GEOMETRIC RIGIDITY OF ×m INVARIANT MEASURES

@inproceedings{Hochman2010GEOMETRICRO,
  title={GEOMETRIC RIGIDITY OF ×m INVARIANT MEASURES},
  author={Michael Hochman},
  year={2010}
}
Let μ be a probability measure on [0, 1] which is invariant and ergodic for Ta(x) = ax mod 1, and 0 < dimμ < 1. Let f be a local diffeomorphism on some open set. We show that if E ⊆ R and (fμ)|E ∼ μ|E , then f (x) ∈ {±a : r ∈ Q} at μ-a.e. point x ∈ fE. In particular, if g is a piecewise-analytic map preserving μ then there is an open g-invariant set U containing supp μ such that g|U is piecewise-linear with slopes which are rational powers of a. In a similar vein, for μ as above, if b is… CONTINUE READING

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