},... ,ip}} ω_{n}(d_{u}φ_{i_{1},... ,i_{p}}) (\frac{1}{k^{p}} )^{s}
= k^{−sp}ω_{n,K_{i_{1},... ,ip}}(φ_{i_{1},... ,i_{p}})
≤ k^{−sp} ( max_{i=1,... ,k} ω_{n,K_{i}}(φ_{i}))^{p}
= k^{−sp}ν^{sp}
< l

for any p-tupel (i_{1}, …, i_{p}) ∈ {1, …, k}^{p}. Since every of the maps φ̃_{i_{1},... ,i_{p}} is C^{1} on a neighbourhood of K̃_{i_{1},... ,i_{p}}, numbers ε_{0}(i_{1}, …, i_{p}) < ε_{p} exist such that for any ε ∈ (0, ε_{0}(i_{1}, …, i_{p})]

μ (φ̃_{i_{1},... ,i_{p}} \binom{}{K_{i_{1},... ,i_{p}}} , d, λl^{\frac{1}{d}} ε) ≤ Clμ (K̃_{i_{1},... ,i_{p}}, d, ε)

holds with λ = \sqrt{n + 1} and C = 2^{n}\sqrt{n + 1} (see [4]). Because of Proposition 2 this is equivalent to

μ (K × [∑_{j=1}^{p} \frac{i_{j} − 1}{k^{j}}, ∑_{j=1}^{p} \frac{i_{j} − 1}{k^{j}} + \frac{i_{p}}{k^{p}} ] , d, λl^{\frac{1}{d}} ε) ≤ Clμ (K̃_{i_{1},... ,i_{p}}, d, ε)

Defining ε_{0} := min_{(i_{1},... ,i_{p})∈{1,... ,k}^{p}} ε_{0}(i_{1}, …, i_{p}) and using the subadditivity of μ from the

previous inequalities we get the estimate

μ (K̃, d, λl^{\frac{1}{d}} ε) ≤ Cl ∑_{(i_{1},... ,i_{p})∈\{1,... ,k\}^{p}} μ (K̃_{i_{1},... ,i_{p}}, d, ε) ,

which is satisfied for any ε ∈ (0, ε_{0}]. For such values of ε we have by assumption

μ(K̃, d, ε) = ∑_{(i_{1},... ,i_{p})∈{1,... ,k}^{p}} μ (K̃_{i_{1},... ,i_{p}}, d, ε) ,

and combining this with the preceeding inequality we get

μ (K̃, d, λl^{\frac{1}{d}} ε) ≤ Clμ (K̃, d, ε) .

Now let l be so small that λl^{\frac{1}{d}} < 1 and Cl < 1 are valid. Since μ is monotone decreasing with respect to ε the inequality

μ(K̃, d, λl^{\frac{1}{d}} ε) ≥ μ(K̃, d, ε)

holds for any ε > 0. This leads to

μ(K̃, d, ε) ≤ Clμ(K̃, d, ε)

with arbitrary small l and ε ∈ (0, ε_{0}], i. e. μ(K̃, d, ε) = 0 for any ε ∈ (0, ε_{0}]. So dim_{H}(K̃) ≤ d and therefore

dim_{H}(K) ≤ d − 1 = n − 1 + s. ■

Since the condition (∗) is not easy to be checked, this condition can be resigned, if the partial maps φ_{i} are invertible, and the inverse maps are C^{1}.

We say that a k -1-map φ : U → M with respect to K and a cover {K_{i}}_{i=1}^{k} has the property (E2’), if it satisfies (E2a), (E2b) and

(E2c’) For any i ∈ {1, …, k} the map φ_{i} := φ|_{K_{i}} can be extended to a C^{1}- diffeomorphism on an open neighbourhood U_{i} ⊂ U of K_{i}.

For the iterates of maps with the properties (E1) and (E2’) we get the following proposition, which can be proved analogously to Proposition 1.

Proposition 3 Let φ : U → M be a k -1-map with respect to a compact set K ⊂ U and a cover {K_{i}}_{i=1}^{k} such that the property (E2’) is satisfied, and let p be an arbitrary natural number. Then the sets

K_{i_{1},... ,i_{p}} := φ^{−p+1}(K_{i_{1}}) ∩ … ∩ φ^{−1}(K_{i_{p−1}}) ∩ K_{i_{p}}

with (i_{1}, …, i_{p}) ∈ {1, …, k}^{p} define a cover of K, so that φ^{p} is a k^{p}-1-map with respect to this cover and satisfies (E2’).

For invariant sets of k -1-maps with the property (E2’) we get the following dimension estimate.

Theorem 2 Suppose that M is a smooth n-dimensional Riemannian manifold, U ⊂ M is an open set, K ⊂ U is compact and φ : U → M is a k -1-map with respect to K and the cover {K_{i}}_{i=1}^{k} of K and satisfies (E2’). Suppose further that there exists a number d ∈ [m, n] such that

max_{i=1,... ,k} ω_{d,K}(φ_{i}^{−1}) < \frac{1}{k}

holds for the partial maps φ_{i} defined in (E2c’). Then the Hausdorff dimension of K can be estimated by

dim_{H}(K) ≤ d.

Proof Denote max_{i=1,... ,k} ω_{d,K}(φ_{i}^{−1}) by ν. Then by assumption ν < \frac{1}{k}. Therefore, for any

l > 0 there is a natural number p_{0} with (kν)^{p} < l for any p > p_{0}. Using the argument of [4], for arbitrary p-tupels (i_{1}, …, i_{p}) ∈ {1, …, k}^{p} we get

ω_{d,K}(φ_{i_{1}}^{−1} ∘ … ∘ φ_{i_{p}}^{−1}) ≤ ν^{p} < \frac{l}{k^{p}}

and therefore

μ ((φ_{i_{1}}^{−1} ∘ … ∘ φ_{i_{p}}^{−1})(K), d, λ (\frac{l}{k^{p}} )^{\frac{1}{d}} ε) ≤ C \frac{l}{k^{p}} μ(K, d, ε)

where λ = \sqrt{d_{0} + 1} and C = 2^{d_{0}} \sqrt{d_{0} + 1}, provided d is written in the form d = d_{0} + s with d_{0} ∈ {m, m + 1, …, n} and s ∈ [0, 1]. It follows from the properties (E1) and (E2c’) that φ_{i}^{−1}(K) = K_{i} for every i = 1, …, k, and with Proposition 3 there is

(φ_{i_{1}}^{−1} ∘ … ∘ φ_{i_{p}}^{−1})(K) = K_{i_{1},... ,i_{p}}

for any p-tupel (i_{1}, …, i_{p}) ∈ {1, …, k}^{p}. Therefore we have the estimation

μ (K_{i_{1},... ,i_{p}}, d, λ (\frac{l}{k^{p}} )^{\frac{1}{d}} ε) ≤ C \frac{l}{k^{p}} μ(K, d, ε).

The subadditivity of μ(K, d, ·) implies

μ (K, d, λ (\frac{l}{k^{p}} )^{\frac{1}{d}} ε) ≤ Clμ(K, d, ε).

Now we choose l so small that λl^{\frac{1}{d}} < 1 and Cl < 1 is satisfied. Because of k ≥ 1 there is k^{\frac{p}{d}} ≥ 1 and therefore λ (\frac{l}{k^{p}} )^{\frac{1}{d}} < 1. Since μ(K, d, ·) is monotone decreasing we have

μ(K, d, ε) ≤ Clμ(K, d, ε)

with Cl < 1. This means μ(K, d, ε) = 0, i. e. dim_{H}(K) ≤ d. ■

Remark Theorem 2 is also applicable to maps which are not really k -1-maps, i. e. for which there exists a cover {K_{i}}_{i=1}^{p} with φ(K_{i}) = K, but the sets K_{i} can overlap each other.

5 Application examples

Continuation of Example 2 Theorem 1 is applicable to the generalized tent map with parameter α = 2. As shown in section 2 this is a 2-1-map with respect to K = [0, 1] × \{0\} and the cover consisting of K_{1} = [0, \frac{1}{2}] × \{0\} and K_{2} = [\frac{1}{2}, 1] × \{0\}. Before applying Theorem 1 it is necessary to check all the other requirements of this theorem.

(E2a) The map φ is continuous on K.
(E2b) Open sets are always mapped onto sets with interior points.

(E2c) The maps φ_{1} and φ_{2} defined by

\begin{aligned}φ_{1}(u, v) & = & (2u, βv) \\ φ_{2}(u, v) & = & (1 − 2u, βv) \end{aligned}

are affine-linear and, therefore, can be extended to C^{1}-maps on the whole space ℝ^{2}.

The subsets K_{i_{1},... ,i_{p}} defined by Proposition 1 are intervals of the form [\frac{j−1}{2^{p}}, \frac{j}{2^{p}} ]×\{0\}, j = 1, …, 2^{p}. This means that the sets K_{i_{1},... ,i_{p}} × [0, 1] are two-dimensional quaders, i. e. μ(K_{i_{1},... ,i_{p}} × [0, 1], d, ε) = 0 for arbitrary d ∈ (2, 3] and ε > 0. Therefore

∑_{(i_{1},... ,i_{p})∈{1,2}^{p}} μ(K_{i_{1},... ,i_{p}} × [0, 1], d, ε) = μ(K × [0, 1], d, ε)

holds for any ε > 0 and d ∈ (2, 3]. Furthermore max_{i=1,2} ω_{2,K_{i}}(φ_{i}) = 2β < 2^{s} has to be

satisfied for a s ∈ (0, 1]. This holds for 0 < β < 2^{s−1}. Especially for β = \frac{1}{2} Theorem 1 can be applied for every s ∈ (0, 1] and gives dim_{H}(K) ≤ 1 + s. For s → 0 this yields dim_{H}(K) ≤ 1. Since the set K = [0, 1] × {0} obviously has dimension 1, this estimation is sharp.

Theorem 2 as well can be applied to the generalized tent map with α = 2. Since the partial maps φ_{1} and φ_{2} are affine-linear, the property (E2c’) is satisfied. Furthermore we have for β > 2

max_{i=1,2} ω_{d,K}(φ_{i}^{−1}) = \{ \begin{matrix}(\frac{1}{2})d & for & 0 ≤ d ≤ 1, \\ \frac{1}{2β^{d−1}} & for & 1 < d ≤ 2. \end{matrix}

This means that max_{i=1,2} ω_{d,K}(φ_{i}^{−1}) < \frac{1}{2} holds for all d > 1. With Theorem 2 we get

dim_{H}(K) ≤ d for any d > 1 and, therefore, dim_{H}(K) ≤ 1.

Example 3 A map with an invariant set with non-integer Hausdorff dimension is φ : (0 − ε, 1 + ε) × (0 − ε, 1 + ε) → ℝ^{2} defined by

φ(u, v) = \{ \begin{matrix}(u, βv − \frac{1−β}{2} ) & for & v ≤ \frac{1}{2} \\ (u, \frac{1+β}{2} − βv) & for & v > \frac{1}{2} \end{matrix}

with parameters ε > 0 and β > 2. This map when applied to the unit square [0, 1] × [0, 1] first stretches the square in v-direction with factor β and then folds it.

[figure: stretching ; ⟶ ; with factor β; \underrightarrow{folding}]

The set K = ⋂_{i=1}^{∞} φ^{−i}([0, 1] × [0, 1]) is a compact invariant set. Considering the cover

consisting of K_{1} = \{(u, v) ∈ K | v ≤ \frac{1}{2}\} and K_{2} = \{(u, v) ∈ K | v ≥ \frac{1}{2}\} we get that φ is a 2-1-map with respect to K and this cover. Since the partial maps are affine-linear the property (E2’) is satisfied as well. Furthermore we have

max_{i=1,2} ω_{d,K}(φ_{i}^{−1}) = \{ \begin{matrix}1^{d} & for & 0 ≤ d ≤ 1, \\ (\frac{1}{β} )d−1 & for & 1 < d ≤ 2. \end{matrix}

So we can apply Theorem 2 if (\frac{1}{β} )^{d−1} < \frac{1}{2} holds, i. e. d > 1+ \frac{ln 2}{ln β}, and therefore we get

dim_{H}(K) ≤ 1+ \frac{ln 2}{ln β}. Using the techniques of [2] to determine the Hausdorff dimension of self-affine sets it can be shown that dim_{H}(K) = 1+ \frac{ln 2}{ln β}, i. e. the dimension estimate of Theorem 2 is sharp.

Example 4 We consider a map similar to the previous example, but now the underlying manifold is the flat cylinder

M := {(ϑ, v) | ϑ ∈ [0, 2π), v ∈ ℝ}.

Let φ be defined on the subset U = {(ϑ, v) | ϑ ∈ [0, 2π), v ∈ (0 − ε, 1 + ε)} as

φ(ϑ, v) = \{ \begin{matrix}((ϑ + αv) mod 2π, βv − \frac{1−β}{2} ) & for & v ≤ \frac{1}{2} \\ ((ϑ + αv) mod 2π, \frac{1+β}{2} − βv) & for & v > \frac{1}{2} \end{matrix}

with parameters α ∈ ℝ and β > 2. The set K = ⋂_{i=1}^{∞} φ^{−i}([0, 1] × [0, 1]) is a com-

pact invariant set independent of α. Analogously to the previous example we have dim_{H}(K) = 1 + \frac{ln 2}{ln β}.

Theorem 2 gives a dimension estimate using the singular values

\begin{aligned}α_{1} = α_{1}(d_{(ϑ,v)}φ_{i}^{−1}) & = & \sqrt{\frac{1 + α^{2} + β^{2}}{2β^{2}} + \sqrt{\frac{(1 + α^{2} + β^{2})^{2}}{4β^{4}} − \frac{1}{β^{2}}}} ≥ 1, \\ α_{2} = α_{2}(d_{(ϑ,v)}φ_{i}^{−1}) & = & \sqrt{\frac{1 + α^{2} + β^{2}}{2β^{2}} − \sqrt{\frac{(1 + α^{2} + β^{2})^{2}}{4β^{4}} − \frac{1}{β^{2}}}} ≥ \frac{1}{β} . \end{aligned}

Therefore we have dim_{H}(K) ≤ d for all d > 1 with α_{1}α_{2}^{d−1} < \frac{1}{2}. Because of α_{2} < 1 this means d > 1 + \frac{ln 2+ln α_{1}}{− ln α_{2}} . So we get dim_{H}(K) ≥ 1 + \frac{ln 2+ln α_{1}}{− ln α_{2}} , which is a sharp estimation only in the case α = 0.

Remark Theorem 2 is also applicable to maps which are not piecewise affine-linear, for instance to the Lorenz map (see [3]) with special parameters.

References

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[5] K. Simon: Hausdorff dimension for non-invertible maps. Ergod. Th. & Dynam. Sys. 13: 199 - 212, 1993.

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[7] H. Wegmann: Die Hausdorff-Dimension von kartesischen Produktmengen in metrischen Räumen. J. Reine und Angew. Math. 234: 163 - 171, 1969.