A new class of positive recurrent functions

In [Sa] Sarig has introduced and explored the concept of positively recurrent functions. In this paper, using the concept of an iterated function system, we construct a natural wide class of positively recurrent functions and we show that they have stronger properties than the general functions considered in [Sa]. In some parts our exposition is similar and follows the approach developed in [MU1] and [Wa], where also the idea of embedding the infinite dimensional shift space into a compact metric space and the Shauder-Tichonov fixed-point theorem have been used. To begin with, let IN be the set of positive integers and let Σ = IN be the infinitely dimensional shift space equipped with the product topology. Let σ : Σ → Σ be the shift transformation (cutting out the first coordinate), σ({xn}n=1) = ({xn} ∞ n=2). Fix β > 0. If φ : Σ → IR and n ≥ 1, we set Vn(φ) = sup{|φ(x) − φ(y)| : x1 = y1, x2 = y2, . . . , xn = yn}.