Limit Theorems for Fast-slow partially hyperbolic systems

We prove several limit theorems for a simple class of partially hyperbolic fast-slow systems. We start with some well know results on averaging, then we give a substantial refinement of known large (and moderate) deviation results and conclude with a completely new result (a local limit theorem) on the distribution of the process determined by the fluctuations around the average. The method of proof is based on a mixture of standard pairs and Transfer Operators that we expect to be applicable in a much wider generality

Potts models on hierarchical lattices and Renormalization Group dynamics

We prove that the generator of the renormalization group of Potts models on hierarchical lattices can be represented by a rational map acting on a finite-dimensional product of complex projective spaces. In this framework we can also consider models with an applied external magnetic field and multiple-spin interactions. We use recent results regarding iteration of rational maps in several complex variables to show that, for some class of hierarchical lattices, Lee-Yang and Fisher zeros belong to the unstable set of the renormalization map.Comment: 21 pages, 7 figures; v3 revised, some issues correcte

Statistical properties of mostly contracting fast-slow partially hyperbolic systems

We consider a class of $\mathcal C^{4}$ partially hyperbolic systems on $\mathbb T^2$ described by maps $F_\varepsilon(x,\theta)=(f(x,\theta),\theta+\varepsilon\omega(x,\theta))$ where $f(\cdot,\theta)$ are expanding maps of the circle. For sufficiently small $\varepsilon$ and $\omega$ generic in an open set, we precisely classify the SRB measures for $F_\varepsilon$ and their statistical properties, including exponential decay of correlation for H\"older observables with explicit and nearly optimal bounds on the decay rate

Abundance of escaping orbitsin a family of anti-integrable limitsof the standard map

We give quantitative results about the abundance of escaping orbits in a family of exact twist maps preserving Lebesgue measure on the cylinder T × R; geometrical features of maps of this family are quite similar to those of the well-known Chirikov-Taylor standard map, and in fact we believe that the techniques presented in this work can be further improved and eventually applied to studying ergodic properties of the standard map itself. We state conditions which assure that escaping orbits exist and form a full Hausdorff dimension set. Moreover, under stronger conditions we can prove that such orbits are not charged by the invariant measure. We also obtain prove that, generically, the system presents elliptic islands at arbitrarily high values of the action variable and provide estimates for their total measure

Marked Length Spectral determination of analytic chaotic billiards with axial symmetries

We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table.Comment: 57 pages, 8 figure