Abundance of Fast Growth of the Number of Periodic Points in 2-Dimensional Area-Preserving Dynamics

Abstract

We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there exists an open subset of the set of smooth area-preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points.1 In the appendix of this paper, we give a simple example for C ∞ case.

Complexities of differentiable dynamical systems

“…The symplectic counterpart of the above result has been proved previously in [Asa16]. Both confirm Conjecture 1.5.…”

Periodic expansiveness of smooth surface diffeomorphisms and applications

“…For 1 ≤ r ≤ ∞ V. Kaloshin [34] has proved that in Newhouse domains (i.e. C r open sets with a dense subset of diffeomorphisms having an homoclinic tangency) generic C r smooth surfaces diffeomorphisms have arbitrarily fast growth of saddle periodic points (see [1] for analytic examples). Kaloshin stated his result for finite r, but his proof also works for r = ∞.…”

Generic family displaying robustly a fast growth of the number of periodic points

“…In favor of this conjecture, there is the local density of conservative analytic maps satisfying (⋆) [3]. Also, Corollary 4.11 of the proof of Theorem A shows the local density of analytic dynamics with a normally hyperbolic periodic circles at which the dynamics is a smooth Diophantine rotation.…”

“…Indeed, an immediate consequence of Gelfreich-Turaev theorem [19] is that a locally tologically generic conservative diffeomorphism of surface displays a fast growth of the number of periodic points. ( 6) More recently, Asaoka [3] used also KAM theory to show the existence of an open set( 7 ) of conservative C ∞ -surface diffeomorphisms in which typically in the sense of Arnold, a map displays a fast growth of the number of periodic points.…”