Dense set of negative Schwarzian maps whose critical points have minimal limit sets

Alexander Blokh; Michał Misiurewicz

doi:10.3934/dcds.1998.4.141

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1998, Volume 4, Issue 1: 141-158. Doi: 10.3934/dcds.1998.4.141

This issue Previous Article A general approach to stability and sensitivity in dynamical systems Next Article Convergence of solitary-wave solutions in a perturbed bi-hamiltonian dynamical system ii. complex analytic behavior and convergence to non-analytic solutions

Dense set of negative Schwarzian maps whose critical points have minimal limit sets

Alexander Blokh^1, and
Michał Misiurewicz^2,

1.
Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, AL 35294-1170
2.
University of Alabama in Birmingham and Indiana University - Purdue University Indianapolis

Received: February 28, 1997

Published: December 1997

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Abstract

We study $C^2$-structural stability of interval maps with negative Schwarzian. It turns out that for a dense set of maps critical points either have trajectories attracted to attracting periodic orbits or are persistently recurrent. It follows that for any structurally stable unimodal map the $\omega$-limit set of the critical point is minimal.

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Mathematics Subject Classification: 58F03, 58F10.

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