单位正方形非1- 1映射的奇异SRB测度
Singular SRB measures for a non 1--1 map of the unit square
摘要：
我们考虑一个非1- 1的单位正方形的映射，例如在\cite{MwM1}中研究的记忆映射。内存映射定义如下：$x_{n+1}=M_α(x_{n-1},x_{n})=τ(α\cdot x_{n}+(1-α)\cdot x_{n-1}),$其中$τ$是$I=[0,1]$上的一维映射，而$0<α<1$确定使用了多少内存。本文用$τ$作为对称帐篷映射。为了研究$M_α$的动力学，我们考虑二维映射$$ G_{α}:[x_{n-1},x_{n}]\mapsto [x_{n},τ(α\cdot x_{n}+(1-α)\cdot x_{n-1})]\, .$$。在\cite{MwM1}中研究了$α\in(0,3/4]$的映射$G_α$。本文证明了对于$α\in(3/4,1)$，映射$G_α$允许奇异Sinai-Ruelle-Bowen测度.我们通过将Rychlik的结果应用于Lozi映射来做到这一点。然而，与Lozi映射不同，映射$G_α$不是可逆的，这产生了我们能够克服的复杂性。
英文摘要：
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.
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