On Hölder maps of cubes

• Published in: Mathematical Notes Vol. 87; no. 5-6; pp. 757 - 767
• Main Author: Shchepin, E. V.
• Format: Journal Article
• Language: English
• Published: Dordrecht SP MAIK Nauka/Interperiodica 01-06-2010
• Subjects: Mathematics; Mathematics and Statistics; Peano-Arnold map; Hausdorff volume; Hausdorff dimension; Lipschitz constant; fractal map; Hölder map
• ISSN: 0001-4346; 1573-8876
• DOI: 10.1134/S0001434610050135

A map of metric spaces
f: X
→
Y
satisfying the inequality
for some
C
and
α
and all
x, y
∈
X
is called a
Hölder map
with
exponent α
. V. I. Arnold posed the following problem: Does there exist a Höldermap from the square onto the cube with exponent 2/3? The firstmain theorem of this paper gives a general method for constructing Höldermaps of compact metric spaces. This construction yields, in particular, a dimension-raising map
f
:
I
n
→
I
m
with Hölder exponent arbitrarily close to
m/n
for
m
>
n
> 1 and a map
I
1
→
I
m
with Hölder exponent 1/
m
. The second main theorem states the nonexistence of a regular fractal map
f
:
I
n
→
I
m
with Hölder exponent
n/m
from the
n
-cube onto the
m
-cube for
m
< 2
n
.