TITLE: Does there exist a autohomeomorphism $\mathbb{R} ^ 2$ transforming the disc into Koch snowflake?
QUESTION [5 upvotes]: Does there exist a uniform autohomeomorphism $\mathbb{R} ^ 2$ transforming the disc $\{(x, y) \mid x ^ 2 + y ^ 2 \leq 1\}$ into Koch snowflake? The Koch snowflake is a closed subset of the plane bounded by the Koch curve.
I understand that this subspace itself is uniformly homeomorphic to a disc.
If instead of the Koch snowflake we take a triangle, a rectangle and, in general, any convex figure containing a unit disk, then the desired autohomeomorphism of the plane is constructed using some stretching of the rays going out from the center of the disk.

REPLY [2 votes]: Of course, according to Schoenflies's theorem, there is an autohomeomorphism of the large disk that turns the small disk in its center into a coch snowflake. Since the disk is compact, the same homeomorphism is uniform.