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.