We consider C^{2} Henon-like families of diffeomorphisms of R^{2} and study the boundary of the region of parameter values for which the nonwandering set is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the loss of hyperbolicity is caused by a first homoclinic or heteroclinic tangency and that uniform hyperbolicity estimates hold uniformly in the parameter up to this bifurcation parameter and even, to some extent, at the bifurcation parameter.
We consider the dynamics of strongly dissipative Hénon-like maps in the plane, around the first bifurcation parameter $a^*$ at which the uniform hyperbolicity is destroyed by the formation of homoclinic or heteroclinic tangencies inside the limit set. In [Takahasi H.: Commun. Math. Phys. {\bf 312} 37-85 (2012)], it was proved that $a^*$ is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under forward iteration. For these parameters, we show that all Lyapunov exponents of all invariant ergodic Borel probability measures a…
We numerically study quasiperiodic normally hyperbolic attracting invariant circles that appear for certain parameter values in a family of three-dimensional Henon-like maps. These parameter values make up contour segments in the parameter space where the invariant circles have constant irrational rotation number. At the edges of these segments we find a breakdown of the hyperbolicity of the invariant circle. We observe the collision and loss of smoothness of two of the invariant Lyapunov bundles while the Lyapunov exponents all remain distinct. This is very similar to the breakdown of quasipe…
We consider complex Henon maps which are quasi-hyperbolic. We show that a quasi-hyperbolic map is uniformly hyperbolic if and only if there are no tangencies between stable and unstable manifolds.
The purpose of this article is to investigate geometric properties of the parameter locus of the Hénon family where the uniform hyperbolicity of a horseshoe breaks down. As an application, we obtain a variational characterization of equilibrium measures "at temperature zero" for the corresponding non-uniformly hyperbolic Hénon maps. The method of the proof also yields that the boundary of the hyperbolic horseshoe locus in the parameter space consists of two monotone pieces, which confirms a conjecture in [AI]. The proofs of these results are based on the machinery developed in [AI] which emplo…
We effect a multifractal analysis for a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We decompose the set of non wandering points on the unstable manifold into level sets of Birkhoff averages of continuous functions, and derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures.
The Henon family has been shown to have period-doubling cascades. We show here that the same occurs for a much larger class: Large perturbations do not destroy cascades. Furthermore, we can classify the period of a cascade in terms of the set of orbits it contains, and count the number of cascades of each period. This class of families extends a general theory explaining why cascades occur.
In dimension three and under certain regularity assumptions, we construct a renormalisation scheme at the heterodimensional tangency of a non-transverse heterodimensional cycle associated with a pair of saddle-foci whose limit dynamic is a center-unstable Hénon-like family displaying blender-horseshoes. As a consequence, the initial cycle can be approximated in higher regularity topologies by diffeomorphisms having blender-horseshoes.
We give an explicit family of polynomial maps called center unstable Hénon-like maps and prove that they exhibits blenders for some parametervalues. Using this family, we also prove the occurrence of blenders near certain non-transverse heterodimensional cycles under high regularity assumptions. The proof involves a renormalization scheme along heteroclinic orbits. We also investigate the connection between the blender and the original heterodimensional cycle.
A blender-horseshoe is a locally maximal transitive hyperbolic set that appears in dimension at least three carrying a distinctive geometrical property: its local stable manifold "behaves" as a manifold of topological dimension greater than the expected one (the dimension of the stable bundle). This property persists under perturbations turning this kind of dynamics an important piece in the global description of robust non-hyperbolic systems. In this paper, we consider a parameterized family of center-unstable Hénon-like of endomorphisms in dimension three and show how blender-horseshoes natu…
For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e., decompose the set of non wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the conti…
In this note we will discuss the notion of robust chaos, and show that (i) there are natural one-parameter families with robust chaos and (ii) hyperbolicity is dense within generic one-parameter families (and so these families are not robustly chaotic).
We develop a thermodynamic formalism for a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any $t\in\mathbb R$ we prove the existence of an invariant Borel probability measure which minimizes the free energy associated with a non continuous geometric potential $-t\log J^u$, where $J^u$ denotes the Jacobian in the unstable direction. Under a mild condition, we show that any accumulation point of these measures as $t\to+\infty$ minimizes the unstable Lyapunov exponent. …
We study the dynamics of strongly dissipative Hénon-like maps, around the first bifurcation parameter $a^*$ at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove that $a^*$ is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under positive iteration. A key ingredient is that $a^*$ corresponds to "non-recurrence of every critical point", reminiscent of Misiurewicz parameters in one-dimensional dynamics. Adapting on the one hand Benedicks & Carleson's parameter exclusi…
In this paper we study arithmetic properties of a one-parameter family ${\mathbf H}$ of Hénon maps over the affine line. Given a family of initial points ${\mathbf P}$ satisfying a natural condition, we show the height function $h_{\mathbf P}$ associated to ${\mathbf H}$ and ${\mathbf P}$ is the restriction of the height function associated to a semipositive adelically metrized line bundle on projective line. We then show various local properties of $h_{\mathbf P}$. Next we consider the set $Σ({\mathbf P})$ consisting of periodic parameter values, and study when $Σ({\mathbf P})$ is an infinite…
We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $\mathbb{P}^2(\mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current …
We study $C^{d,r}$-Hénon-like families $(f_{a\, b})_{a\, b}$ with two parameters $(a,b)\in \mathbb R^2$. We show the existence of an open set of parameters $(a,b)\in \mathcal D$, so that a renormalization chart conjugates an iterate of $f_{a\, b}$ to a perturbation of $(x,y)\mapsto ((x^2+c_1)^2+c_2,0)$. We prove that the map $(a,b)\in \mathcal D\mapsto (c_1,c_2)$ is a $C^d$-diffeomorphism; as first numerically conjectured by Milnor in 1992. Furthermore, we show the existence of an open set of parameters $(a,b)$ so that $f_{a\, b}$ displays exactly two different renormalized Hénon-like maps who…
The famous MLC Conjecture states that the Mandelbrot set is locally connected, and it is considered by many to be the central conjecture in one-dimensional complex dynamics. Among others, it implies density of hyperbolicity in the quadratic family $\{z^2+c\}_{c\in\mathbb{C}}$. We describe recent advances on MLC and the relations between MLC, the Density of Hyperbolicity Conjecture, the Rigidity Conjecture, the No Invariant Line Fields Conjecture, and the Triviality of Fibers Conjecture. We treat families of unicritical polynomials and rational maps as well as the exponential family and familie…
We consider a scenario when a stable and unstable manifolds of compact center manifold of a saddle-center coincide. The normal form of the ODE governing the system near the center manifold is derived and so is the normal form of the return map to the neighbourhood of the center manifold. The limit dynamics of the return map is investigated by showing that it might take the form of a Henon-like map possessing a Lorenz-like attractor or satisfy 'cone-field condition' resulting in partial hyperbolicity. We consider also motivating example from game theory.
In a previous work, we proved an index theorem for families of asymptotically hyperbolic discrete dynamical systems and obtained applications to bifurcation theory. A weaker and far more common assumption than asymptotic hyperbolicity is the existence of an exponential dichotomy. In this paper we generalize all our previous results to the latter setting, which requires substantial modifications of our arguments. In addition, we generalize previous results on continuity and differentiability of Nemitski operators for discrete dynamical systems to obtain even better bifurcation results.
Standard bifurcation theory is concerned with families of vector fields $dx/dt = f(x,λ)$, $x \in \R^n$, involving one or several constant real parameters $λ$. Viewed as a differential equation for the pair $(x,λ)$, we observe a foliation of the total phase space by constant $λ$. Frequently, the presence of a trivial stationary solution $x=0$ is also imposed: $0 = f(0,λ)$. Bifurcation without parameters, in contrast, discards the foliation by a constant parameter $λ$. Instead, we consider systems $dx/dt = f(x,y), dy/dt = g(x,y)$. Standard bifurcation theory then corresponds to the special case …
