Geometric rigidity of Xm invariant measures

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^-1E. 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 C². 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 C²-conjugate to T_a, Tb then they have no common measure of positive dimension that is ergodic for both.