Sets of Non-differentiability for Conjugacies between Expanding Interval Maps

Metric stability for random walks with applications in renormalization theory

Consider deterministic random walks F : I × Z → I × Z, defined by F (x, n) = (f(x), ψ(x) + n), where f is an expanding Markov map on the interval I and ψ : I → Z. We study the universality (stability) of ergodic (for instance, recurrence and transience), geometric and multifractal properties in the class of perturbations of the type F̃ (x, n) = (fn(x), ψ̃(x, n) +n) which are topologically conjuga...

Statistical Properties of Nonequilibrium Dynamical Systems

We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infinite SRB measure. In particular, we obtain a linear response formula for both finite and infinite SRB measures. We apply our results to interval maps with a neutral fixed p...

Explosion of smoothness for conjugacies between multimodal maps

Let f and g be smooth multimodal maps with no periodic attractors and no neutral points. If a topological conjugacy h between f and g is C at a point in the nearby expanding set of f , then h is a smooth diffeomorphism in the basin of attraction of a renormalization interval of f . In particular, if f : I → I and g : J → J are Cr unimodal maps and h is C at a boundary of I then h is Cr in I.

Quasi-symmetry of Conjugacies between Interval Maps

Smooth Cocycle Rigidity for Expanding Maps, and an Application to Mostow Rigidity

We consider the Livsic cocycle equation, with values in compact Lie groups, and dynamics given by a piecewise-smooth expanding Markov map. We prove that any measurable solution to this equation must necessarily have a smooth version. We deduce a smooth rigidity theorem for conjugacies between piecewise-conformal expanding Markov maps, and apply this to give a variation on the proof of Mostow's ...