'22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

"Transitions in dynamical systems from one- to two-dimensions"

Ou, Dyi-Shing

Studies show that the possible dynamical behavior of a system is constrained by the dimension of its phase space. Here, we investigate how the constraint is lifted as the dimension increases. In particular, we view the Hénon [H] and the Lozi [L] families as perturbations of the unimodal and the tent families in two dimensions. We introduce a renormalization model and use the model to explain the following phenomena:  1.) The Hénon attractor does not depend on the parameters continuously. In fact, the prime end rotation number [KP] of the attractor is discontinuous. (with J. Boronski)  2.) On continuous interval mappings, all possible trajectories can be classified by a forcing relation [G,CE] based on the kneading theory. However, the forcing relation does not have a continuation in the Hénon and the Lozi families, even when the families are arbitrary close to one dimension [Ou].  3.) A two-dimensional system can have infinitely many periodic attractors [N1,N2,R], whereas a sufficiently smooth one-dimensional system can not [S].  4.) There are no Fibonacci maps [LM] in two dimensions. $$\phantom{---}$$ [H] Hénon, A two-dimensional mapping with a strange attractor, Commun. Math. Phys. 50 (1976) 69--77.  [L] Lozi, Un attracteur étrange (?) du type attracteur de Hénon, Le Journal De Physique Colloques. 39 (1978) C5--9.  [KP] Koropecki and Passeggi, A Poincar\'{e}--Bendixson theorem for translation lines and applications to prime ends, Commentarii Mathematici Helvetici. 94 (2019) 141--183.  [N1] Newhouse, Diffeomorphisms with infinitely many sinks, Topology. 13 (1974) 9--18.  [N2] Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publications Mathématiques De IHÉS. 50 (1979) 101--151.  [R] Robinson, Bifurcation to infinitely many sinks, Commun. Math. Phys. 90 (1983) 433--459.  [S] Singer, Stable orbits and bifurcation of maps of the interval, SIAM Journal on Applied Mathematics. 35 (1978) 260--267.  [G] Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Commun. Math. Phys. 70 (1979) 133--160.  [CE] Collet and Eckmann, Iterated maps on the interval as dynamical systems, Birkhäuser (1980).  [Ou] Ou, Critical points in higher dimensions, I: Reverse order of periodic orbit creations in the Lozi family, arXiv:2203.02326 (2022).  [LM] Lyubich and Milnor, The Fibonacci unimodal map, Journal of the American Mathematical Society. 6 (1993) 425--457.

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