Generalized iterated function systems (GIFSs for short) of some order $m \in \mathbb{N}$ were introduced in 2008 by Miculescu and Mihail and represent extensions of classical iterated function systems (IFSs for short), in the sense that, instead of considering finite families of contractions that are selfmaps of a complete metric space $X$, one takes the domain to be a Cartesian product space $X^m$, endowed with the maximum metric. The theory of IFSs was broadened further in 2014 by Secelean and also by Maślanka and Strobin in 2018, when they considered families of maps defined on the space of all bounded sequences of elements from $X$ and called them GIFSs of infinite order. They proved that many classical results concerning GIFSs of finite order, such as the existence of the attractor and the coding map, have analogues in this new frame.

In this talk, we see how one can define the Markov operator for a GIFS of infinite order and we establish the existence and uniqueness of the invariant measure, a fact which is one of the open questions posed by Strobin in [Qual. Theory Dyn. Syst. \textbf{19}, 85 (2020)], where he proved the existence and uniqueness of invariant measures for GIFSs of finite order. Moreover, we show that the invariant measure for a GIFS of infinite order $\mathcal{F}$ can be realized as the limit of the sequence of invariant measures of certain GIFSs of finite order, that are naturally linked with $\mathcal{F}$.