## Abstract

Let μ be a probability measure on [0, 1] which is invariant and ergodic for T_{a}(x) = ax mod 1, and 0 < dim μ < 1. Let f be a local diffeomorphism on some open set. We show that if E R and (f μ)|E ~ μ|E, then f'(x) ∈ {±a^{r} : r ∈ Q} at μ-a.e. point x ∈ f^{-1}E. 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 another integer and a, b are not powers of a common integer, and if ν is a T_{b}- invariant measure, then fμ ν for all local diffeomorphisms f of class C^{2}. This generalizes the Rudolph-Johnson Theorem and shows that measure rigidity of T_{a}, T_{b} is a property not of the structure of the abelian action, but rather of their smooth conjugacy classes: if U, V are maps of R/Z which are C^{2}-conjugate to T_{a}, Tb then they have no common measure of positive dimension that is ergodic for both.

Original language | American English |
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Pages (from-to) | 1539-1563 |

Number of pages | 25 |

Journal | Journal of the European Mathematical Society |

Volume | 14 |

Issue number | 5 |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

## Keywords

- Fractal geometry
- Geometric measure theory
- Interval map
- Invariant measure
- Measure rigidity
- Scenery flow