Finitely-additive measures on the asymptotic foliations of a Markov compactum

Let ρ ≥ 2 be an integer, let M be a compact orientable surface of genus ρ, and let ω be a holomorphic one-form on M . Denote by m = (ω ∧ ω)/2i the area form induced by ω and assume that m(M) = 1. Let ht be the vertical flow on M (i.e., the flow corresponding to R(ω)); let h−t be the horizontal flow on M (i.e., the flow corresponding to I(ω)). The flows ht , h − t preserve the area m and are uniquely ergodic. Take x ∈ M , t1, t2 ∈ R+ and assume that the closure of the set