Controlled Hahn–Mazurkiewicz Theorem and some new dimension functions of Peano continua

Abstract

Given a metric Peano continuum X we introduce and study the Hölder Dimension Hö-dim(X) = inf{d: there is a 1 d -Hölder onto map f : [0, 1] → X} of X as well as its topological counterpart Hö-dim(X) = inf{Hö-dim(X, d): d is an admissible metric for X}. We show that for each convex metric continuum X the dimension Hö-dim(X) equals the fractal dimension of X. The topological Hölder dimension Hö-dim(M n ) of the n-dimensional universal Menger cube M n equals n. On the other hand, there are 1-dimensional rim-finite Peano continua X with arbitrary prescribed Hö-dim(X) 1.In this paper we are interested in controlled versions of two important topological results: Theorem 0.1 (Alexandroff-Urysohn, 1929). Each metric compact space is a continuous image of the Cantor set.Theorem 0.2 (Hahn-Mazurkiewicz, 1914). Each metric compact connected locally connected space X is a continuous image of the arc [0, 1].In fact, the latter theorem can be derived from the former: given a locally connected metric continuum X first find a continuous surjective map f : C → X from the Cantor set and then use the (local) path-connectedness to extend the map f onto the whole interval [0, 1] ⊃ C.The controlled versions of the Alexandroff-Urysohn and Hahn-Mazurkiewicz Theorems we are interested in treat possible continuity moduli of surjective maps from a Cantor set or [0, 1] onto a given metric space.

Counterexamples for IFS-attractors

“…On the other hand, Shark teeth type spaces were constructed in [5]. They are parametrized through an infinite non-decreasing sequence (n k ) ∞ k=1 .…”

The shark teeth is a topological IFS-attractor

“…Spaces called shark teeth are constructed in [2] and are parametrized by an infinite non-decreasing sequence (n k ) ∞ k=1 . Let I = [0, 1] × {0} be the bone of shark teeth, and for every k ∈ N let M k = t, 1 k ϕ n k (t) : t ∈ [0, 1] be the kth row of teeth.…”

A 1-dimensional Peano continuum which is not an IFS attractor

“…In order to prove Theorem 1 we shall observe that each connected IFS-attractor has finite S-dimension. This dimension was introduced and studied in [2].…”

“…Our space M is a partial case of the spaces constructed in [2] and called "shark teeth". Consider the piecewise linear periodic function…”