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Sensitivity and historic behavior for continuous maps on Baire metric spaces

Part of: Smooth dynamical systems: general theory Ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  07 February 2023

MARIA CARVALHO Open the ORCID record for MARIA CARVALHO [Opens in a new window] ,
VINICIUS COELHO Open the ORCID record for VINICIUS COELHO [Opens in a new window] ,
LUCIANA SALGADO Open the ORCID record for LUCIANA SALGADO [Opens in a new window]  and
PAULO VARANDAS Open the ORCID record for PAULO VARANDAS [Opens in a new window]
MARIA CARVALHO
Affiliation:
CMUP & Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, Porto 4169-007, Portugal (e-mail: mpcarval@fc.up.pt)
VINICIUS COELHO
Affiliation:
Centro Multidisciplinar de Bom Jesus da Lapa, Universidade Federal do Oeste da Bahia, Av. Manoel Novais, 1064, Centro, Bom Jesus da Lapa 47600-000, Bahia, Brazil (e-mail: viniciuscs@ufob.edu.br)
LUCIANA SALGADO*
Affiliation:
Departamento de Matemática, Universidade Federal do Rio de Janeiro, Avenida Athos da Silveira Ramos 149 Cidade Universitária, P.O. Box 68530, Rio de Janeiro 21941-909, Rio de Janeiro, Brazil
PAULO VARANDAS
Affiliation:
CMUP & Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, Porto 4169-007, Portugal (e-mail: mpcarval@fc.up.pt) Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, Salvador 40170-110, Bahia, Brazil (e-mail: paulo.varandas@ufba.br)
*
e-mail: lsalgado@im.ufrj.br
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Abstract

We introduce a notion of sensitivity with respect to a continuous real-valued bounded map which provides a sufficient condition for a continuous transformation, acting on a Baire metric space, to exhibit a Baire generic subset of points with historic behavior (also known as irregular points). The applications of this criterion recover, and extend, several known theorems on the genericity of the irregular set, in addition to yielding a number of new results, including information on the irregular set of geodesic flows, in both negative and non-positive curvature, and semigroup actions.

Keywords

Baire metric spacehistoric behaviortransitivitysensitivitygroup and semigroup action

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators 37C10: Vector fields, flows, ordinary differential equations 37C40: Smooth ergodic theory, invariant measures 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 1 , January 2024 , pp. 1 - 30
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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