Variational Principle for Nonhyperbolic Ergodic Measures: Skew Products and Elliptic Cocycles
For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems we consider are partially hyperbolic with one-dimensional central direction for which there are positive entropy ergodic measu...
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| Veröffentlicht in: | Communications in mathematical physics 2022-08, Vol.394 (1), p.73-141 |
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| creator | Díaz, L. J. Gelfert, K. Rams, M. |
| description | For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems we consider are partially hyperbolic with one-dimensional central direction for which there are positive entropy ergodic measures whose central Lyapunov exponent is negative, zero, or positive. We construct ergodic measures with zero central Lyapunov exponent whose entropy is positive and arbitrarily close to the topological entropy of the set of points with central Lyapunov exponent zero. This provides a restricted variational principle for nonhyperbolic (zero exponent) ergodic measures. The result is applied to the setting of
SL
(
2
,
R
)
matrix cocycles and provides a counterpart to Furstenberg’s classical result: for an open and dense subset of elliptic
SL
(
2
,
R
)
cocycles we construct ergodic measures with upper Lyapunov exponent zero and with metric entropy arbitrarily close to the topological entropy of the set of infinite matrix products with subexponential growth of the norm. |
| doi_str_mv | 10.1007/s00220-022-04406-w |
| format | Article |
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SL
(
2
,
R
)
matrix cocycles and provides a counterpart to Furstenberg’s classical result: for an open and dense subset of elliptic
SL
(
2
,
R
)
cocycles we construct ergodic measures with upper Lyapunov exponent zero and with metric entropy arbitrarily close to the topological entropy of the set of infinite matrix products with subexponential growth of the norm.</description><identifier>ISSN: 0010-3616</identifier><identifier>EISSN: 1432-0916</identifier><identifier>DOI: 10.1007/s00220-022-04406-w</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Chaos theory ; Classical and Quantum Gravitation ; Complex Systems ; Entropy ; Ergodic processes ; Hyperbolic systems ; Liapunov exponents ; Mathematical and Computational Physics ; Mathematical Physics ; Physics ; Physics and Astronomy ; Principles ; Quantum Physics ; Relativity Theory ; Theoretical ; Topology</subject><ispartof>Communications in mathematical physics, 2022-08, Vol.394 (1), p.73-141</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022</rights><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-3a20639d25cdd5adb14d61b758a9ba00bf0a4e3b8017b2b01fda035e127acdfb3</citedby><cites>FETCH-LOGICAL-c319t-3a20639d25cdd5adb14d61b758a9ba00bf0a4e3b8017b2b01fda035e127acdfb3</cites><orcidid>0000-0002-5123-4611</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00220-022-04406-w$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00220-022-04406-w$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Díaz, L. J.</creatorcontrib><creatorcontrib>Gelfert, K.</creatorcontrib><creatorcontrib>Rams, M.</creatorcontrib><title>Variational Principle for Nonhyperbolic Ergodic Measures: Skew Products and Elliptic Cocycles</title><title>Communications in mathematical physics</title><addtitle>Commun. Math. Phys</addtitle><description>For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems we consider are partially hyperbolic with one-dimensional central direction for which there are positive entropy ergodic measures whose central Lyapunov exponent is negative, zero, or positive. We construct ergodic measures with zero central Lyapunov exponent whose entropy is positive and arbitrarily close to the topological entropy of the set of points with central Lyapunov exponent zero. This provides a restricted variational principle for nonhyperbolic (zero exponent) ergodic measures. The result is applied to the setting of
SL
(
2
,
R
)
matrix cocycles and provides a counterpart to Furstenberg’s classical result: for an open and dense subset of elliptic
SL
(
2
,
R
)
cocycles we construct ergodic measures with upper Lyapunov exponent zero and with metric entropy arbitrarily close to the topological entropy of the set of infinite matrix products with subexponential growth of the norm.</description><subject>Chaos theory</subject><subject>Classical and Quantum Gravitation</subject><subject>Complex Systems</subject><subject>Entropy</subject><subject>Ergodic processes</subject><subject>Hyperbolic systems</subject><subject>Liapunov exponents</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Principles</subject><subject>Quantum Physics</subject><subject>Relativity Theory</subject><subject>Theoretical</subject><subject>Topology</subject><issn>0010-3616</issn><issn>1432-0916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kEtLxDAUhYMoOI7-AVcF19GbpE2n7mQYHzA-wMdOQl4dM9amJi3D_HujFdy5OYcL5ztwD0LHBE4JQHkWASgFnARDngPHmx00ITlLZ0X4LpoAEMCME76PDmJcA0BFOZ-g1xcZnOydb2WTPQTXatc1Nqt9yO58-7btbFC-cTpbhJU3yW-tjEOw8Tx7fLebhHgz6D5msjXZomlc16fQ3Outbmw8RHu1bKI9-vUper5cPM2v8fL-6mZ-scSakarHTFLgrDK00MYU0iiSG05UWcxkpSSAqkHmlqkZkFJRBaQ2ElhhCS2lNrViU3Qy9nbBfw429mLth5BeioLyipaM0RlNKTqmdPAxBluLLrgPGbaCgPieUYwziiTiZ0axSRAboZjC7cqGv-p_qC_Mq3cB</recordid><startdate>20220801</startdate><enddate>20220801</enddate><creator>Díaz, L. J.</creator><creator>Gelfert, K.</creator><creator>Rams, M.</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-5123-4611</orcidid></search><sort><creationdate>20220801</creationdate><title>Variational Principle for Nonhyperbolic Ergodic Measures: Skew Products and Elliptic Cocycles</title><author>Díaz, L. J. ; Gelfert, K. ; Rams, M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-3a20639d25cdd5adb14d61b758a9ba00bf0a4e3b8017b2b01fda035e127acdfb3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Chaos theory</topic><topic>Classical and Quantum Gravitation</topic><topic>Complex Systems</topic><topic>Entropy</topic><topic>Ergodic processes</topic><topic>Hyperbolic systems</topic><topic>Liapunov exponents</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Principles</topic><topic>Quantum Physics</topic><topic>Relativity Theory</topic><topic>Theoretical</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Díaz, L. J.</creatorcontrib><creatorcontrib>Gelfert, K.</creatorcontrib><creatorcontrib>Rams, M.</creatorcontrib><collection>CrossRef</collection><jtitle>Communications in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Díaz, L. J.</au><au>Gelfert, K.</au><au>Rams, M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Variational Principle for Nonhyperbolic Ergodic Measures: Skew Products and Elliptic Cocycles</atitle><jtitle>Communications in mathematical physics</jtitle><stitle>Commun. Math. Phys</stitle><date>2022-08-01</date><risdate>2022</risdate><volume>394</volume><issue>1</issue><spage>73</spage><epage>141</epage><pages>73-141</pages><issn>0010-3616</issn><eissn>1432-0916</eissn><abstract>For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems we consider are partially hyperbolic with one-dimensional central direction for which there are positive entropy ergodic measures whose central Lyapunov exponent is negative, zero, or positive. We construct ergodic measures with zero central Lyapunov exponent whose entropy is positive and arbitrarily close to the topological entropy of the set of points with central Lyapunov exponent zero. This provides a restricted variational principle for nonhyperbolic (zero exponent) ergodic measures. The result is applied to the setting of
SL
(
2
,
R
)
matrix cocycles and provides a counterpart to Furstenberg’s classical result: for an open and dense subset of elliptic
SL
(
2
,
R
)
cocycles we construct ergodic measures with upper Lyapunov exponent zero and with metric entropy arbitrarily close to the topological entropy of the set of infinite matrix products with subexponential growth of the norm.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00220-022-04406-w</doi><tpages>69</tpages><orcidid>https://orcid.org/0000-0002-5123-4611</orcidid></addata></record> |
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| language | eng |
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| subjects | Chaos theory Classical and Quantum Gravitation Complex Systems Entropy Ergodic processes Hyperbolic systems Liapunov exponents Mathematical and Computational Physics Mathematical Physics Physics Physics and Astronomy Principles Quantum Physics Relativity Theory Theoretical Topology |
| title | Variational Principle for Nonhyperbolic Ergodic Measures: Skew Products and Elliptic Cocycles |
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