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

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 .