Inductive topological Hausdorff dimensions and fibers of generic continuous functions

TL;DR

This paper introduces the inductive topological Hausdorff dimension to analyze the Hausdorff dimension of fibers of generic continuous functions on compact metric spaces, extending previous work on level set dimensions.

Contribution

It defines the $n$th inductive topological Hausdorff dimension and establishes its role in determining the Hausdorff dimension of fibers of generic functions from $K$ to $\

Findings

The supremum of fiber dimensions equals $ ext{dim}_{t^nH} K - n$ for generic functions.

The supremum is attained, not just an upper bound.

Characterization of spaces where fiber dimensions are exactly $ ext{dim}_{t^nH} K - n$.

Abstract

In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$. The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from $K$ to $\mathbb{R}^n$. In order to do so, we define the $n$th inductive topological Hausdorff dimension, $\dim_{t^nH} K$. Let $\dim_H K$, $\dim_t K$ and $C_n(K)$ denote the Hausdorff and topological dimension of $K$ and the Banach space of the continuous functions from $K$ to $\mathbb{R}^n$. We show that $\sup_{y\in \mathbb{R}^n} \dim_{H}f^{-1}(y) = \dim_{t^nH} K -n$ for the generic $f \in C_n(K)$, provided that $\dim_t K\geq n$, otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces $K$ for which $\dim_{H} f^{-1}(y)=\dim_{t^nH}K-n$ for the generic $f\in C_n(K)$ and the generic $y\in f(K)$. We also generalize a result of Kirchheim by showing that if $K$ is self-similar and $\dim_t K\geq n$ then $\dim_{H} f^{-1}(y)=\dim_{t^nH}K-n$ for the generic $f\in C_n(K)$ for every $y\in \inter f(K)$.