Geodesic Flow on the Quotient Space of the Action of ⟨z + 2;−1/z 〉 on the Upper Half Plane

Let G be the group generated by z ↦ z+2 and z → -1/z , z ∈ ℂ. This group acts on the upper half plane and the associated quotient surface is topologically a sphere with two cusps. Assigning a “geometric” code to an oriented geodesic not going to cusps, with alphabets in ℤ \ {0}, enables us to conjugate the geodesic ow on this surface to a special ow over the symbolic space of these geometric codes. We will show that for k ≥ 1, a subsystem with codes from ℤ \ {0; ±1; ±2;… ; ±k} is a TBS: topologically Bernouli scheme. For similar codes for geodesic ow on modular surface, this was true for k ≥ 3. We also give bounds for the entropy of these subsystems.