Symbolic dynamics for the geodesic flow on Hecke surfaces

Abstract

In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross section for which the first return map factors through a simple (explicit) map given in terms of the generating map of a particular continued fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times.

Nearest 𝜆_{𝑞}-multiple fractions

“…We also write x y for x ≺ y or x = y. This is indeed an order on regular λ q -CF's, since Lemmas 22 and 23 in [14] imply: The authors of [14] introduce a process called "rewriting" of λ q -CF's where forbidden blocks in the λ q -CF are replaced by allowed ones without changing its value. The rules for "rewriting" are based on the interpretation of a λ q -CF in terms of Möbius transformations given by group elements of the Hecke group, see (2.2.4), and applying the group relations (2.1.3).…”

“…The rules for "rewriting" are based on the interpretation of a λ q -CF in terms of Möbius transformations given by group elements of the Hecke group, see (2.2.4), and applying the group relations (2.1.3). We refer in particular to Lemma 11 and Lemma 13 in [14] for the details. A simple example for this rewriting is used in the proof of Lemma 2.2.2.…”

“…Lemma 2.5.2 shows that g x can be written as an infinite λ q -CF with regular tail satisfying the assumptions of Lemma 2.5.3. Using the rewriting rules in Lemmas 11 and 13 of [14] we can recursively rewrite the λ q -CF of g x into a regular λ q -CF from the left to the right. We procede to the next forbidden block if rewriting does not lead to a new forbidden block.…”

“…The transfer operator for G q . The authors of [14] have constructed a Poincaré section Σ for the geodesic flow Φ t : S 1 G q \H → S 1 G q \H on the Hecke surfaces G q \H for which the Poincaré map P : Σ → Σ is basically given by the natural extension F q of the map f q : I q → I q . The periodic orbits of this geodesic flow can therefore be characterized by the periodic orbits of F q and therefore also by the periodic orbits of the map f q respectively its periodic points which determine the ones of F q uniquely.…”

Fine ergodic properties of partially hyperbolic dynamical systems

“…However, D. Ornstein and B. Weiss [14] proved that the geodesic flow in a negatively curved surface is Bernoulli and the technique they used was generalized in [43], and by Y. Pesin in [62]. Rudolph on extension of Bernoulli shifts ( [54], [53]) has been used by several authors (for example [13], [35], [49], [50], [17]) to obtain Bernoulli property for different types of dynamical systems using the Kolmogorov property. However, the method of Ornstein and Weiss does not apply when we have a center direction with nontrivial behavior as for diffeomorphisms in PH r m (T 3 ), r > 1.…”