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: $x_{n+1}=M_α(x_{n-1},x_{n})=τ(α\cdot x_{n}+(1-α)\cdot x_{n-1}),$ where $τ$ is a one-dimensional map on $I=[0,1]$ and $0<α<1$ determines how much memory is being used. In this paper we let $τ$ to be the symmetric tent map. To study the dynamics of $M_α$, we consider the two-dimensional map $$ G_{α}:[x_{n-1},x_{n}]\mapsto [x_{n},τ(α\cdot x_{n}+(1-α)\cdot x_{n-1})]\, .$$ The map $G_α$ for $α\in(0,3/4]$ was studied in \cite{MwM1}. In this paper we prove that for $α\in(3/4,1)$ the map $G_α$ 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 $G_α$ are not invertible which creates complications that we are able to overcome.