Are Level Sets Open for Continuous Functions

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Topological Hausdorff dimension and level sets of generic continuous functions on fractals

Abstract

In an earlier paper we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space K let dim _H K and dim _tH K denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on K, namely $\sup \{{\dim }_H{f}^{-1}(y):y\in \mathbb{R}\}={\dim }_{\mathit{tH}}K-1$ for the generic f  ∈ C(K), provided that K is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if K is not totally disconnected and sufficiently homogeneous then dim _H f ⁻¹(y)   =   dim _tH K  −   1 for the generic f  ∈ C(K) and the generic y  ∈ f(K). The most important goal of this paper is to make these theorems more precise.

As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension.

As for the second result, we characterize those compact metric spaces for which for the generic f  ∈ C(K) and the generic y  ∈ f(K) we have dim _H f ⁻¹(y)   =   dim _tH K  −   1. We also generalize a result of B. Kirchheim by showing that if K is self-similar then for the generic f  ∈ C(K) for every $y\in \mathrm{int}f(K)$ we have dim _H f ⁻¹(y)   =   dim _tH K  −   1.

Finally, we prove that the graph of the generic f  ∈ C(K) has the same Hausdorff and topological Hausdorff dimension as K.

Highlights

► We examine a new fractal dimension, the so called topological Hausdorff dimension. ► The generic continuous function has a level set of maximal Hausdorff dimension. ► This maximal dimension is the topological Hausdorff dimension minus one. ► Homogeneity implies that "most" level sets are of this dimension. ► We calculate the various dimensions of the graph of the generic function.

Introduction

We recall first the definition of the (small inductive) topological dimension.

Definition 1.1

Set dim _t   ∅   =   −1. The topological dimension of a non-empty metric space X is defined by induction as ${\dim }_{t}X=\inf \{d:X\text{has a basis}\mathcal{U}\text{s.t.}{\dim }_t\partial U⩽d-1\text{for every}U\in \mathcal{U}\}\text{.}$

For more information on this concept see [3] or [6].

We introduced the topological Hausdorff dimension for compact metric spaces in [1]. It is defined analogously to the topological dimension. However, it is not inductive, and it can attain non-integer values as well. The Hausdorff dimension of a metric space X is denoted by dim _H X, see e.g. [5] or [9]. In this paper we adopt the convention that dim _H   ∅   =   −1.

Definition 1.2

Set dim _tH   ∅   =   −1. The topological Hausdorff dimension of a non-empty metric space X is defined as ${\dim }_{\mathit{tH}}X=\inf \{d:X\text{has a basis}\mathcal{U}\text{s.t.}{\dim }_H\partial U⩽d-1\text{for every}U\in \mathcal{U}\}\text{.}$

Both notions of dimension can attain the value ∞ as well.

Let K be a compact metric space, and let C(K) denote the space of continuous real-valued functions equipped with the supremum norm. Since this is a complete metric space, we can use Baire category arguments. If dim _t K  =   0 then the generic f  ∈ C(K) is well-known to be one-to-one (see Lemma 2.6), so every non-empty level set is a singleton.

Assume dim _t K  >   0. The following results from [1] show the connection between the topological Hausdorff dimension and the level sets of the generic f  ∈ C(K).

Theorem 1.3

If K is a compact metric space with dim _t K  > 0 then for the generic f  ∈ C(K)

(i)

dim _H f ⁻¹ (y)  ⩽ dim _tH K  − 1 for every $y\in \mathbb{R}$ ,

(ii)

for every ε  > 0 there exists a non-degenerate interval I _f,ε such that dim _H f ⁻¹ (y)  ⩾ dim _tH K  − 1  − ε for every y  ∈ I _f,ε .

Corollary 1.4

If K is a compact metric space with dim _t K  > 0 then for the generic f  ∈ C(K) $\sup \{{\dim }_H{f}^{-1}(y):y\in \mathbb{R}\}={\dim }_{\mathit{tH}}K-1\text{.}$

If K is also sufficiently homogeneous, for example self-similar, then we can actually say more.

Theorem 1.5

If K is a self-similar compact metric space with dim _t K  > 0 then for the generic f  ∈ C(K) and the generic y  ∈ f(K) ${\dim }_H{f}^{-1}(y)={\dim }_{\mathit{tH}}K-1\text{.}$

Theorem 1.3, Theorem 1.5 are the starting points of this paper, our primary aim is to make these theorems more precise.

In the Preliminaries section we introduce some notation and definitions, cite some important properties of the topological Hausdorff dimension, and prove several technical lemmas.

In Section 3 we prove a partial converse of Theorem 1.5. We show that for the generic f  ∈ C(K) for the generic y  ∈ f(K) we have dim _H f ⁻¹(y)   =   dim _tH K  −   1 iff K is homogeneous for the topological Hausdorff dimension, that is for every non-empty closed ball B(x, r)   ⊆ K we have dim _tH   B(x, r)   =   dim _tH K. If K is (weakly) self-similar then much more is true: For the generic f  ∈ C(K) for every y  ∈   int f(K) we have dim _H f ⁻¹(y)   =   dim _tH K  −   1. This generalizes a result of B. Kirchheim. He proved in [8] that for the generic f  ∈ C([0,   1] ^d ) for every y  ∈   int f([0,   1] ^d ) we have dim _H f ⁻¹(y)   = d  −   1.

In Section 4 we prove that the generic f  ∈ C(K) has at least one level set of maximal Hausdorff dimension. Hence the supremum is attained in Corollary 1.4. We construct an attractor of an iterated function system $K\subseteq {\mathbb{R}}^2$ such that the generic f  ∈ C(K) has a unique level set of Hausdorff dimension dim _tH K  −   1. This shows that the above theorem is sharp.

Finally, in Section 5 we prove that the graph of the generic f  ∈ C(K) has the same Hausdorff and topological Hausdorff dimension as K. This generalizes a result of R.D. Mauldin and S.C. Williams which states that the graph of the generic f  ∈ C([0,   1]) is of Hausdorff dimension one, see [11].

Section snippets

Notation and definitions

Let (X, d) be a metric space, and let A, B  ⊆ X be arbitrary sets. We denote by intA and ∂A the interior and boundary of A. The diameter of A is denoted by diam A. We use the convention diam∅   =   0. The distance of the sets A and B is defined by dist (A, B)   =   inf{d(x, y) : x  ∈ A, y  ∈ B}. Let B(x, r)   =   {y  ∈ X : d(x, y)   ⩽ r} and U(x, r)   =   {y  ∈ X : d(x, y)   < r}. More generally, we define B(A, r)   =   {x  ∈ X : dist (x, A)   ⩽ r} and U(A, r)   =   {x  ∈ X : dist (x,A)   < r}.

For two metric spaces (X, d _X ) and (Y,d _Y ) a function f : X  → Y is Lipschitz if there

Level sets on fractals

Let K be a compact metric space. If dim _t K  =   0 then the generic continuous function is one-to-one on K by Lemma 2.6, hence every non-empty level set is a single point.

Thus in the sequel we assume that dim _t K  >   0.

Definition 3.1

If K is a compact metric space then let $\mathrm{supp}K=\left\{x\in K:\forall r>0\text{,}{\dim }_{\mathit{tH}}B(x\text{,}r)={\dim }_{\mathit{tH}}K\right\}\text{.}$ We say that K is homogeneous for the topological Hausdorff dimension if supp K  = K.

Remark 3.2

The stability of the topological Hausdorff dimension for closed sets clearly yields supp K  ≠   ∅. If K is self-similar then it is also

Level sets of maximal dimension

Let K be a compact metric space. If dim _t K  =   0 then the generic f  ∈ C(K) is one-to-one by Lemma 2.6, thus every non-empty level set is a single point.

Assume dim _t K  >   0. Corollary 1.4 states that for the generic f  ∈ C(K) we have ${\sup }_{y\in \mathbb{R}}{\dim }_H{f}^{-1}(y)={\dim }_{\mathit{tH}}K-1$ . First we prove that in this statement the supremum is attained.

Theorem 4.1

Let K be a compact metric space with dim _t K  > 0. Then for the generic f  ∈ C(K) ${\displaystyle \underset{y\in \mathbb{R}}{\max }}{\dim }_H{f}^{-1}(y)={\dim }_{\mathit{tH}}K-1\text{.}$

Proof

By Theorem 1.3 it is sufficient to prove that for the generic f  ∈ C(K) there exists a level

The dimension of the graph of the generic continuous function

The graph of the generic f  ∈ C([0,   1]) is of Hausdorff dimension one, this is a result of R.D. Mauldin and S.C. Williams [11, Theorem 2]. We generalize the cited theorem for arbitrary compact metric spaces. Let K be a compact metric space, then for the generic f  ∈ C(K) the graph of f is of Hausdorff dimension dim _H K. We prove an analogous theorem for the topological Hausdorff dimension, for the generic f  ∈ C(K) the graph of f is of topological Hausdorff dimension dim _tH K.

Definition 5.1

If f  ∈ C(K) let us define $\tilde{f}:K\to \mathrm{graph}$

Acknowledgment

We are indebted to an anonymous referee for his valuable comments and for suggesting Example 3.9.

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