Prevalence of uniform hyperbolicity at the first internal-tangency bifurcation of Hénon-like families

At the first bifurcation parameter $a^*$ of a strongly dissipative Hénon-like family, the uniform hyperbolicity of the horseshoe is destroyed by a quadratic homoclinic tangency whose orbit lies inside the limit set (an internal tangency bifurcation). Takahasi (2012) proved that escape dynamics is of full Lebesgue density at $a^*$, and in the same passage asked whether uniform hyperbolicity is itself prevalent at $a^*$, and whether the resulting good-parameter maps retain a weak form of hyperbolicity as a memory of the hyperbolic regime. The weak-hyperbolicity memory question is substantially answered at the level of Lyapunov exponents by Takahasi's preprint arXiv:1308.4199 (every ergodic invariant measure of every good-parameter map is hyperbolic, with exponents uniformly bounded away from zero over all parameters), while the prevalence of uniform hyperbolicity at $a^*$ remains open. The open problem is to determine the lower one-sided Lebesgue density at $a^*$ of the parameters at which the limit set is uniformly hyperbolic: positive (prevalence) or zero.

Background

The Hénon family $f_a(x,y)=(1-ax^2+by,\sqrt{b}\,x)$ with $0<b\ll1$ is the standard two-dimensional model for the transition from Smale's horseshoe to chaotic attractors. A Hénon-like family is a perturbation $f_a(x,y)=(1-ax^2,0)+b\cdot\Phi(a,b,x,y)$, where $\Phi$ is bounded, continuous in $(a,b,x,y)$ and $C^4$ in $(a,x,y)$; such arcs are embedded in generic one-parameter unfoldings of quadratic homoclinic tangencies of dissipative saddle fixed points of surface diffeomorphisms, so they model the local anatomy of how uniform hyperbolicity dies in dimension two. A compact invariant set is uniformly hyperbolic if it carries a continuous invariant splitting into a stable direction that is exponentially contracted by all forward iterates and an unstable direction that is exponentially contracted by all inverse iterates. Writing $K_a=\{z:\{f_a^n z\}_{n\in\mathbb{Z}}\text{ is bounded}\}$ for the two-sided trapped set (the limit set of the family), $K_a$ is a uniformly hyperbolic horseshoe — an invariant Cantor set on which the dynamics is conjugate to the full two-symbol shift — for every $a>a^*$, where $a^*=a^*(b)$ near $2$ is the first bifurcation parameter, at which the family generically unfolds a quadratic homoclinic or heteroclinic tangency between the invariant manifolds of the two fixed saddles. Unlike the classical bifurcations treated by Palis and Takens, where the orbit of the tangency stays outside the limit set, here the tangency point, accumulated by transverse homoclinic points, lies inside the limit set; this is called an internal tangency bifurcation, and it blocks the standard reduction of the bifurcation to the intersection geometry (thickness, a numerical measure of the size of a Cantor set that controls whether two such sets must meet) of two invariant Cantor sets. Below $a^*$ the structure of $K_a$ depends discontinuously on $a$: $a^*$ is accumulated from the left by parameter intervals of attracting periodic orbits (sinks) and by positive-measure sets of parameters with nonuniformly hyperbolic strange attractors constructed by Benedicks and Carleson and by Mora and Viana. The relevant frequency notion is Palis's prevalence: a dynamical phenomenon $P$ is prevalent at $a^*$ if

$\liminf_{\varepsilon\to+0}\frac{1}{\varepsilon}\operatorname{Leb}\{a\in[a^*-\varepsilon,a^*]:\ f_a \text{ displays } P\}>0.$

For unfoldings of tangencies whose orbit leaves the limit set, Palis and Takens proved prevalence of hyperbolicity under conditions on the limiting Cantor sets, and Rios proved prevalence of hyperbolicity for arcs of surface diffeomorphisms destroying three-branch ("type 3") horseshoes; the expansion-rate and curvature hypotheses of that result fail for strongly dissipative Hénon-like families. Takahasi (2012) proved that at an internal tangency bifurcation escape is not merely prevalent but of full density: there is a set $\Delta\subset(a^{**},a^*]$ of good parameters, where $a^{**}<a^*$ is a parameter identified in the source, with $\lim_{\varepsilon\to+0}\varepsilon^{-1}\operatorname{Leb}(\Delta\cap[a^*-\varepsilon,a^*])=1$ such that for every $a\in\Delta$ the forward trapped set $K_a^+$ has zero Lebesgue measure (Lebesgue-almost every initial point diverges to infinity under forward iteration) and $f_a$ is transitive on $K_a$; physically observable chaos near $a^*$ is thus chaotic transient, not sustained. This contrasts sharply with the one-dimensional quadratic family, whose corresponding first bifurcation parameter is a full density point of stochastic parameters with absolutely continuous invariant measures. In the same passage the paper states two things it cannot decide: whether the uniform hyperbolicity of the horseshoe regime — $K_a$ being a uniformly hyperbolic set, the property holding for every $a>a^*$ — is prevalent at $a^*$, and whether the maps $f_a$ with $a\in\Delta$ retain some weak form of hyperbolicity, statistical or geometric, as a memory of the uniformly hyperbolic regime for $a>a^*$.

Problem Statement

For a Hénon-like family $(f_a)$ — $f_a(x,y)=(1-ax^2,0)+b\cdot\Phi(a,b,x,y)$ with $0<b\ll1$, where $\Phi$ is bounded, continuous in $(a,b,x,y)$ and $C^4$ in $(a,x,y)$ — with first internal-tangency bifurcation parameter $a^*$ (the parameter near $2$ at which the family generically unfolds a quadratic tangency whose orbit lies inside the limit set), decide whether uniform hyperbolicity is prevalent at $a^*$ in Palis's sense:

$\liminf_{\varepsilon\to+0}\frac{1}{\varepsilon}\operatorname{Leb}\{a\in[a^*-\varepsilon,a^*]:\ K_a \text{ is a uniformly hyperbolic set}\}>0\ ?$

Here $K_a=\{z:\{f_a^n z\}_{n\in\mathbb{Z}}\text{ is bounded}\}$ is the two-sided trapped (limit) set, and a compact invariant set is uniformly hyperbolic if it carries a continuous invariant splitting into a stable direction exponentially contracted by all forward iterates and an unstable direction exponentially contracted by all inverse iterates; the source paper establishes that $K_a$ is a uniformly hyperbolic set for every $a>a^*$, so the question asks whether this property has positive one-sided density at the parameter where it is destroyed. Equivalently, determine the lower one-sided Lebesgue density at $a^*$, from the left, of the parameters at which the limit set is uniformly hyperbolic: is it positive (uniform hyperbolicity is prevalent at the death of the horseshoe), or zero (not prevalent: a quantified Newhouse-type statement that non-hyperbolicity dominates arbitrarily close to the boundary of the horseshoe locus)? The companion question raised in the same passage of the source — whether the good-parameter maps $f_a$, $a\in\Delta$ (where $\Delta$ is the full-density good-parameter set constructed in the source), retain a quantified weak form of hyperbolicity — has been substantially answered at the Lyapunov-exponent level by Takahasi's preprint arXiv:1308.4199 (every ergodic invariant measure of every $f_a$ with $a\in\Delta$ is hyperbolic with exponents uniformly bounded away from zero, uniformly over parameters), so the surviving open core of the source question is the uniform-hyperbolicity prevalence problem above.

References

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