Arnold posed a problem (1988-5 in "Arnold's problems") if there is a surjective map [0,1]^2 -> [0,1]^3 with Holder exponent 2/3. E.V.Shchepin proved that one can get arbitrarily close to that (and to n/m in generic case [0,1]^n->[0,1]^m). See:

Shchepin, E.V. On Hölder maps of cubes. Math Notes 87, 757–767 (2010). https://doi.org/10.1134/S0001434610050135

The problem of construction of a map that attains exact exponent n/m -- remains open, as far as I know.

Arnold posed a problem (1988-5 in "Arnold's problems") if there is a surjective map $[0,1]^2 \to [0,1]^3$ with Holder exponent $2/3$. E. V. Shchepin proved that one can get arbitrarily close to that (and to $n/m$ in generic case $[0,1]^n\to[0,1]^m$). See:

Shchepin, E.V. On Hölder maps of cubes. Math Notes 87, 757–767 (2010). https://doi.org/10.1134/S0001434610050135

The problem of construction of a map that attains exact exponent n/m -- remains open, as far as I know.