Ergodic theorem for a Loeb space and hyperfinite approximations of dynamical systems
Although the G.Birkhoff Ergodic Theorem (BET) is trivial for finite spaces, this does not help in proving it for hyperfinite Loeb spaces. The proof of the BET for this case, suggested by T. Kamae, works, actually, for arbitrary probability spaces, as it was shown by Y. Katznelson and B. Weiss. In th...
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| creator | Glebsky, L. Yu Gordon, E. I Henson, C. W |
| description | Although the G.Birkhoff Ergodic Theorem (BET) is trivial for finite spaces,
this does not help in proving it for hyperfinite Loeb spaces. The proof of the
BET for this case, suggested by T. Kamae, works, actually, for arbitrary
probability spaces, as it was shown by Y. Katznelson and B. Weiss.
In this paper we discuss the reason why the usual approach, based on transfer
of some simple facts about arbitrary large finite spaces on infinite spaces
using nonstandard analysis technique, does not work for the BET. We show that
the the BET for hyperfinite spaces may be interpreted as some qualitative
result for very big finite spaces. We introduce the notion of a hyperfinite
approximation of a dynamical system and prove the existence of such an
approximation. The standard versions of the results obtained in terms of
sequences of finite dynamical systems are formulated. |
| doi_str_mv | 10.48550/arxiv.1104.0237 |
| format | Article |
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this does not help in proving it for hyperfinite Loeb spaces. The proof of the
BET for this case, suggested by T. Kamae, works, actually, for arbitrary
probability spaces, as it was shown by Y. Katznelson and B. Weiss.
In this paper we discuss the reason why the usual approach, based on transfer
of some simple facts about arbitrary large finite spaces on infinite spaces
using nonstandard analysis technique, does not work for the BET. We show that
the the BET for hyperfinite spaces may be interpreted as some qualitative
result for very big finite spaces. We introduce the notion of a hyperfinite
approximation of a dynamical system and prove the existence of such an
approximation. The standard versions of the results obtained in terms of
sequences of finite dynamical systems are formulated.</description><identifier>DOI: 10.48550/arxiv.1104.0237</identifier><language>eng</language><subject>Mathematics - Classical Analysis and ODEs ; Mathematics - Dynamical Systems ; Mathematics - Logic</subject><creationdate>2011-04</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1104.0237$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1104.0237$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Glebsky, L. Yu</creatorcontrib><creatorcontrib>Gordon, E. I</creatorcontrib><creatorcontrib>Henson, C. W</creatorcontrib><title>Ergodic theorem for a Loeb space and hyperfinite approximations of dynamical systems</title><description>Although the G.Birkhoff Ergodic Theorem (BET) is trivial for finite spaces,
this does not help in proving it for hyperfinite Loeb spaces. The proof of the
BET for this case, suggested by T. Kamae, works, actually, for arbitrary
probability spaces, as it was shown by Y. Katznelson and B. Weiss.
In this paper we discuss the reason why the usual approach, based on transfer
of some simple facts about arbitrary large finite spaces on infinite spaces
using nonstandard analysis technique, does not work for the BET. We show that
the the BET for hyperfinite spaces may be interpreted as some qualitative
result for very big finite spaces. We introduce the notion of a hyperfinite
approximation of a dynamical system and prove the existence of such an
approximation. The standard versions of the results obtained in terms of
sequences of finite dynamical systems are formulated.</description><subject>Mathematics - Classical Analysis and ODEs</subject><subject>Mathematics - Dynamical Systems</subject><subject>Mathematics - Logic</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotz71qwzAUBWAtHUravVPQC9iVdGUrHkNIf8DQxbu5sq4SQWwZyZT47Zu0nQ5nOZyPsRcpSr2rKvGK6Rq-SymFLoUC88i6YzpFFwa-nCkmGrmPiSNvI1meZxyI4-T4eZ0p-TCF5dbnOcVrGHEJcco8eu7WCccw4IXnNS805if24PGS6fk_N6x7O3aHj6L9ev887NsC68oUSliUogIYhG5qV5MgDZUyGgBUbb2TBGYAo7xvUGpntWp21ggjvdEkETZs-zf7q-rndDuV1v6u6-86-AEjDkpL</recordid><startdate>20110401</startdate><enddate>20110401</enddate><creator>Glebsky, L. Yu</creator><creator>Gordon, E. I</creator><creator>Henson, C. W</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20110401</creationdate><title>Ergodic theorem for a Loeb space and hyperfinite approximations of dynamical systems</title><author>Glebsky, L. Yu ; Gordon, E. I ; Henson, C. W</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a657-20ba10533c0496d6e0e43527433326bfd1e37c372ff9a14db4298b7071f74e1a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Mathematics - Classical Analysis and ODEs</topic><topic>Mathematics - Dynamical Systems</topic><topic>Mathematics - Logic</topic><toplevel>online_resources</toplevel><creatorcontrib>Glebsky, L. Yu</creatorcontrib><creatorcontrib>Gordon, E. I</creatorcontrib><creatorcontrib>Henson, C. W</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Glebsky, L. Yu</au><au>Gordon, E. I</au><au>Henson, C. W</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Ergodic theorem for a Loeb space and hyperfinite approximations of dynamical systems</atitle><date>2011-04-01</date><risdate>2011</risdate><abstract>Although the G.Birkhoff Ergodic Theorem (BET) is trivial for finite spaces,
this does not help in proving it for hyperfinite Loeb spaces. The proof of the
BET for this case, suggested by T. Kamae, works, actually, for arbitrary
probability spaces, as it was shown by Y. Katznelson and B. Weiss.
In this paper we discuss the reason why the usual approach, based on transfer
of some simple facts about arbitrary large finite spaces on infinite spaces
using nonstandard analysis technique, does not work for the BET. We show that
the the BET for hyperfinite spaces may be interpreted as some qualitative
result for very big finite spaces. We introduce the notion of a hyperfinite
approximation of a dynamical system and prove the existence of such an
approximation. The standard versions of the results obtained in terms of
sequences of finite dynamical systems are formulated.</abstract><doi>10.48550/arxiv.1104.0237</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Classical Analysis and ODEs Mathematics - Dynamical Systems Mathematics - Logic |
| title | Ergodic theorem for a Loeb space and hyperfinite approximations of dynamical systems |
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