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On the finiteness of attractors for piecewise$C^{2}$ maps of the interval

Published online by Cambridge University Press:  04 December 2017

P. BRANDÃO ,
J. PALIS  and
V. PINHEIRO
P. BRANDÃO
Affiliation:
Impa, Estrada Dona Castorina 110, Rio de Janeiro, Brazil email [email protected], [email protected]
J. PALIS
Affiliation:
Impa, Estrada Dona Castorina 110, Rio de Janeiro, Brazil email [email protected], [email protected]
V. PINHEIRO
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil email [email protected]
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Abstract

We consider piecewise $C^{2}$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^{2}$ non-flat map of the interval displays only a finite number of non-periodic attractors.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 7 , July 2019 , pp. 1784 - 1804
Copyright
© Cambridge University Press, 2017 

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