Hausdorff dimension estimates for invariant sets of piecewise smooth maps

Volker Reitmann^{∗}  Uwe Schnabel
reitmann@urzdfn.mathematik.tu-dresden.de schnabel@math.tu-dresden.de

Abstract. Upper Hausdorff dimension estimates for the invariant sets of a class of piecewise smooth and non-invertible maps on manifolds with controllable preimage sets are given in terms of the singular values of the corresponding tangent maps. These estimates generalize some well-known Douady-Oesterlé-type results for differentiable maps in a linear space. A special class of piecewise smooth non-injective maps is considered for which a factor describing the ”degree of non-injectivity” is introduced. This factor is included in the dimension estimate in order to weaken the condition for the singular values of the tangent map. The contraction of a singular value function of iterated maps is investigated with respect to a given partition of the invariant set into ”good” and ”bad” parts (relative to the long time behavior).

Keywords: Hausdorff dimension estimates, tangent map, singular values, noninjective maps, non-differentiability AMS subject classification: Primary 1.58F12, secondary 2.58F08

1 Introduction

For the first time upper Hausdorff dimension estimates for compact sets which are invariant under C^{1}-maps in ℝ^{n} have been published in [5]. The main idea is to show that some iteration of the considered map acts contracting with respect to a given d-dimensional Hausdorff outer measure of the set. It follows that d is an upper bound for the Hausdorff dimension. The contraction constant can be estimated by means of a certain singular value function of the linearized map.

In [9] these contraction conditions of certain Hausdorff outer measures of invariant sets in ℝ^{n} are weakened using Lyapunov-type functions.

In practice the considered maps often act on physical phase spaces which are in general manifolds ([1, 2, 8, 11, 13]). For instance the second order digital filter ([1, 8]) is the fundamental building block of the realization of arbitrary digital filters. Such a secondorder digital filter with two’s complement (modulo 2) nonlinearity can be modeled as piecewise smooth difference equation on a manifold as phase space which is the twodimensional torus T^{2}. For certain parameters such systems have fractal-like invariant sets ([8]). In order to characterize the fractality of such sets it is necessary to get informations on the dimension (e.g. Hausdorff dimension) of the sets. For a special class of differentiable maps on the cylinder considered as feedback systems this is done in [13].

Typeset by L^{A}TEX 2ε

∗Supported by the DFG-Schwerpunktprogramm ”Dynamik: Analysis, effiziente Simulation und Ergodentheorie”

One possibility to handle non-differentiable maps on n-dimensional manifolds is to suppose that the set of non-differentiable points is a finite union of submanifolds having topological dimension less than the dimension of the manifold and to consider only orbits being without contact to the set of non-differentiability (see [1]). In contrast to this in the present article we consider an other class of piecewise smooth maps. The maps under consideration are supposed to be differentiable on the elements of a partition of the given set such that the preimages of these elements under the iterated map are controllable in a certain sense (assumption (A)).

In concrete physical or technical systems the considered maps are often non-injective (see for instance [2, 3, 12]). For such non-injective maps it is possible to include into dimension estimates some information about the degree of non-injectivity. For the first time such Hausdorff and fractal dimension estimates using non-injectivity information have been considered in [3, 12]. We apply the approach of [3] to our class of piecewise smooth maps.

In general the singular value function computed for the differential of the basic map needs not to satisfy the contraction condition on all parts of the invariant set. Nevertheless in such a situation it may be possible to get a contraction for the outer measure if we consider higher iterates of the given map. For this purpose we divide the phase space into ”good” and ”bad” parts in order to compensate the growth of the outer measure elements in bad parts by the decrease of such elements in good parts of the phase space. Investigating the asymptotic behavior of the system we have to guarantee that any orbit stays sufficiently long in good parts and not too long in bad parts of the phase space.

2 Hausdorff dimension and singular value function

First let us recall some notation. Suppose that K is a bounded set in a metric space (X, ρ). For arbitrary real numbers ε > 0 and d ≥ 0 we consider the d-dimensional Hausdorff outer measure at level ε 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 B(u_{i}, r_{i}) = {v ∈ X | ρ(u_{i}, v) ≤ r_{i}} with center u_{i} ∈ X and 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 d-dimensional Hausdorff outer measure of K. For every bounded set K ⊂ M there exists a critical number d^{∗}(K) ≥ 0 such that

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

This critical number is called Hausdorff dimension of K and denoted by dim_{H}(K).

Let E and E′ be two n-dimensional Euclidean spaces with scalar products ⟨·, ·⟩_{E} and ⟨·, ·⟩_{E′}, respectively, and let L : E → E′ be a linear operator. Then L^{∗} denotes the adjoint operator, i. e. the unique linear operator L^{∗} : E′ → E satisfying ⟨L^{∗}u, v⟩_{E} = ⟨u, Lv⟩_{E′} for all u ∈ E′, v ∈ E. We denote by α_{1}(L) ≥ … ≥ α_{n}(L) ≥ 0 the singular values of L, i.e. the eigenvalues of the positive semidefinite operator (L^{∗}L)^{\frac{1}{2}} , ordered with respect to size and algebraic multiplicity. For an arbitrary integer k ∈ {0, 1, …, n} we define the singular value function

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

Furthermore, for an arbitrary number d ∈ [0, n] written in the form d = d_{0} + s with d_{0} ∈ {0, 1, …, n − 1} and s ∈ [0, 1] we define

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

3 A class of piecewise smooth maps

Let (M, g) be an n-dimensional smooth Riemannian manifold and let ρ(·, ·) denote the geodesic distance between two arbitrary points of M. Now we characterize a class of piecewise smooth maps φ : U ⊂ M → M with the property that certain subsets of the set U have controllable preimage sets.

We say that a map φ : U ⊂ M → M satisfies assumption (A) if there exists a partition U = ⋃_{i∈I} U_{i} with U_{i} ∩ U_{j} = ∅ for i ≠ j and I ⊂ {1, 2, … } finite or infinite such that the following conditions hold :

(A1) For an arbitrary m ∈ ℕ there exists a number ε̃_{1}(m) > 0 such that for every mtuple (i_{1}, …, i_{m}) ∈ I^{m} := I × … × I_{m times} two arbitrary points u, v ∈ φ^{−m+1}(U_{i_{m}}) ∩

φ^{−m+2}(U_{i_{m−1}}) ∩ … ∩ U_{i_{1}} with ρ(u, v) < ε̃_{1}(m) can be connected by a continuous curve which lies except of u and v completely in B(u, ρ(u, v)) ∩ int (φ^{−m+1}(U_{i_{m}}) ∩ φ^{−m+2}(U_{i_{m−1}}) ∩ … ∩ U_{i_{1}}). Further, we assume that φ|_{int U_{i}} is C^{1} for every i ∈ I and for arbitrary m ∈ ℕ the map φ^{m} can be extended to a C^{1}-map on φ^{−m+1}(U_{i_{m}}) ∩ φ^{−m+2}(U_{i_{m−1}}) ∩ … ∩ U_{i_{1}}. Let d_{u}φ^{m} : T_{u}M → T_{φ^{m}(u)}M denote the tangent map of this extension for all points u ∈ U.

(A2) There is a natural number k̃ with the property that for arbitrary m ∈ ℕ there exists ε̃_{2}(m) > 0 such that for all ε ∈ (0, ε̃_{2}(m)] and all u ∈ M the number of all m-tuples (i_{1}, …, i_{m}) ∈ I^{m} with

φ^{−m+1}(U_{i_{m}}) ∩ φ^{−m+2}(U_{i_{m−1}}) ∩ … ∩ U_{i_{1}} ∩ B(u, ε) ≠ ∅

is not greater than k̃.

The following two examples show that certain well-known classes of maps satisfy the condition (A).

Example 3.1 Consider the square U = (0, 1] × [0, 1] ⊂ ℝ^{2} and the map φ : U → U (baker’s transformation) defined by

φ(x_{1}, x_{2}) = \{ \begin{matrix}(2x_{1}, λx_{2}) & for & 0 < x_{1} ≤ \frac{1}{2}, 0 ≤ x_{2} ≤ 1, \\ (2x_{1} − 1, λx_{2} + \frac{1}{2}) & for & \frac{1}{2} < x_{1} ≤ 1, 0 ≤ x_{2} ≤ 1 \end{matrix}

with λ ∈ (0, \frac{1}{2}) as a parameter (see [6]). In order to satisfy condition (A) we choose the sets U_{1} = (0, \frac{1}{2}] × [0, 1], U_{2} = (\frac{1}{2}, 1] × [0, 1] and I = \{1, 2\}. It is easy to see that (A) is satisfied with ε̃_{1}(m) = ε̃_{2}(m) = (\frac{1}{2})^{m+1} and k̃ = 2 for every m ∈ ℕ. □

Example 3.2 Consider the set U = (0, 1] × [0, 1] ⊂ ℝ^{2} and the map φ : U → U defined by

φ(x_{1}, x_{2}) = \{ \begin{matrix}(l_{1}x_{1}, r_{1}x_{2} + a_{1}) & for & 0 < x_{1} ≤ \frac{1}{l_{1}} , 0 ≤ x_{2} ≤ 1, \\ (l_{i}(x_{1} − ∑_{k=1}^{i−1} \frac{1}{l_{k}}), r_{i}x_{2} + a_{i}) & for & ∑_{k=1}^{i−1} \frac{1}{l_{k}} < x_{1} ≤ ∑_{k=1}^{i} \frac{1}{l_{k}} , i = 2, …, m̃, \\ & & 0 ≤ x_{2} ≤ 1. \end{matrix}

Here it is assumed that m̃ ≥ 2 is a natural number, l_{i}, r_{i} and a_{i} are real with l_{i} > 1, 0 ≤ a_{i} < 1, 0 < r_{i} ≤ 1 − a_{i} (i = 1, 2, …, m̃) and ∑m̃ ; i=1 \frac{1}{l_{i}} = 1. Suppose further that a_{i} + r_{i} < a_{i+1} for i = 1, 2, …, m̃ − 1. We choose the sets U_{1} = (0, \frac{1}{l_{1}} ] × [0, 1] and U_{i} = ( ∑_{k=1}^{i−1} \frac{1}{l_{k}} , ∑_{k=1}^{i} \frac{1}{l_{k}} ]×[0, 1] (i = 2, 3, …, m̃) and the index set I = \{1, 2, …, m̃\}. Obviously the condition (A) is satisfied with k̃ = 2 and ε̃_{1}(m) = ε̃_{2}(m) = \frac{1}{2}(max_{i=1,... ,m̃} l_{i})^{−m} for every m ∈ ℕ. □

Since the maps from these examples are not everywhere differentiable known methods from [3, 5, 9, 14, 16] can not be applied to get dimension estimates of the invariant sets. We will come back to these examples in the next section.

For a map φ : U ⊂ M → M satisfying (A) and a number d ∈ [0, n] we consider for an arbitrary u ∈ U the linear operator d_{u}φ : T_{u}M → T_{φ(u)}M and define for any bounded set K ⊂ U the function

ω_{d,K}(φ) := sup_{u∈K} ω_{d}(d_{u}φ).

We formulate the following Lemma which can be proved using the methods of [5] and [14].

Lemma 3.1 Let φ : U ⊂ M → M be a map satisfying (A). Furthermore, let K ⊂ U be a bounded and φ-invariant set (i.e. φ(K) = K) and d ∈ [0, n] an arbitrary number. Then the following two statements are true:

a) For arbitrary u ∈ K there holds

ω_{d}(d_{u}φ^{2}) ≤ ω_{d}(d_{φ(u)}φ) · ω_{d}(d_{u}φ).

b) For arbitrary m ∈ ℕ = {1, 2, … } there holds

ω_{d,K}(φ^{m}) ≤ (ω_{d,K}(φ))^{m}.

4 Douady-Oesterlé-type estimates

In [5, 14] Douady-Oesterlé-type dimension estimates for invariant sets of smooth maps are given. In this section we generalize some of these results for our class of piecewise smooth maps.

Theorem 4.1 Suppose that φ : U ⊂ M → M satisfies (A). Let K ⊂ U be a bounded and φ-invariant set (i.e. φ(K) = K) and suppose that for a number d ∈ (0, n] the inequality ω_{d,K}(φ) < 1 is satisfied. Then we have dim_{H}(K) ≤ d.

Corollary 4.1 Suppose that φ : U ⊂ M → M satisfies (A). Let K ⊂ U be bounded and φ-invariant and suppose that there exists a number d̃ ∈ (0, n] with ω_{d̃,K}(φ) = 1 and ω_{d,K}(φ) < 1 for all d ∈ (d̃, n]. Then we have dim_{H}(K) ≤ d̃.

Proof By Theorem 4.1 we have dim_{H}(K) ≤ d for all d > d̃. For d → d̃ we get dim_{H}(K) ≤ d̃. ■

Corollary 4.2 Suppose that φ : U ⊂ M → M satisfies (A). Let K ⊂ U be a bounded and φ-invariant set and suppose that for a certain number d ∈ (0, n] and a continuous function p : K → ℝ_{+} = {x ∈ ℝ : x > 0} the condition

sup_{u∈K} (\frac{p(φ(u))}{p(u)} ω_{d}(d_{u}φ)) < 1

is satisfied. Then we have dim_{H}(K) ≤ d.

Proof The proof is based on a technique considering p as Lyapunov-type function as it is done in [9]. We set v := sup_{u∈K} (\frac{p(φ(u))}{p(u)}ω_{d}(d_{u}φ)). By assumption the value v is less than one. Using the chain rule and applying Lemma 3.1 we have

ω_{d}(d_{u}φ^{j}) ≤ ω_{d}(d_{φ^{j−1}(u)}φ) · . . . · ω_{d}(d_{u}φ) ≤ v \frac{p(φ^{j−1}(u))}{p(φ^{j}(u))} · . . . · v \frac{p(u)}{p(φ(u))}
= v^{j} \frac{p(u)}{p(φ^{j}(u))} ≤ v^{j} \frac{sup_{u∈K} p(u)}{inf_{u∈K} p(u)}

for all j ∈ ℕ and for all u ∈ K. Thus, there exists an integer j such that sup_{u∈K} ω_{d}(d_{u}φ^{j}) < 1 holds. Now we can apply Theorem 4.1 to the map φ^{j} and we obtain dim_{H}(K) ≤ d. ■

Before proving Theorem 4.1 we formulate some lemmata. The first one generalizes a result of [14] for the case of bounded sets. Moreover the steps of its proof can be used for the proof of the subsequent Lemma 4.2.

Lemma 4.1 Let φ : U ⊂ M → M be a map and ε̃ > 0 be a number such that two arbitrary points u, v ∈ U with ρ(u, v) < ε̃ can be connected by a continuous curve which lies except of u and v completely in B(u, ρ(u, v))∩int U. Suppose that the restriction φ|_{int U} is continuously differentiable, and φ can be extended to a C^{1}-map on U. Let d ∈ (0, n] be a number written in the form d = d_{0} + s with d_{0} ∈ {0, 1, …, n − 1} and s ∈ (0, 1], and define C = 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} and λ = 2\sqrt{d_{0} + 1}. Suppose that for a bounded set K ⊂ U and a number k > 0 the inequality ω_{d,K}(φ) ≤ k is satisfied. Then for every l > k there exists a number ε_{0} > 0 such that for every ε ∈ (0, ε_{0}] the inequality

μ(φ(K), d, λl^{\frac{1}{d}} ε) ≤ Clμ(K, d, ε)

holds.

Proof Let ε_{1} < ε̃ be a positive number such that for the set V := U ∩ ⋃_{u∈K} B(u, ε_{1}) the relation k′ := sup_{u∈V} ω_{d}(d_{u}φ) < l holds. We choose a number m > 0 such that k′ < m^{d} and sup_{u∈V} ||d_{u}φ|| ≤ m is satisfied, where || · || denotes the operator norm for the given linear operator. We can find a number η > 0 satisfying

[1 + (\frac{m^{d_{0}}}{k′} )^{1/s} η]^{d} k′ = l.  (1)

Further we take ε_{0} < \frac{1}{2}ε_{1} small enough such that

||τ_{φ(v)}^{φ(u)}d_{v}φτ_{u}^{v} − d_{u}φ|| ≤ η

is satisfied for all u, v ∈ V with ρ(u, v) ≤ 2ε_{0}. Hereby τ_{u}^{v} denotes the isometry between the tangent spaces T_{u}M and T_{v}M defined by parallel transport along the geodesic for points lying sufficiently near to each other.

Let us fix a number ε ≤ ε_{0} and a finite cover of K with balls {B(u_{i}, r̃_{i})}_{i} of radius r̃_{i} ≤ ε, where each ball B(u_{i}, r̃_{i}) contains at least one point v_{i} of K. Then the inclusion B(u_{i}, r̃_{i}) ⊂ B(v_{i}, 2r̃_{i}) holds. Thus, K can be covered by balls {B(v_{i}, r_{i})}_{i} of radius r_{i} ≤ 2ε and with center v_{i} ∈ K.

Now let i be fixed and consider an arbitrary point u ∈ K ∩B(v_{i}, r_{i}). We have u, v_{i} ∈ K and therefore u, v_{i} ∈ U. Resulting from our assumption u and v_{i} can be connected by a continuous curve which lies except of u and v_{i} completely in B(v_{i}, ρ(u, v_{i})) ∩ int U ⊂ V. The Taylor’s formula for the differentiable map φ on this continuous curve provides that

∥ exp_{φ(v_{i})}^{−1} φ(u) − d_{v_{i}}φ(exp_{v_{i}}^{−1}(u))∥ ≤ sup_{w∈B(v_{i},r_{i})} ||τ_{φ(w)}^{φ(v_{i})}d_{w}φτ_{v_{i}}^{w} − d_{v_{i}}φ|| ∥ exp_{v_{i}}^{−1}(w)∥

holds. Thus, we have the inclusion

φ(B(v_{i}, r_{i}) ∩ K) ⊂ exp_{φ(v_{i})} (d_{v_{i}}φ(B(O_{v_{i}}, r_{i})) + B(O_{φ(v_{i})}, ηr_{i})) ,

where O_{v} denotes the origin of the tangent space T_{v}M. Following the way of [14] we get μ(φ(K), d, λl^{\frac{1}{d}} ε) ≤ Clμ(K, d, ε). ■

Lemma 4.2 Suppose that φ : U ⊂ M → M satisfies (A). Let K ⊂ U be a bounded and φ-invariant set. Let C and λ be defined as in Lemma 4.1, let ε̃_{1}(m), ε̃_{2}(m) and k̃ be the constants from property (A2). For a number d ∈ (0, n] and a natural number m we assume that ω_{d,K}(φ^{m}) ≤ k. Then for every l > k there exists a number ε_{0} > 0 such that for all ε ∈ (0, ε_{0}] the inequality

μ(φ^{m}(K), d, λl^{\frac{1}{d}} ε) ≤ k̃Clμ(K, d, ε)

holds.

Proof Setting ε̃:= min{ε̃_{1}(m), ε̃_{2}(m)} we choose ε_{1} < ε̃ such that for the set V := U ∩ ⋃_{u∈K} B(u, ε_{1}) the relation k′ := sup_{u∈V} ω_{d}(d_{u}φ^{m}) holds. In analogous manner as in the proof of Lemma 4.1 we choose numbers m > 0 and η > 0 such that k′ < m^{d}, sup_{u∈V} ||d_{u}φ^{m}|| ≤ m and equation (1) are satisfied. We take ε_{0} < \frac{1}{2}ε_{1} so small such that

||τ_{φ^{m}(v)}^{φ^{m}(u)}d_{v}φ^{m}τ_{u}^{v} − d_{u}φ^{m}|| ≤ η

is satisfied for all u, v ∈ V with ρ(u, v) ≤ 2ε_{0}.

For a fixed number ε ≤ ε_{0} we consider a finite cover of K with balls {B(u_{j}, r̃_{j})}_{j} of radius r̃_{j} ≤ ε. As in the proof of Lemma 4.1 the set K can be covered by balls {B(v_{j}, r_{j})}_{j} of radius r_{j} ≤ 2ε and with center v_{j} ∈ K. By assumption there holds K = φ(K) = … = φ^{m−1}(K) ⊂ U = ⋃_{i∈I} U_{i}. Therefore, we have

K = ⋃_{(i_{1},... ,i_{m})∈I^{m}}(φ^{−m+1}(U_{i_{m}}) ∩ … ∩ U_{i_{1}} ∩ K)

and

φ^{m}(K) = ⋃_{(i_{1},... ,i_{m})∈I^{m}} φ^{m}(φ^{−m+1}(U_{i_{m}}) ∩ … ∩ U_{i_{1}} ∩ K).

Let j be fixed. As in the proof of Lemma 4.1 condition (A1) and Taylors formula provide that for (i_{1}, …, i_{m}) ∈ I^{m} with φ^{−m+1}(U_{i_{m}}) ∩ … ∩ U_{i_{1}} ∩ B(v_{j}, r_{j}) ∩ K ≠ ∅

∥ exp_{φ(v_{j})}^{−1} φ(u) − d_{v_{j}}φ(exp_{v_{j}}^{−1}(u))∥ ≤ sup_{w∈B(v_{j},r_{j})} ||τ_{φ(w)}^{φ(v_{j})}d_{w}φτ_{v_{j}}^{w} − d_{v_{j}}φ|| ∥ exp_{v_{j}}^{−1}(w)∥

holds for all u ∈ φ^{−m+1}(U_{i_{m}}) ∩ … ∩ U_{i_{1}} ∩ B(v_{j}, r_{j}) ∩ K. Thus, we have the inclusion

φ^{m} (B(v_{j}, r_{j}) ∩ φ^{−m+1}(U_{i_{m}}) ∩ … ∩ U_{i_{1}} ∩ K) ⊂ exp_{φ^{m}(v_{j})} (d_{v_{j}}φ^{m}(B(O_{v_{j}}, r_{j})) + B(O_{φ^{m}(v_{j})}, ηr_{j})) .

Using assumption (A2) and following the method of [14] we get

μ(φ^{m}(K), d, λl^{\frac{1}{d}} ε) ≤ k̃Clμ(K, d, ε).
■

Proof of Theorem 4.1 From ω_{d,K}(φ) < 1 and the statement b) of Lemma 3.1 it follows that ω_{d,K}(φ^{m}) becomes arbitrary small for sufficiently large m. Therefore, the number l from Lemma 4.2 can be made so small that the inequalities λl^{\frac{1}{d}} < 1 and k̃Cl < 1 are satisfied, where λ and C are defined as in Lemma 4.1. Using K = φ(K) = … = φ^{m}(K), μ(K, d, ε) < +∞ and Lemma 4.2 we obtain

μ(K, d, ε) ≤ μ(K, d, λl^{\frac{1}{d}} ε) = μ(φ^{m}(K), d, λl^{\frac{1}{d}} ε) ≤ k̃Clμ(K, d, ε)

for all ε ∈ (0, ε_{0}]. Because of k̃Cl < 1 we get as a consequence μ(K, d, ε) = 0 for all ε ∈ (0, ε_{0}]. This implies μ(K, d) = 0 and therefore we get dim_{H} K ≤ d. ■

Now we apply the results of Theorem 4.1 to some examples. In all these cases the Hausdorff dimension estimates of the considered invariant sets, given on the basis of Corollary 4.1, are sharp.

Example 4.1 (Example 3.1 continued) We consider the baker’s transformation φ described in Example 3.1. An invariant set K of φ is given by K = ⋂_{j=1}^{∞} φ^{j}(U). The singular values of the linearization d_{(x_{1},x_{2})}φ are α_{1} = 2 and α_{2} = λ independently of (x_{1}, x_{2}) ∈ K. Therefore, we have ω_{1+s,K}(φ) < 1 for all s ∈ ( − \frac{ln 2}{ln λ}, 1]. With Corollary 4.1 we get the estimate dim_{H}(K) ≤ 1 − \frac{ln 2}{ln λ}. This is a sharp estimation because of the well-known fact that dim_{H}(K) = 1 − \frac{ln 2}{ln λ} (see for example [6]). □

Example 4.2 (Example 3.2 continued) For the map φ defined in example 3.2 a φinvariant set is given by K = ⋂_{k=1}^{∞} φ^{k}(U).

Suppose that the number s̃ ∈ (0, 1] is determined by the condition ∑m̃ ; i=1 rs̃ ; i = 1, and furthermore l_{i} = r_{i}^{−s̃} for i = 1, …, m̃ is satisfied. For all i = 1, 2, …, m̃ and s > s̃ we have ω_{1+s,K∩U_{i}}(φ) = l_{i}r_{i}^{s} < l_{i}rs̃ ; i = 1. It follows that ω_{1+s,K}(φ) < 1, and by Corollary 4.1 we get dim_{H}(K) ≤ 1 + s̃.

Note that our invariant set may be represented in the form K = A × B, where A = (0, 1] and B is a modified Cantor set. Since A and B are Borel sets we have ([6], p. 94) dim_{H}(A)+dim_{H}(B) ≤ dim_{H}(A×B). We pay our attention to the Hausdorff dimension of the set B. With the estimate above and dim_{H}(A) = 1 we obtain dim_{H}(B) ≤ s̃. Resulting from [7] or [6] (Theorem 9.3) we have dim_{H}(B) = s̃. So our estimate is sharp.

For the special case m̃ = 2, l_{1} = l_{2} = 2, r_{1} = r_{2} = \frac{1}{3}, a_{1} = 0 and a_{2} = \frac{2}{3} we see that B is the standard Cantor set. Our estimate gives dim_{H}(B) ≤ s̃ = \frac{ln 2}{ln 3} which coincides with the well-known value dim_{H}(B) = \frac{ln 2}{ln 3}. □

Example 4.3 Consider the map

φ(x_{1}, x_{2}, x_{3}) = \{ \begin{matrix}(3x_{1}, \frac{1}{2}x_{2}, \frac{1}{2}x_{3}) & for & 0 < x_{1} ≤ \frac{1}{3}, (x_{2}, x_{3}) ∈ G, \\ (3x_{1} − 1, \frac{1}{2}x_{2} + \frac{1}{2}, \frac{1}{2}x_{3}) & for & \frac{1}{3} < x_{1} ≤ \frac{2}{3}, (x_{2}, x_{3}) ∈ G, \\ (3x_{1} − 2, \frac{1}{2}x_{2} + \frac{1}{4}, \frac{1}{2}x_{3} + \frac{\sqrt{3}}{4}) & for & \frac{2}{3} < x_{1} ≤ 1, (x_{2}, x_{3}) ∈ G, \end{matrix}

where G = G_{1} ∪ G_{2} ⊂ ℝ^{2} is given by

\begin{matrix}G_{1} = \{(x_{2}, x_{3}) : 0 ≤ x_{2} ≤ \frac{1}{2}, 0 ≤ x_{3} ≤ \sqrt{3}x_{2}\} and \\ G_{2} = \{(x_{2}, x_{3}) : \frac{1}{2} ≤ x_{2} ≤ 1, 0 ≤ x_{3} ≤ \sqrt{3} − \sqrt{3}x_{2}\}. \end{matrix}

Analogously to Example 4.2 an invariant set K can be represented as K = A × B. In the given situation we have A = (0, 1], and B is the well-known Sierpiński gasket (see [6], Example 9.4). It can be shown that Corollary 4.1 gives the estimate dim_{H}(B) ≤ \frac{ln 3}{ln 2}. As to be seen in [6] this estimation is also sharp. □

5 Consideration of the degree of non-injectivity

In order to use the Douady-Oesterlé-type conditions of the previous section for dimension estimates it is necessary that the contraction condition of the singular value function ω_{d,K}(φ) < 1 is satisfied. This may result in strong restrictions for the parameters as it happens for instance in Example 4.1 where λ ∈ (0, \frac{1}{2}) is required. The question raises how we can get a dimension estimate for the parameters λ > \frac{1}{2}, i. e. for the case ω_{d,K}(φ) > 1.

It turns out that in certain cases, where the contraction condition of the singular value function is not fulfilled an additional non-injectivity condition allows to get a contraction for the outer measures. Using the approach of [3] we investigate piecewise C^{1}-maps on ndimensional smooth manifolds having the following property. Suppose that for the given d-dimensional Hausdorff outer measure μ(., d, ε) at level ε there exists a number 0 < a < 1 and a family {K_{j}}_{j≥j_{0}} of subsets of the φ-invariant set K such that for all j ≥ j_{0} this outer measure of K_{j} is at most a^{j} times the outer measure of K and the outer measures of the sets φ^{j}(K_{j}) and φ^{j}(K) are equal. A map with such a property can be considered as piecewise expansive with respect to the Hausdorff measure on K and the factor a^{−1} is the expansion parameter which describes the ”degree of non-injectivity”.

Theorem 5.1 Let (M, g) be a smooth n-dimensional Riemannian manifold. Suppose that φ : U ⊂ M → M satisfies (A). Let K ⊂ U be a bounded and φ-invariant set. Suppose that for some numbers a > 0 and d ∈ (0, n] of the form d = d_{0} +s with d_{0} ∈ {0, …, n−1} and s ∈ (0, 1] the following conditions are satisfied:

(a) ω_{d,K}(φ) < \frac{1}{a};

(b) there are numbers l with ω_{d,K}(φ) < l < \frac{1}{a} and m_{0} ∈ ℕ such that for every natural number m > m_{0} there exist a set K_{m} ⊂ K and a number ε_{m} > 0 such that

μ(φ^{m}(K_{m}), d, ε) = μ(φ^{m}(K), d, ε),

μ(K_{m}, d, 2^{−1}(d_{0} + 1)^{− \frac{1}{2}} l^{− \frac{m}{d}} ε) ≤ a^{m}μ(K, d, ε)

hold for all ε ∈ (0, ε_{m}].

Then we have dim_{H}(K) ≤ d.

Proof The proof is contributed by methods of [3] and Lemma 4.2. By Lemma 3.1 we have ω_{d,K}(φ^{m}) ≤ ω_{d,K}^{m}(φ) < l^{m} for all m ∈ ℕ. Thus, for every δ > 0 we find an integer m_{δ} > m_{0} such that k̃2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} (al)^{m} < δ will be true for all m > m_{δ}. Let m > m_{δ} be fixed. Using the invariance of the set K and condition (b) we obtain

μ(K, d, ε) = μ(φ^{m}(K), d, ε) = μ(φ^{m}(K_{m}), d, ϵ)  (2)

for all ε ∈ (0, ε_{m}]. Following the method of the proof of Lemma 4.2 we can show that there exists a number ε_{0} > 0 such that

μ(φ^{m}(K_{m}), d, ε) ≤ k̃ 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} l^{m}μ(K_{m}, d, 2^{−1}(d_{0} + 1)^{− \frac{1}{2}} l^{− \frac{m}{d}} ε)  (3)

holds for every ε ≤ 2(d_{0} + 1)^{\frac{1}{2}} l^{\frac{m}{d}} ε_{0}. For ε ∈ (0, min\{ε_{m}, 2(d_{0} + 1)^{\frac{1}{2}} l^{\frac{m}{d}} ε_{0}\}] condition (b),

(2) and (3) imply

μ(K, d, ε) ≤ k̃ 2^{d_{0}}(d_{0} + 1)^{\frac{d}{2}} (la)^{m}μ(K, d, ε) < δμ(K, d, ε).

The number δ > 0 can be chosen arbitrarily small. Since μ(K, d, ε) is finite we can conclude that μ(K, d, ε) = 0 holds for all ε ∈ (0, min\{ε_{m}, 2(d_{0} + 1)^{\frac{1}{2}} l^{\frac{m}{d}} ε_{0}\}]. Hence we get μ(K, d) = 0 and dim_{H}(K) ≤ d. ■

Example 5.1 Consider the set U = [0, 1] × [−1, 1) ⊂ ℝ^{2} and the map φ : U → U defined by

φ(x_{1}, x_{2}) = \{ \begin{matrix}(2x_{1}, λx_{2}) & for & 0 ≤ x_{1} ≤ \frac{1}{2}, −1 ≤ x_{2} < 1, \\ (2 − 2x_{1}, λx_{2}) & for & \frac{1}{2} < x_{1} ≤ 1, −1 ≤ x_{2} < 1, \end{matrix}

where λ ∈ (0, 1) is a parameter.

In order to satisfy condition (A) we choose U_{1} = [0, \frac{1}{2}] × [−1, 1), U_{2} = (\frac{1}{2}, 1] × [−1, 1) and the index set I = {1, 2}. The condition (A) is satisfied if we choose ε̃_{1}(m) = ε̃_{2}(m) = (\frac{1}{2})^{m+1} for m ∈ ℕ and k̃ = 2. It is easy to see that U ⊃ φ(U) ⊃ … ⊃ φ^{m}(U) holds for every m ∈ ℕ, so we can define a φ-invariant bounded set K = ⋂_{m=1}^{∞} φ^{m}(U) = [0, 1]×{0}. The singular values of d_{u}φ^{m} are α_{1}(d_{u}φ^{m}) = 2^{m} and α_{2}(d_{u}φ^{m}) = λ^{m}.

Let us define K_{1} := K ∩ [0, \frac{1}{2}] × \{0\} and K_{m} := K_{m−1} ∩ φ^{−1}(K_{m−1}) (m = 2, 3, …). For arbitrary m ∈ ℕ we have φ^{m}(K_{m}) = K = φ^{m}(K). Thus, for all ε > 0 and d ∈ (1, 2] the relation μ(φ^{m}(K_{m}), d, ε) = μ(φ^{m}(K), d, ε) holds. If we cover the set K by balls of radius smaller than 1 the set K_{m} can be covered by linear copies of elements covering the set K_{m−1} scaled by the factor \frac{1}{2}. Thus, we get

μ(K_{m}, d, ε) ≤ \frac{1}{2^{d}} μ(K_{m−1}, d, 2ε)

for arbitrary m ∈ ℕ, arbitrary d ∈ (1, 2] and all ε < 1. For ε_{m} < 1 it follows that μ(K_{m}, d, 2^{−m}ε) ≤ 2^{−dm}μ(K, d, ε) holds for all ε ∈ (0, ε_{m}]. For a = 2^{−d} and l := 2 we have ω_{d,K}(φ) = 2λ^{d−1} < l < \frac{1}{a}. Further, there holds 2^{−1− \frac{1}{2}} l^{− \frac{m}{d}} < 2^{−m} for every m ∈ ℕ. Because of the property of the outer measure we conclude

μ(K_{m}, d, 2^{−1− \frac{1}{2}} l^{− \frac{m}{d}} ε) ≤ a^{m}μ(K, d, ε)

for all ε ∈ (0, ε_{m}] and m = 1, 2, . . .. So all conditions of Theorem 5.1 are satisfied and we get dim_{H}(K) ≤ d for arbitrary d ∈ (1, 2]. Applying Corollary 4.1 this yields to dim_{H}(K) ≤ 1, which is a sharp estimate. □

6 Introduction of long time behavior information

In this section we investigate maps φ for which the singular value function of the tangent map satisfies a contraction condition on a subset of the φ-invariant set K only. Using information about the long time behavior of the system we derive dimension estimates.

Consider again a map φ : U ⊂ M → M satisfying (A) which possesses a bounded invariant set K ⊂ U. Suppose that there exists a partition of K of the form K = (⋃_{i=1}^{l} K_{i}) ∪ (⋃_{i=1}^{h} K′_{i}) such that ω_{d,K_{i}}(φ) < 1 (i = 1, …, l) and ω_{d,K′_{i}}(φ) ≥ 1 (i = 1, …, h) hold. For i = 1, …, l we define the numbers

p_{i} = lim inf_{m→∞} inf_{u∈K} \frac{#\{k : 0 ≤ k ≤ m − 1, φ^{k}(u) ∈ K_{i}\}}{m}

and for i = 1, …, h the numbers

p′_{i} = lim sup_{m→∞} sup_{u∈K} \frac{#\{k : 0 ≤ k ≤ m − 1, φ^{k}(u) ∈ K′_{i}\}}{m}.

Note that in the particular case that {φ^{m}} has an invariant ergodic probability measure ν on U then for an arbitrary measurable set A ⊂ U and for ν-almost every u ∈ U the relation

lim_{m→∞} \frac{1}{m} ∑_{k=0}^{m−1} χ_{A}(φ^{k}(u)) = lim_{m→∞} \frac{#\{k : 0 ≤ k ≤ m − 1, φ^{k}(u) ∈ A\}}{m} = ν(A)

holds, where χ_{A} denotes the characteristic function of A. The following theorem is a generalization of a result of [15] for the case of piecewise smooth maps on manifolds.

Theorem 6.1 Let (M, g) be a smooth n-dimensional Riemannian manifold. Suppose that φ : U ⊂ M → M satisfies assumption (A) and let K ⊂ U be a bounded and φ-invariant set. Suppose that {p_{i}}_{i=1}^{l} and {p′_{i}}_{i=1}^{h} are the numbers defined above for φ with respect to the partition K = (⋃_{i=1}^{l} K_{i}) ∪ (⋃_{i=1}^{h} K′_{i}). Let \overline{p}_{i} and \overline{p}′_{i} be numbers with \overline{p}_{i} ≤ p_{i} (i = 1, …, l) and \overline{p}′_{i} ≥ p′_{i} (i = 1, …, h) such that

(∏_{i=1}^{l} ω_{d,K_{i}}(φ)^{\overline{p}^{i}}) (∏_{i=1}^{h} ω_{d,K′_{i}}(φ)^{\overline{p}′^{i}}) < 1  (4)

is satisfied. Then dim_{H}(K) ≤ d.

Proof Without loss of generality we suppose that the inequality (4) is satisfied with numbers \overline{p}_{i} < p_{i} (i = 1, …, l) and \overline{p}′_{i} > p_{i} (i = 1, …, h). Let us take a sufficiently large m such that

\overline{p}_{i} < \frac{1}{m} inf_{u∈K} #\{k : 0 ≤ k ≤ m − 1, φ^{k}(u) ∈ K_{i}\} (i = 1, …, l)

and

\overline{p}′_{i} > \frac{1}{m} sup_{u∈K} #\{k : 0 ≤ k ≤ m − 1, φ^{k}(u) ∈ K′_{i}\} (i = 1, …, h)

are satisfied. It follows that an arbitrary orbit of length m starting in u ∈ K passes the set K_{i} more frequently than \overline{p}_{i}m-times (i = 1, …, l) and the set K′_{i} less than \overline{p}′_{i}m-times (i = 1, …, h). For the tangent map d_{u}φ^{m} in an arbitrary point u ∈ K the chain rule

d_{u}φ^{m} = d_{φ^{m−1}(u)}φ ∘ … ∘ d_{u}φ

holds. Using Lemma 3.1 we get the inequality

ω_{d,k}(φ^{m}) ≤ sup_{u∈K} [ω_{d}(d_{φ^{m−1}(u)}φ)ω_{d}(d_{φ^{m−2}(u)}φ) … ω_{d}(d_{u}φ)]
≤ (∏_{i=1}^{l} ω_{d,K_{i}}^{m\overline{p}_{i}}(φ)) (∏_{i=1}^{h} ω_{d,K′_{i}}^{m\overline{p}′_{i}}(φ))

= [(∏_{i=1}^{l} ω_{d,K_{i}}^{\overline{p}_{i}}(φ)) (∏_{i=1}^{h} ω_{d,K′_{i}}^{\overline{p}′_{i}}(φ))]^{m} < 1.

On the basis of Theorem 4.1 we conclude dim_{H}(K) ≤ d. ■

Example 6.1 ([15]) Consider the set U = (0, 1]×[0, 1] and the map φ : U → U (modified baker’s transformation) defined by

φ(x_{1}, x_{2}) = \{ \begin{matrix}(2x_{1}, λ_{2}x_{2}) & for & 0 < x_{1} ≤ \frac{1}{2}, \frac{1}{2} ≤ x_{2} ≤ 1, \\ (2x_{1}, λ_{1}x_{2}) & for & 0 < x_{1} ≤ \frac{1}{2}, 0 ≤ x_{2} < \frac{1}{2}, \\ (2x_{1} − 1, λ_{1}x_{2} + \frac{1}{2}) & for & \frac{1}{2} < x_{1} ≤ 1, 0 ≤ x_{2} ≤ 1, \end{matrix}

where λ_{1}, λ_{2} are parameters with 0 < λ_{1} < λ_{2} < \frac{1}{2}. The singular values of the tangent map are α_{1}(d_{u}φ) = 2 and α_{2}(d_{u}φ) ∈ {λ_{1}, λ_{2}}. On the basis of Theorem 4.1 we get dim_{H}(K) ≤ 1 + \frac{ln 2}{| ln λ_{2}|}.

We want to improve this estimate using Theorem 6.1. For this purpose we consider the sets U′_{1} = [0, \frac{1}{2}] × [\frac{1}{2}, 1] and U_{1} = U \ U′_{1}. From φ(U′_{1}) ⊂ U_{1} it follows that every point u ∈ U′_{1} satisfies φ(u) ∉ U′_{1}. So we can conclude that

lim sup_{m→∞} sup_{u∈U′_{1}} \frac{#\{k : 0 ≤ k ≤ m − 1, φ^{k}(u) ∈ U′_{1}\}}{m} ≤ \frac{1}{2}

and

lim inf_{m→∞} inf_{u∈U_{1}} \frac{#\{k : 0 ≤ k ≤ m − 1, φ^{k}(u) ∈ U_{1}\}}{m} ≥ \frac{1}{2}

are satisfied. For the invariant set K = ⋂_{k=1}^{∞} φ^{k}(U) we define K_{1} = U_{1} ∩ K and K′_{1} = U′_{1} ∩ K. Note that ω_{d,K_{1}}(φ) = 2λ_{1}^{s} and ω_{d,K′_{1}}(φ) = 2λ_{2}^{s} hold for any d = 1 + s with s ∈ [0, 1]. Now we define \overline{p}_{1} := \frac{1}{2} ≤ p_{1} and \overline{p}′_{1} := \frac{1}{2} ≥ p′_{1}. The condition of Theorem 6.1 is fulfilled if (2λ_{1}^{s})^{\frac{1}{2}} (2λ_{2}^{s})^{\frac{1}{2}} < 1 holds. So we get the estimate dim_{H}(K) ≤ 1 + \frac{ln 2}{| ln \sqrt{λ_{1}λ_{2}}|}. □

Recently there have been done numerical investigations in order to approximate essential dynamics of a given system (e.g. [4]). We expect that these methods can be applied to get estimates of the values \overline{p_{i}}, \overline{p_{i}}′ from Theorem 6.1 numerically.

Acknowledgment

We would like to thank Astrid Franz and Katrin Gelfert from the University of Technology Dresden for their fruitful discussions, support and help.

References

[1] V. S. Afraimovich : On the Lyapunov dimension of invariant sets in a model of active medium. In: ”Methods of Qualitative Theory of Differential Equations”, Gorki State University, Gorki (1986), pp. 19 - 29. (Russian).

[2] V. N. Belykh : Models of discrete systems of phase synchronization. In: V. V. Shakhgil’dyan and L. N. Belyustina (Eds.): Systems of phase synchronization. Radio i Svyaz’, Moscow (1982), pp. 161 - 176. (Russian).

[3] V. A. Boichenko, A. Franz, G. A. Leonov and V. Reitmann : Hausdorff and fractal dimension estimates for invariant sets of non-injective maps. ZAA 17 (1998) 1, 1-17 [4] M. Dellnitz and O. Junge : On the Approximation of Complicated Dynamical

Behavior. Preprint 62/96 of the DFG-Schwerpunktprogramm ”Dynamik: Analysis, effiziente Simulation und Ergodentheorie” (1996).

[5] A. Douady and J. Oesterlé : Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris Ser. A 290 (1980), 1135 - 1138.

[6] K. J. Falconer : Fractal Geometry. Mathematical Foundations and Applications. John Wiley & Sons, Chichester (1990).

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

[8] L. Kocarev and L. O. Chua : On chaos in Digital Filters : Case b = −1. IEEE Transaction on Circuits and systems. II : Analog and Digital Signal Processing, 40 (1993) 6, 404-407.

[9] G. A. Leonov and V. A. Boichenko : Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math. 26 (1992), 1-60.

[10] G. A. Leonov and V. Reitmann : Lokalisierung der Lösung diskreter Systeme mit instationärer periodischer Nichtlinearität. ZAMM 66 (1986) 2, 103-111.

[11] G. A. Leonov, V. Reitmann and V. B. Smirnova : Non-local methods for pendulum-like feed-back systems. Teubner-Texte zur Mathematik, Bd. 132, B. G. Teubner Stuttgart- Leipzig (1992).

[12] A. Mirle : Hausdorff dimension estimates for invariant sets of k -1-maps. Preprint 25/95 of the DFG-Schwerpunktprogramm ”Dynamik: Analysis, effiziente Simulation und Ergodentheorie” (1995).

[13] A. Noack : Hausdorff Dimension Estimates for time-discrete Feedback Control Systems. ZAMM 77 (1997) 12, 891-899.

[14] A. Noack and V. Reitmann : Hausdorff dimension of invariant sets of timedependent vector fields. ZAA 15 (1996) 2, 457 - 473.

[15] G. Schmidt : Dimension estimates for invariant sets of differential equations with non-smooth right part and of locally expanding dynamical systems. Diploma thesis, University of Technology Dresden (1996).

[16] R. Temam : Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988).