Dynamics of a quasi-quadratic map

Detalhes bibliográficos
Autor(a) principal: Azevedo, Assis
Data de Publicação: 2014
Outros Autores: Carvalho, Maria, Machiavelo, António
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/1822/24729
Resumo: We consider the map $\cchi:\Q\to\Q$ given by $ \cchi(x)= x\ceil{x}$, where $\ceil{x}$ denotes the smallest integer greater than or equal to $x$, and study the problem of finding, for each rational, the smallest number of iterations by $\cchi$ that sends it into an integer. Given two natural numbers $M$ and $n$, we prove that the set of numerators of the irreducible fractions that have denominator$M$ and whose orbits by $\cchi$ reach an integer in exactly $n$ iterations is a disjoint union of congruence classes modulo $M^{n+1}$. Moreover, we establish a finite procedure to determine them. We also describe an efficient algorithm to decide if an orbit of a rational number bigger than one fails to hit an integer until a prescribed number of iterations have elapsed, and deduce that the probability that such an orbit enters $\Z$ is equal to one.
id RCAP_cdfcae027e1df95b129e247974002d4a
oai_identifier_str oai:repositorium.sdum.uminho.pt:1822/24729
network_acronym_str RCAP
network_name_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository_id_str 7160
spelling Dynamics of a quasi-quadratic mapDiscrete dynamical systemCeiling functionDensityCovering systemScience & TechnologyWe consider the map $\cchi:\Q\to\Q$ given by $ \cchi(x)= x\ceil{x}$, where $\ceil{x}$ denotes the smallest integer greater than or equal to $x$, and study the problem of finding, for each rational, the smallest number of iterations by $\cchi$ that sends it into an integer. Given two natural numbers $M$ and $n$, we prove that the set of numerators of the irreducible fractions that have denominator$M$ and whose orbits by $\cchi$ reach an integer in exactly $n$ iterations is a disjoint union of congruence classes modulo $M^{n+1}$. Moreover, we establish a finite procedure to determine them. We also describe an efficient algorithm to decide if an orbit of a rational number bigger than one fails to hit an integer until a prescribed number of iterations have elapsed, and deduce that the probability that such an orbit enters $\Z$ is equal to one.Fundação para a Ciência e a Tecnologia (FCT)Taylor and FrancisUniversidade do MinhoAzevedo, AssisCarvalho, MariaMachiavelo, António20142014-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/1822/24729eng1023-619810.1080/10236198.2013.805754http://dx.doi.org/10.1080/10236198.2013.805754info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-07-21T12:49:18Zoai:repositorium.sdum.uminho.pt:1822/24729Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T19:47:44.832128Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Dynamics of a quasi-quadratic map
title Dynamics of a quasi-quadratic map
spellingShingle Dynamics of a quasi-quadratic map
Azevedo, Assis
Discrete dynamical system
Ceiling function
Density
Covering system
Science & Technology
title_short Dynamics of a quasi-quadratic map
title_full Dynamics of a quasi-quadratic map
title_fullStr Dynamics of a quasi-quadratic map
title_full_unstemmed Dynamics of a quasi-quadratic map
title_sort Dynamics of a quasi-quadratic map
author Azevedo, Assis
author_facet Azevedo, Assis
Carvalho, Maria
Machiavelo, António
author_role author
author2 Carvalho, Maria
Machiavelo, António
author2_role author
author
dc.contributor.none.fl_str_mv Universidade do Minho
dc.contributor.author.fl_str_mv Azevedo, Assis
Carvalho, Maria
Machiavelo, António
dc.subject.por.fl_str_mv Discrete dynamical system
Ceiling function
Density
Covering system
Science & Technology
topic Discrete dynamical system
Ceiling function
Density
Covering system
Science & Technology
description We consider the map $\cchi:\Q\to\Q$ given by $ \cchi(x)= x\ceil{x}$, where $\ceil{x}$ denotes the smallest integer greater than or equal to $x$, and study the problem of finding, for each rational, the smallest number of iterations by $\cchi$ that sends it into an integer. Given two natural numbers $M$ and $n$, we prove that the set of numerators of the irreducible fractions that have denominator$M$ and whose orbits by $\cchi$ reach an integer in exactly $n$ iterations is a disjoint union of congruence classes modulo $M^{n+1}$. Moreover, we establish a finite procedure to determine them. We also describe an efficient algorithm to decide if an orbit of a rational number bigger than one fails to hit an integer until a prescribed number of iterations have elapsed, and deduce that the probability that such an orbit enters $\Z$ is equal to one.
publishDate 2014
dc.date.none.fl_str_mv 2014
2014-01-01T00:00:00Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/1822/24729
url http://hdl.handle.net/1822/24729
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 1023-6198
10.1080/10236198.2013.805754
http://dx.doi.org/10.1080/10236198.2013.805754
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Taylor and Francis
publisher.none.fl_str_mv Taylor and Francis
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron:RCAAP
instname_str Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron_str RCAAP
institution RCAAP
reponame_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
collection Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository.name.fl_str_mv Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
repository.mail.fl_str_mv
_version_ 1799133052098576384