Ergodicity: from another point of view

Aliasghar Sarizadeh

Published 2015-10-29Version 1

In the present work, we study iterated function systems (IFSs) and their induced IFSs. We introduce a weaker concept of minimality for induced IFSs so that this property together whit $C^1$-regularity of generators implies that the Lebesgue measure is ergodic for our original IFS. As a consequence of mentioned results, we obtain $C^1$-ergodicity for some cascade systems. Moreover, we also obtain that the Lebesgue measure is ergodic for a dense subset of the space $\overline{\mathcal{O}}^\infty(\mathbb{T}^2)$ where $\overline{\mathcal{O}}^\infty(\mathbb{T}^2)$ is the $C^\infty$-closure of the conjugancy class of translations of $\mathbb{T}^2$.