Markov measures and extended zeta functions

We find new representations of $q$-families of algebras. These algebras are built on generators and relations. They are C*-algebras and their representations are a part of non-commutative harmonic analysis. Our approach is amenable to applications in problems from dynamics and mathematical physics: We introduce a deformation parameter $q$, and an associated family of $q$-relations where the number $q$ is a "quantum-deformation," and also a parameter in a scale of (Riemann-Ruelle) zeta functions. Our representations are used in turn in a derivation of formulas for this $q$-zeta function.