Ergodic Theory and Dynamical Systems

Homeomorphic restrictions of smooth endomorphisms of an interval

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We describe the asymptotic dynamics of homeomorphisms obtained as restrictions of generic C2 endomorphisms of an interval with finitely many critical points, all of which are non-flat, and with all periodic points hyperbolic. The ω -limit set of such a restricted endomorphism cannot be infinite, except when the restriction of the endomorphism to the closure of the orbit of some critical point is a minimal homeomorphism of an infinite set. © 1992, Cambridge University Press. All rights reserved.