TY - JOUR
T1 - A continuous bowen-mañé type phenomenon
AU - Muñoz-Young, Esteban
AU - Navas, Andrés
AU - Pujals, Enrique
AU - Vásquez, Carlos H.
PY - 2008/3
Y1 - 2008/3
N2 - In this work we exhibit a one-parameter family of C1- diffeomorphisms Fα of the 2-sphere, where α > 1, such that the equator S1 is an attracting set for every F α and Fα|S1 is the identity. For α > 2 the Lebesgue measure on the equator is a non ergodic physical measure having uncountably many ergodic components. On the other hand, for 1 < α < 2 there is no physical measure for Fα. If α < 2 this follows directly from the fact that the ω-limit of almost every point is a single point on the equator (and the basin of each of these points has zero Lebesgue measure). This is no longer true for α = 2, and the non existence of physical measure in this critical case is a more subtle issue.
AB - In this work we exhibit a one-parameter family of C1- diffeomorphisms Fα of the 2-sphere, where α > 1, such that the equator S1 is an attracting set for every F α and Fα|S1 is the identity. For α > 2 the Lebesgue measure on the equator is a non ergodic physical measure having uncountably many ergodic components. On the other hand, for 1 < α < 2 there is no physical measure for Fα. If α < 2 this follows directly from the fact that the ω-limit of almost every point is a single point on the equator (and the basin of each of these points has zero Lebesgue measure). This is no longer true for α = 2, and the non existence of physical measure in this critical case is a more subtle issue.
KW - Ergodic components
KW - Physical measures
UR - http://www.scopus.com/inward/record.url?scp=44249096121&partnerID=8YFLogxK
U2 - 10.3934/dcds.2008.20.713
DO - 10.3934/dcds.2008.20.713
M3 - Article
AN - SCOPUS:44249096121
VL - 20
SP - 713
EP - 724
JO - Discrete and Continuous Dynamical Systems
JF - Discrete and Continuous Dynamical Systems
SN - 1078-0947
IS - 3
ER -