Artin-Mazur Zeta Functions of Generalized Beta-Transformations

Abstract

Abstract. In this paper, we study the Artin-Mazur zeta function of a generalization of the well-known β-transformation introduced by Góra [Invariant densities for generalized β-maps. Ergodic Theory Dynam. Systems 27 (2007), [1583][1584][1585][1586][1587][1588][1589][1590][1591][1592][1593][1594][1595][1596][1597][1598]. We show that the ArtinMazur zeta function can be extended to a meromorphic function via an expansion of 1 defined by using the transformation. As an application, we relate its analytic properties to the algebraic properties of β.

Density of periodic measures and large deviation principle for generalised mod one transformations

“…This family has recently attracted attention in the setting beyond specification, since T β does not satisfy the specification property for Lebesgue almost parameter β > 1 [2,26]. After that, various generalisations of T β has been considered by many authors: (α, β)-transformations [4,5,9,11,24], (−β)-transformations [18][19][20]27], generalised β-transformations [9,10,29,32] (see also figure 1).…”

Density of periodic measures and large deviation principle for generalized $(α, β)$-transformations

“…This family has recently attracted attention in the setting beyond specification, since T β does not satisfy the specification property for Lebesgue almost parameter β > 1 ( [2,22]). Inspired by β-transformations, many authors have considered various generalizations of T β : (α, β)-transformations ( [4,5,7,9,20]), (−β)-transformations ( [15,16,17,23]), generalized βtransformations ( [7,8,24,26]).…”