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] |
| Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: Singular SRB Measures for a Non 1-1 Map of the Unit Square. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Góra%2C+Paweł%22">Góra, Paweł</searchLink><relatesTo>1</relatesTo><i> pawel.gora@concordia.ca</i><br /><searchLink fieldCode="AR" term="%22Boyarsky%2C+Abraham%22">Boyarsky, Abraham</searchLink><relatesTo>1</relatesTo><i> abraham.boyarsky@concordia.ca</i><br /><searchLink fieldCode="AR" term="%22Li%2C+Zhenyang%22">Li, Zhenyang</searchLink><relatesTo>2</relatesTo><i> zhenyangemail@gmail.com</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Statistical+Physics%22">Journal of Statistical Physics</searchLink>. Oct2016, Vol. 165 Issue 2, p409-433. 25p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Piecewise+linear+approximation%22">Piecewise linear approximation</searchLink><br /><searchLink fieldCode="DE" term="%22Memory+maps+%28Computer+science%29%22">Memory maps (Computer science)</searchLink><br /><searchLink fieldCode="DE" term="%22Ruelle+operators%22">Ruelle operators</searchLink><br /><searchLink fieldCode="DE" term="%22Dynamic+storage+allocation+%28Computer+science%29%22">Dynamic storage allocation (Computer science)</searchLink><br /><searchLink fieldCode="DE" term="%22Maps%22">Maps</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Statistical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10955-016-1620-y Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 25 StartPage: 409 Subjects: – SubjectFull: Piecewise linear approximation Type: general – SubjectFull: Memory maps (Computer science) Type: general – SubjectFull: Ruelle operators Type: general – SubjectFull: Dynamic storage allocation (Computer science) Type: general – SubjectFull: Maps Type: general Titles: – TitleFull: Singular SRB Measures for a Non 1-1 Map of the Unit Square. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Góra, Paweł – PersonEntity: Name: NameFull: Boyarsky, Abraham – PersonEntity: Name: NameFull: Li, Zhenyang IsPartOfRelationships: – BibEntity: Dates: – D: 15 M: 10 Text: Oct2016 Type: published Y: 2016 Identifiers: – Type: issn-print Value: 00224715 Numbering: – Type: volume Value: 165 – Type: issue Value: 2 Titles: – TitleFull: Journal of Statistical Physics Type: main |
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