Geodesic flows, interval maps, and symbolic dynamics

The Dynamics of Geodesic Flows*

Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits

0. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I. Renewal theorems in symbolic dynamics . . . . . . . . . . . . 1. Background: Shifts, suspension flows, thermodynamic formalism . 2. 3. 4. 5. 6. 7. 1 5 5 Renewal measures and renewal theorems . . . . . . . . . . . . . . 7 A modification for finite sequences . . . . . . . . . . . . . . . . . 10 Equidistribution theo...

Symbolic extensions of smooth interval maps

In this course we will present the full proof of the fact that every smooth dynamical system on the interval or circle X, constituted by the forward iterates of a function f : X → X which is of class Cr with r > 1, admits a symbolic extension, i.e., there exists a bilateral subshift (Y, S) with Y a closed shift-invariant subset of Λ, where Λ is a finite alphabet, and a continuous surjection π :...

Geodesic Flows with Positive Topological Entropy, Twist Maps and Hyperbolicity

We prove a perturbation lemma for the derivative of geodesic flows in high dimension. This implies that a C generic riemannian metric has a non-trivial hyperbolic basic set in its geodesic flow.

Symbolic dynamics for Lozi maps

We study the family of the Lozi maps La,b : R2 → R2, La,b(x, y) = (1+y−a|x|, bx), and their strange attractors Λa,b. We introduce the set of kneading sequences for the Lozi map and prove that it determines the symbolic dynamics for that map. We also introduce two other equivalent approaches. 2010 Mathematics Subject Classification: 37B10, 37D45, 37E30, 54H20