The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there are interactions exhibiting a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant changes in the low-temperature behavior of its Gibbs measures. For some one-dimensional XY models we exhibit sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit sensitive dependence of Gibbs measures for an interaction on a classical lattice system with finite-state space. This interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we solve some problems stated by Chazottes and Hochman.

Abstract. We give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function. This implies that several dimension spectra and large deviation rate functions associated to this map are not (expected to be) real analytic, in contrast to the uniformly hyperbolic case. The quadratic map we study has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.

We show that every linearly repetitive Delone set in the Euclidean d-space R d , with d ≥ 2, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice Z d . In the particular case when the Delone set X in R d comes from a primitive substitution tiling of R d , we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from X to the lattice lattice βZ d for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.

This paper concerns self-similar tilings of the Euclidean plane. We consider the number of occurrences of a given tile in any domain bounded by a Jordan curve. For a large class of self-similar tilings, including many well-known examples, we give estimates of the oscillation of this number of occurrences around its average frequency times the total number of tiles in the domain, which depend only on the Jordan curve.Date: November 12, 2018.