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The $C^{1}$ property of convex carrying simplices for competitive maps

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  17 October 2018

JANUSZ MIERCZYŃSKI Open the ORCID record for JANUSZ MIERCZYŃSKI [Opens in a new window]
JANUSZ MIERCZYŃSKI*
Affiliation:
Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland email mierczyn@pwr.edu.pl
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Abstract

For a class of competitive maps there is an invariant one-codimensional manifold (the carrying simplex) attracting all non-trivial orbits. In this paper it is shown that its convexity implies that it is a $C^{1}$ submanifold-with-corners, neatly embedded in the non-negative orthant. The proof uses the characterization of neat embedding in terms of inequalities between Lyapunov exponents for ergodic invariant measures supported on the boundary of the carrying simplex.

Keywords

smooth dynamics

MSC classification

Primary: 37C65: Monotone flows 37D10: Invariant manifold theory 37D30: Partially hyperbolic systems and dominated splittings
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 5 , May 2020 , pp. 1335 - 1350
Copyright
© Cambridge University Press, 2018

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