A few problems connected with invariantmeasures of Markov maps : verication of some claims and opinions that circulate inthe literature

2020
journal article
article
cris.lastimport.wos2024-04-10T00:05:49Z
dc.abstract.enWe begin with the celebrated result of [3]. The authors were well aware that their result cannot be extended to expanding transformations with countably many one-to-one pieces in a simple way (see Th. 2, and the comment below on Cond. (17) there). The real task in that period of time was to find reasonable additional conditions which would guarantee the existence of density invariant under the action of expanding map with countably many one-to-one pieces. Several attempts was made to accomplish that task (for more details see e.g. a review article [4], and also [5], or [6], Sect. 6). One of the mentioned attempts was published in [7], as Adler’s Theorem. Since no proof was given there, the question arose whether it is true [8]. A solution was published in [1], and [2]. After the above two notes and a few other ones, related with them, were published, some further claims and opinions concerning the existence of invariant densities and their lower and upper bounds for Markov Maps appear in the literature. Those claims and opinions reveal that their authors were unacquainted with the essence of the problem. That problem is rather of delicate nature. It involves, among other things, the so-called measure-theoretic recurrence property. In this note we clear up, in a systematic way, the essence of the problems with the aid of examples, comments and some published results.pl
dc.affiliationWydział Matematyki i Informatykipl
dc.contributor.authorBugiel, Piotr - 127471 pl
dc.contributor.authorWędrychowicz, Stanisławpl
dc.contributor.authorRzepka, Beatapl
dc.date.accessioned2020-05-25T21:09:10Z
dc.date.available2020-05-25T21:09:10Z
dc.date.issued2020pl
dc.date.openaccess0
dc.description.accesstimew momencie opublikowania
dc.description.number1pl
dc.description.physical1607-1616pl
dc.description.versionostateczna wersja wydawcy
dc.description.volume9pl
dc.identifier.doi10.1515/anona-2020-0221pl
dc.identifier.eissn2191-950Xpl
dc.identifier.issn2191-9496pl
dc.identifier.projectROD UJ / OPpl
dc.identifier.urihttps://ruj.uj.edu.pl/xmlui/handle/item/156519
dc.languageengpl
dc.language.containerengpl
dc.rightsUdzielam licencji. Uznanie autorstwa 4.0 Międzynarodowa*
dc.rights.licenceCC-BY
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/legalcode.pl*
dc.share.typeotwarte czasopismo
dc.subject.enMarkov mapspl
dc.subject.eninvariant measurepl
dc.subject.enexpanding transformationpl
dc.subject.enrecurrencepl
dc.subject.enaperiodicitypl
dc.subtypeArticlepl
dc.titleA few problems connected with invariantmeasures of Markov maps : verication of some claims and opinions that circulate inthe literaturepl
dc.title.journalAdvances in Nonlinear Analysispl
dc.typeJournalArticlepl
dspace.entity.typePublication
cris.lastimport.wos
2024-04-10T00:05:49Z
dc.abstract.enpl
We begin with the celebrated result of [3]. The authors were well aware that their result cannot be extended to expanding transformations with countably many one-to-one pieces in a simple way (see Th. 2, and the comment below on Cond. (17) there). The real task in that period of time was to find reasonable additional conditions which would guarantee the existence of density invariant under the action of expanding map with countably many one-to-one pieces. Several attempts was made to accomplish that task (for more details see e.g. a review article [4], and also [5], or [6], Sect. 6). One of the mentioned attempts was published in [7], as Adler’s Theorem. Since no proof was given there, the question arose whether it is true [8]. A solution was published in [1], and [2]. After the above two notes and a few other ones, related with them, were published, some further claims and opinions concerning the existence of invariant densities and their lower and upper bounds for Markov Maps appear in the literature. Those claims and opinions reveal that their authors were unacquainted with the essence of the problem. That problem is rather of delicate nature. It involves, among other things, the so-called measure-theoretic recurrence property. In this note we clear up, in a systematic way, the essence of the problems with the aid of examples, comments and some published results.
dc.affiliationpl
Wydział Matematyki i Informatyki
dc.contributor.authorpl
Bugiel, Piotr - 127471
dc.contributor.authorpl
Wędrychowicz, Stanisław
dc.contributor.authorpl
Rzepka, Beata
dc.date.accessioned
2020-05-25T21:09:10Z
dc.date.available
2020-05-25T21:09:10Z
dc.date.issuedpl
2020
dc.date.openaccess
0
dc.description.accesstime
w momencie opublikowania
dc.description.numberpl
1
dc.description.physicalpl
1607-1616
dc.description.version
ostateczna wersja wydawcy
dc.description.volumepl
9
dc.identifier.doipl
10.1515/anona-2020-0221
dc.identifier.eissnpl
2191-950X
dc.identifier.issnpl
2191-9496
dc.identifier.projectpl
ROD UJ / OP
dc.identifier.uri
https://ruj.uj.edu.pl/xmlui/handle/item/156519
dc.languagepl
eng
dc.language.containerpl
eng
dc.rights*
Udzielam licencji. Uznanie autorstwa 4.0 Międzynarodowa
dc.rights.licence
CC-BY
dc.rights.uri*
http://creativecommons.org/licenses/by/4.0/legalcode.pl
dc.share.type
otwarte czasopismo
dc.subject.enpl
Markov maps
dc.subject.enpl
invariant measure
dc.subject.enpl
expanding transformation
dc.subject.enpl
recurrence
dc.subject.enpl
aperiodicity
dc.subtypepl
Article
dc.titlepl
A few problems connected with invariantmeasures of Markov maps : verication of some claims and opinions that circulate inthe literature
dc.title.journalpl
Advances in Nonlinear Analysis
dc.typepl
JournalArticle
dspace.entity.type
Publication
Affiliations

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