Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics

Abstract

We consider semigroup actions on the unit interval generated by strictly increasing C r -maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor, and the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on f ′′ f ′ and the Schwarzian derivative, the above semigroups exhibit, C r -generically for any r ≥ 3, arbitrarily fast growth of the number of periodic points as a function of the period. We also show that a C r -generic semigroup from the class under consideration supports an ultimately complicated behavior called universal dynamics.

Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors

“…They also showed the C 1 -robustness of heterodimensional cycles -a C 1 -small perturbation of a system with a heterodimensional cycle can always be constructed such that the perturbed system gets into a C 1 -open domain in the space of dynamical systems where systems with heterodimensional cycles are dense (in C ∞ or C ω sense). A general higher smoothness version of this result is missing and a C r theory (with r > 1) of perturbations of heterodimensional cycles is much less developed (see, however, [4,5,[10][11][12]24]).…”

“…A computer-assisted proof for the existence of Lorenz attractor in system (4) for the values of parameters (σ, ρ, β) close to σ = 10, ρ = 28, β = 8/3 was given in [43,44] and, in [8], for system (5) for an open set of (α, λ) near α = 0.606, λ = 1.045. Recall that by Lorenz attractor we mean the attractor in the sense of Afraimovich-Bykov-Shilnikov (ABS) model, see [2,3].…”

“…Observe that c i k (i = 1, 2) have different signs and d i k always have the same sign as d. It follows that for class (4) where cdx + > 0 and dy − > 0 one can obtain the desired class (2) homoclinic tangency by picking i such that c i k < 0. If the original tangency belongs to class (3) where cdx + > 0 and dy − < 0, then we can first obtain a class (1) tangency by choosing i such that c i k > 0.…”

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

“…Let (f µ ) µ∈∆ be a C ∞ family in Diff ∞ Ω (Σ) and θ a real number. We call a family (ψ µ ) µ∈∆ a family of KAM circles with rotation number θ if (ψ µ ) µ∈∆ is a C ∞ family of embeddings from S 1 × [−1, 1] to Σ such that 3. Let (f µ ) µ∈∆ be a C ∞ family of maps in Diff r Ω (Σ) which admits a family of KAM circles.…”

“…In higher dimension, as far as the author's knowledge, there was no known real-analytic example which exhibits superexponential growth of the number of periodic points. 3 The first main result of this paper is the density of maps with super-exponential growth in an open subset of the set of real-analytic Hamiltonian diffeomorphisms on the twodimensional torus. Let T 2 be the two-dimensional torus R 2 /Z 2 .…”