From greedy to lazy expansions and their driving dynamics

In this paper we study the ergodic properties of non-greedy series expansions to non-integer bases β > 1. It is shown that the so-called ‘lazy’ expansion is isomorphic to the ‘greedy’ expansion. Furthermore, a class of expansions to base β > 1, β , ‘in between’ the lazy and the greedy expansions are introduced and studies. It is shown that these expansions are isomorphic to expansions of the form Tx = βx + α (mod 1).