Singular SRB Measures for a Non 1-1 Map of the Unit Square.
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| Title: | Singular SRB Measures for a Non 1-1 Map of the Unit Square. |
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| Authors: | Góra, Paweł1 pawel.gora@concordia.ca, Boyarsky, Abraham1 abraham.boyarsky@concordia.ca, Li, Zhenyang2 zhenyangemail@gmail.com |
| Source: | Journal of Statistical Physics. Oct2016, Vol. 165 Issue 2, p409-433. 25p. |
| Subjects: | Piecewise linear approximation, Memory maps (Computer science), Ruelle operators, Dynamic storage allocation (Computer science), Maps |
| Abstract: | We consider a map of the unit square which is not 1-1, such as the memory map studied in Góra (Statistical and deterministic dynamics of maps with memory, ). 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 The map $$G_\alpha $$ for $$\alpha \in (0,3/4]$$ was studied in Góra (Statistical and deterministic dynamics of maps with memory, ). 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. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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