**"Transitions in dynamical systems from one- to two-dimensions"**
#### Ou, Dyi-ShingStudies 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:
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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)
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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].
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3.) A two-dimensional system can have infinitely many periodic attractors
[N1,N2,R], whereas a sufficiently smooth
one-dimensional system can not [S].
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4.) There are no Fibonacci maps [LM] in two dimensions.
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[H] Hénon, A two-dimensional mapping with a strange
attractor, Commun. Math. Phys. 50 (1976) 69--77.
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[L] Lozi, Un attracteur étrange (?) du type attracteur
de Hénon, Le Journal De Physique Colloques. 39 (1978) C5--9.
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[KP] Koropecki and Passeggi, A Poincar\'{e}--Bendixson
theorem for translation lines and applications to prime ends, Commentarii
Mathematici Helvetici. 94 (2019) 141--183.
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[N1] Newhouse, Diffeomorphisms with infinitely
many sinks, Topology. 13 (1974) 9--18.
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[N2] Newhouse, The abundance of wild hyperbolic
sets and non-smooth stable sets for diffeomorphisms, Publications
Mathématiques De IHÉS. 50 (1979) 101--151.
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[R] Robinson, Bifurcation to infinitely many sinks,
Commun. Math. Phys. 90 (1983) 433--459.
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[S] Singer, Stable orbits and bifurcation of maps
of the interval, SIAM Journal on Applied Mathematics. 35 (1978) 260--267.
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[G] Guckenheimer, Sensitive dependence to initial
conditions for one dimensional maps, Commun. Math. Phys. 70 (1979)
133--160.
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[CE] Collet and Eckmann, Iterated maps on the interval
as dynamical systems, Birkhäuser (1980).
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[Ou] Ou, Critical points in higher dimensions,
I: Reverse order of periodic orbit creations in the Lozi family, arXiv:2203.02326
(2022).
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[LM] Lyubich and Milnor, The Fibonacci unimodal
map, Journal of the American Mathematical Society. 6 (1993) 425--457. |