• Corpus ID: 16201760

Persistent homoclinic tangencies and infinitely many sinks for residual sets of automorphisms of low degree in C^{3}

@article{Biebler2016PersistentHT,
  title={Persistent homoclinic tangencies and infinitely many sinks for residual sets of automorphisms of low degree in C^\{3\}},
  author={S'ebastien Biebler},
  journal={arXiv: Dynamical Systems},
  year={2016}
}
We show that there exists a polynomial automorphism $f$ of $\mathbb{C}^{3}$ of degree 5 such that for every automorphism $g$ sufficiently close to $f$, $g$ admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each $d \ge 5$, there exists an open set of polynomial automorphisms of degree at most $d$ in which the automorphisms having infinitely many sinks are dense. To prove these results, we give a complex analogous to the notion of blender… 

Figures from this paper

Robust Degenerate Unfoldings of Cycles and Tangencies

We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index $$c>0$$ c > 0 and homoclinic tangencies of arbitrary codimension $$c>0$$ c > 0 . These type of sets are

Blenders near polynomial product maps of $\mathbb{C}^2$

  • Johan Taflin
  • Mathematics
    Journal of the European Mathematical Society
  • 2017
In this paper we show that if $p$ is a polynomial which bifurcates then the product map $(z,w)\mapsto(p(z),q(w))$ can be approximated by polynomial skew products possessing special dynamical objets

Newhouse Laminations of polynomials on $\mathbb{C}^2$

It has been recently discovered that in smooth unfoldings of maps with a rank-one homoclinic tangency there are codimension two laminations of maps with infinitely many sinks. Indeed, these

Newhouse Laminations.

Newhouse laminations occur in unfoldings of rank-one homoclinic tangencies. Namely, in these unfoldings, there exist codimension $2$ laminations of maps with infinitely many sinks which move

Blenders near polynomial product maps of C

In this paper we show that if p is a polynomial in the bifurcation locus then the product map (z, w) 7→ (p(z), q(w)) can be approximated by polynomial skew products possessing special dynamical

Lattès maps and the interior of the bifurcation locus

It is shown that a Latt{\`e}s map of sufficiently high degree can be perturbed to exhibit this geometry.

A complex Gap lemma

Inspired by the work of Newhouse in one real variable, we introduce a relevant notion of thickness for dynamical Cantor sets in the plane associated to a holomorphic IFS. Our main result is a complex

Complexities of differentiable dynamical systems

  • P. Berger
  • Computer Science
    Journal of Mathematical Physics
  • 2020
A dictionary between the previously explained theory on entropy and the ongoing one on emergence is proposed, proposing the fast growth of the number of periodic points, the positive entropy andThe high emergence.

Robust Degenerate Unfoldings of Cycles and Tangencies

We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index c>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}

References

SHOWING 1-10 OF 30 REFERENCES

Stability and bifurcations for dissipative polynomial automorphisms of $${{\mathbb {C}}^2}$$C2

We study stability and bifurcations in holomorphic families of polynomial automorphisms of $${{\mathbb {C}}^2}$$C2. We say that such a family is weakly stable over some parameter domain if periodic

Infinitely many periodic attractors for holomorphic maps of $2$ variables

An important development in the study of discrete dynamical systems was Newhouse's use of persistent homoclinic tangencies to show that a large set of C2 diffeomorphisms of compact surfaces have

Heterodimensional tangencies

We consider C1-diffeomorphisms f defined on three-dimensional manifolds having a pair of saddles Pf and Qf (of unstable indices one and two) whose homoclinic classes coincide persistently. We prove

Generic family with robustly infinitely many sinks

We show, for every $$r>d\ge 0$$r>d≥0 or $$r=d\ge 2$$r=d≥2, the existence of a Baire generic set of $$C^d$$Cd-families of $$C^r$$Cr-maps $$(f_a)_{a\in {\mathbb {R}}^k}$$(fa)a∈Rk of a manifold M of

Non density of stability for holomorphic mappings on P^k

A well-known theorem due to Mane-Sad-Sullivan and Lyubich asserts that J-stable maps are dense in any holomorphic family of rational maps in dimension 1. In this paper we show that the corresponding

HÉNON-LIKE MAPPINGS IN C²

We study the dynamics of a class of nonalgebraic holomorphic diffeomorphisms, topo- logical analogues in the unit bidisk of complex H6non mappings in C2. In particular a dynamical degree is defined,

DYNAMIQUE DES APPLICATIONS RATIONNELLES DES

We study the dynamics of rational mappings f of C by compactifying them in multiprojective spaces P1 × · · ·×Ps . We focus on maps of the surface P × P. We follow the approach of [Si 99] and

Diffeomorphisms with positive metric entropy

We obtain a dichotomy for C1$C^{1}$-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e.

Diffeomorphisms with infinitely many sinks

Henon mappings in the complex domain II: projective and inductive limits of polynomials

Let H: C^2 -> C^2 be the Henon mapping given by (x,y) --> (p(x) - ay,x). The key invariant subsets are K_+/-, the sets of points with bounded forward images, J_+/- = the boundary of K_+/-, J = the