Abstract: Consider a purely unrectifiable set S ⊂ R2 (i.e. the intersection of S with every rectifiable curve has length 0). Good examples to keep in mind are the four corner Cantor set (of finite, positive H1 measure) and the Koch curve (of non σ-finite H1 measure). Using the Besicovitch-Federer projection theorem, it is essentially an easy exercise to show that if H1(S) < ∞ and S is compact, there exists a sequence fn : R2 → R2 of 1-Lipschitz maps such that
1. ||fn − Id ||∞ → 0,
2. fn(S) is a finite set.
The natural question is whether H1(S) < ∞ is an assumption that can be dropped. The answer is obviously negative: if S is path-connected and nontrivial (e.g. the Koch curve) then for any f : R2 → R2 continuous and sufficiently close to identity, f(S) is connected and non-trivial, hence infinite (even positive H1 measure). Instead, we prove that if S is compact and purely unrectifiable, then for any Borel measure μ supported on S, there exists a sequence fn : R2 → R2 of 1-Lipschitz maps such that
1. ||fn − Id ||∞ → 0,
2. fn#μ is a 0-dimensional measure.
Attempting to prove higher-dimensional variants of this statement in the generality of metric spaces naturally leads to various definitions of tangent fields to metric (measure) spaces (where for a purely-unrectifiable space, we would expect 0-dimensional tangents).
This is joint work with Giovanni Alberti, David Bate and Andrea Marchese.
