Schedule for: 23w5038 - Dynamics of Hénon Maps: Real, Complex and Beyond

I will describe extending 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 "Henon-like" perturbations.

I will discuss recent work with Roland Roeder concerning rational surface maps that we call `toric'. These include all monomial maps and some recent examples with Bell and Jonsson of rational maps that have transcendental first dynamical degrees. Our main theorem is an equidistribution result for preimages of curves that works precisely when the map in question is "bad" in the sense that it cannot be birationally conjugated to an algebraically stable map.

To study of the meromorphic degeneration of dynamics, the theory of hybrid spaces, established by Boucksom and Jonsson, is known to be a strong tool. In this talk, we apply this theory to the dynamics of Hénon maps; for a famiy of Héenon maps$\{H_t\}_t$ parametrized by a unit punctured disk meromorphically degenerating at the origin, we show that as $t\to 0$, the family of the invariant measures $\{\mu_t\}$ “weakly converges” to the measure on the Berkovich affine plane which is naturally defined by the family $\{H_t\}_t$, in the sense of the theory of hybrid spaces.

We show that every Hölder observable satisfies the Central Limit Theorem (CLT) with respect to the measure of maximal entropy of every complex Hénon map. The proof is based on a general criterion to ensure the validity of the CLT, valid for abstract measurable dynamical systems. This is a joint work with Tien-Cuong Dinh.

For hyperbolic polynomial automorphisms with disconnected Julia sets, we give a description of the connected components of J which is strongly reminiscent of the classical Branner-Hubbard theory for 1D polynomials. This is joint work with Misha Lyubich.

I will talk about recent results on topological dynamics of H\’enon maps obtained in joint work with Jan Boro\’nski. For a parameter set generalizing the Benedicks-Carleson parameters (the Wang-Young parameter set) we obtain the following: The existence of wandering domains (answering a question of Lyubich, Martens and van Strien); The pruning front conjecture (due to Cvitanovi\’c, Gunaratne, and Procacci); A kneading theory (realizing a conjecture by Benedicks and Carleson); A classification: two H\’enon maps are conjugate on their strange attractors if and only if their sets of kneading sequences coincide, if and only if their folding patterns coincide. The classification result relies on a further development of the authors’ recent inverse limit description of H\’enon attractors in terms of densely branching trees. (Joint work with Jan P. Boro\’nski)

In this talk I will discuss a joint work with Gabriel Vigny where we study stability properties of marked points in algebraic families of Hénon maps. A marked point is forward-stable (resp. backward-stable) if its forward orbit (seen as a sequence of holomorphic functions of the parameter) is equicontinuous. In analogy with the case of families of endomorphisms of the projective spaces, we show that a marked point is forward-stalbe if and only if it is backward-stable if and only if it is persistently periodic, provided the family is algebraic and not trivial.

We discuss the construction and the convergence properties of Green's function, for (appropriate) non-autonomous families of Hénon maps. Further, we discuss the proof of Bedford's conjecture on uniform attracting basins in $\mathbb{C}^2$, which is an important consequence of the above results. Finally if time permits we discuss the generalisation of the above technique to any $n$-dimensions, using weak shift-like maps.

In a work in progress with Avila and Lyubich, we show that there are maps in the complex Hénon family with a stable homoclinic tangency. This is due to a new mechanism on the stable intersections between two dynamical Cantor sets generated by two classes of conformal IFSs on the complex plane.

In this talk we discuss the following two topics: (i) the construction of a hyperbolic complex Henon map with connected Julia set which is non-planar, (ii) a numerical algorithm to detect the disconnectivity of Julia sets. As a consequence, we obtain a certain topological property of the connectedness locus for the complex Henon family. This is a joint work in progress with Zin Arai (Tokyo Institute of Technology).

Per_n is a (nodal) Riemann surface parametrizing degree-2 rational functions with an n-periodic critical point. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic critical point (i.e. to the period-n components of the Mandelbrot set). Two long-standing open questions are: (1) Is Per_n connected? (2) Is G_n is irreducible over Q? We show that if G_n is irreducible over Q, then Per_n is connected. In order to do this, we find a smooth point with Q-coordinates on a compactification of Per_n. This smooth Q-point represents a special degeneration of degree-2 rational maps, and as such admits an interpretation in terms of tropical geometry.

A horseshoe is a very classic example in hyperbolic dynamics. In this talk, I will consider holomorphic dynamical systems where bifurcations of complex horseshoes appear. In this context, there is a link between - a counting problem of tangencies and - the variations of the Lyapunov exponent of the maximum entropy measure. Moreover, we will see that this counting problem is easy to solve explicitly.

We will give an overview of renormalization for real Henon-like maps, both dissipative and area-preserving, concentrating on rigidity and geometry of invariant sets.

J.w with Alexandra Kuznetsova. A bimeromorphic map f on a polarized family of abelian varieties over the unit disk is always conjugated to a pseudo-automorphism. We shall discuss when it is possible to find a relatively ample model over which f becomes regular.

We generalize the renormalization theory of unimodal intervals maps to dimension two, so that it can be applied to the study of real Hénon maps. The key step is to identify the higher-dimensional analog of critical orbits using Pesin theory. As the main result, we will give an explicit and complete description of the non-uniform partial hyperbolicity of Hénon maps that are properly dissipative, infinitely renormalizable and unimodal.