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Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding |
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| Authors: | Jifa Jiang Janusz Mierczyński Yi Wang |
| Institution: | a Department of Mathematics, Shanghai Normal University, Shanghai 200234, PR China b Institute of Mathematics and Computer Science, Wroc?aw University of Technology, Wybrze?e Wyspiańskiego 27, PL-50-370 Wroc?aw, Poland c Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China d Department of Mathematics and Statistics, University of Helsinki, FIN-00014, Helsinki, Finland |
| Abstract: | The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C1 submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the Cr (r?1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C1 but not neatly embedded, the other in which S is not of class C1. |
| Keywords: | primary 34C12 37C65 secondary 34C30 34D08 34D35 37D10 37D25 37N25 92D25 |
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