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Robust transitivity and domination for endomorphisms displaying critical points

Part of: Hamiltonian and Lagrangian mechanics Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems

Published online by Cambridge University Press:  14 April 2023

C. LIZANA ,
R. POTRIE ,
E. R. PUJALS  and
W. RANTER
C. LIZANA
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatística, Universidade Federal da Bahia, Salvador 40170-110, Bahia, Brazil (e-mail: clizana@ufba.br)
R. POTRIE
Affiliation:
Centro de matemática, Facultad de Ciencias, Universidad de la República, Montevideo 11400, Montevideo, Uruguay (e-mail: rpotrie@cmat.edu.uy)
E. R. PUJALS
Affiliation:
Graduate Center, City University of New York, New York, NY 10016, USA (e-mail: epujals@gc.cuny.edu)
W. RANTER*
Affiliation:
Instituto de Matemática, Universidade Federal de Alagoas, Maceió 57072-090, Alagoas, Brazil
*
e-mail: wagnerranter@im.ufal.br
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Abstract

We show that robustly transitive endomorphisms of a closed manifold must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result, we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation.

Keywords

endomorphismcritical pointsdominated splittingrobust transitivity

MSC classification

Primary: 70H08: Nearly integrable Hamiltonian systems, KAM theory
Secondary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 2 , February 2024 , pp. 594 - 629
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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