In 1979, to each of a wide variety of groups, Rufus Bowen and Caroline Series constructed an associated function on the unit circle. Each Bowen – Series function is “expanding", "Markov” and satisfies “Renyi’s condition”. The function thus has a unique ergodic invariant measure equivalent to Lebesgue measure. After sketching the necessary background, we will define a Markov function family S_\alpha using a Bowen-Series function associated with any cocompact hyperbolic triangle group. Lastly, we will sketch the proof that S_\alpha and the Bowen-Series function have the same measure theoretic entropy.