Symbolic Dynamics for Hyperbolic Systems

◦ Geodesic flows on compact manifolds with negative sectional curvature. Later, we relax uniform hyperbolicity to an asymptotic one, called non-uniform hyperbolicity. Two examples of such systems are: ◦ Slow down of fA : T → T, see [9]. ◦ Geodesic flows on surfaces with nonpositive curvature. Introductory example: Smale’s horseshoe [21]. Let g : K → K be Smale’s horseshoe map, and σ : Σ→ Σ the full shift, Σ = {0, 1}Z. Although g and σ seem different, they are the same: ∃ bijection π : Σ→ K s.t. π ◦ σ = g ◦ π. The map σ is much easier to understand: (1) Easy iteration. (2) Counting of periodic orbits. (3) Invariant measures. The pair of maps σ : Σ→ Σ and π : Σ→ K is called a symbolic model for g.