A Controlled Hahn-Mazurkiewicz Theorem and its Applications

Abstract

For a metric Peano continuum $X$, let $S_X$ be a Sierpi\'nski function assigning to each $\varepsilon>0$ the smallest cardinality of a cover of $X$ by connected subsets of diameter $\le \varepsilon$. We prove that for any increasing function $\Omega:\mathbb R_+\to\mathbb R_+$ with $(0,1]\subseteq\Omega[\mathbb R_+]$ and $s:=\sum_{n=1}^\infty S_X(2^{-n})\sum_{m=n}^\infty S_X(2^{-m})\,\Omega^{-1}(\min\{1,2^{6-m}\})<\infty$ there exists a continuous surjective function $f:[0,s]\to X$ with continuity modulus $\omega_f\le\Omega$. This controlled version of the classical Hahn-Mazurkiewicz Theorem implies that $SDim(X)\le HDim(X)\le 2{\cdot}SDim(X)$, where $SDim(X)=\limsup_{\varepsilon\to 0}\frac{\ln(S_X(\varepsilon))}{\ln(1/\varepsilon)}$ is the $S$-dimension of $X$, and $HDim(X)=\inf\{\alpha\in (0,\infty]:$ there is a~surjective $\frac1\alpha$-H\"older map $f:[0,1]\to X\}$ is the $H\ddot older$ $dimension$ of $X$.