Singular SRB measures for a non 1--1 map of the unit square

Abstract

We consider a map of the unit square which is not 1--1, such as the memory map studied in \cite{MwM1}. Memory maps are defined as follows: $x_{n+1}=M_{\alpha }(x_{n-1},x_n)=\tau (\alpha \cdot x_n+(1-\alpha )\cdot x_{n-1}),$ where $\tau$ is a one-dimensional map on $I=[0,1]$ and $0<\alpha <1$ determines how much memory is being used. In this paper we let $\tau$ to be the symmetric tent map. To study the dynamics of $M_{\alpha }$, we consider the two-dimensional map $$G_{\alpha }:[x_{n-1},x_n]\mapsto [x_n,\tau (\alpha \cdot x_n+(1-\alpha )\cdot x_{n-1})].$$ The map $G_{\alpha }$ for $\alpha \in (0,3/4]$ was studied in \cite{MwM1}. In this paper we prove that for $\alpha \in (3/4,1)$ the map $G_{\alpha }$ admits a singular Sinai-Ruelle-Bowen measure. We do this by applying Rychlik's results for the Lozi map. However, unlike the Lozi map, the maps $G_{\alpha }$ are not invertible which creates complications that we are able to overcome.