No Arabic abstract
In this paper, we consider the renormalization operator $mathcal R$ for multimodal maps. We prove the renormalization operator $mathcal R$ is a self-homeomorphism on any totally $mathcal R$-invariant set. As a corollary, we prove the existence of the full renormalization horseshoe for multimodal maps.
As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension 3, we introduce a $C^2$-open set of diffeomorphisms of whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly $1$. Our proof employs the thicknesses of Cantor sets.
In this note, we extend the renormalization horseshoe we have recently constructed with N. Goncharuk for analytic diffeomorphisms of the circle to their small two-dimensional perturbations. As one consequence, Herman rings with rotation numbers of bounded type survive on a codimension one set of parameters under small two-dimensional perturbations.
We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical circle maps.
In this paper, we consider weak horseshoe with bounded-gap-hitting times. For a flow $(M,phi)$, it is shown that if the time one map $(M,phi_1)$ has weak horseshoe with bounded-gap-hitting times, so is $(M,phi_tau)$ for all $tau eq 0$. In addition, we prove that for an affine homeomorphsim of a compact metric abelian group, positive topological entropy is equivalent to weak horseshoe with bounded-gap-hitting times.
The main purpose of this paper is to present a kneading theory for two-dimensional triangular maps. This is done by defining a tensor product between the polynomials and matrices corresponding to the one-dimensional basis map and fiber map. We also define a Markov partition by rectangles for the phase space of these maps. A direct consequence of these results is the rigorous computation of the topological entropy of two-dimensional triangular maps. The connection between kneading theory and subshifts of finite type is shown by using a commutative diagram derived from the homological configurations associated to $m-$modal maps of the interval.