Dynamics of geodesic flows on hyperbolic compact surfaces with some elliptic points

摘要

We extend the results in [R. Adler, L. Flatto, Geodesic flows, interval maps and symbolic dynamics, Bull Am Math Soc. 1975;25(42): 229-334.] to compact surfaces of genus greater than one and with some elliptic points. Similar to hyperbolic case, a map T_R conjugate to the cross-section map and defined on R, a union of finite rectangles in the plane, exists. This gives rise to a subshift of finite type and a sofic shift σ_T_R and their projections to axis known as Bowen-Series maps. Additionally, we will give a simple geometrical presentation of synchronizing and Markov words for σ_T_R , and will show that when considered as a sofic map, it is not of almost finite type. Moreover, if h is the entropy of any of those shift spaces, then ln (8g + 2r - 7) ≤ h ≤ ln (8g + 2r - 5), where g is the genus and r is the number of elliptic points.