Explicit constants and certified Jacobian thresholds for the finite-time renormalizability criteria in the Hénon family

The regular Hénon-like renormalization theory of Crovisier–Lyubich–Pujals–Yang proves that, for dissipative Hénon-like maps with bounded-type combinatorics, infinite regular renormalizability is characterized by finite-time checkable conditions — the quantitative inequalities (3.3)–(3.5) of Theorem B in Yang's 'On Regular Hénon-like Renormalization' — involving constants $d$, $C$, $K$ whose values the paper leaves entirely implicit. Making these constants explicit, and thereby obtaining a definite certified lower bound $\lambda_0$ on the Jacobian for which the Hénon family provably contains infinitely renormalizable maps of every bounded type, together with a finite-depth verification protocol able in principle to reach Jacobians near 1, is an explicitly acknowledged open problem (Remark 3.1 and Examples 3.2–3.3 of the source). As of August 2026, no work computing these constants or producing such a threshold is known in the literature. The problem remains open.

Background

A smooth interval map with a single folding point is called unimodal; the canonical examples are the quadratic maps $f_a(x)=x^2+a$. Renormalization, introduced in the mid-1970s to explain universal constants observed in cascades of period-doubling bifurcations, is the operation that takes the first return map of a unimodal map to a small restrictive interval around its critical point and rescale it back to the original scale. A map is infinitely renormalizable when this operation can be iterated forever, and its combinatorics is of bounded type when the ratios $R_n/R_{n-1}$ of successive return times stay bounded by a fixed constant $\mathbf{b}\ge 2$ — a combinatorial parameter, distinct from the Jacobian $b$ of a Hénon map introduced below. The hyperbolicity of the one-dimensional renormalization operator (Lyubich 1999; extended to $C^r$ unimodal maps by de Faria–de Melo–Pinto 2006) implies that infinitely renormalizable unimodal maps of bounded type converge under renormalization to a universal attractor.

The Hénon family $\mathcal{H}=\{F_{a,b}(x,y)=(x^2+a-by,\,x)\}$ is the standard two-dimensional analogue: a planar diffeomorphism whose Jacobian equals $b$, dissipative for $|b|<1$ and degenerating to the quadratic family at $b=0$. In the late 1980s Gambaudo–Tresser proposed the conjectural picture, stated in the source paper as two conjectures: first, from every infinitely renormalizable bounded-type quadratic polynomial there extends a real-analytic curve $\gamma(b)$, $b\in[0,1)$, of infinitely renormalizable Hénon maps converging to the same universal renormalization limit; second, the boundary of chaos (the boundary of the zero-topological-entropy locus) in $\mathcal{H}$ is a piecewise real-analytic curve consisting of infinitely renormalizable maps. In the perturbative regime $|b|\ll 1$ this picture was verified for the period-doubling case by de Carvalho–Lyubich–Martens (2005; see also Lyubich–Martens 2011 for the sequel on the heteroclinic web) and for arbitrary stationary combinatorics by Hazard (2011), but the perturbative arguments require the Jacobian to be extremely small and say nothing about the regimes closer to the area-preserving limit $b=1$.

The source paper, together with three companion papers by Crovisier–Lyubich–Pujals–Yang, develops a non-perturbative renormalization theory. Its objects are Hénon-like maps — diffeomorphisms $F(x,y)=(f(x,y),x)$ on a rectangle such that each $f(\cdot,y)$ is unimodal — and the renormalization is defined through first return maps to periodic subdomains, composed with a straightening chart (a coordinate change that puts the return back into Hénon-like form). The key novelty is the use of Pesin theory: a point is $M$-times forward (respectively backward) regular along a tangent direction if the iterates of $DF$ (respectively $DF^{-1}$) contract (respectively expand) that direction with a dominant exponential rate for all times up to $M$ — a quantitative form of non-uniform hyperbolicity. A return map is called $(L,\varepsilon,\lambda)$-regular, with marginal exponent $\varepsilon$ and contraction base $\lambda$, when this regularity holds uniformly with controlled constants. Under such regularity assumptions, the companion papers establish quantitative estimates on invariant manifolds (Crovisier–Lyubich–Pujals–Yang, 'Quantitative Estimates on Invariant Manifolds for Surface Diffeomorphisms'), a priori bounds — uniform distortion control of the return maps at every renormalization depth ('A Priori Bounds for Hénon-like Renormalization') — and the axiomatic theory of regularly unicritical systems ('Renormalization of Unicritical Diffeomorphisms of the Disk').

On this foundation the source paper proves: renormalization convergence — the renormalizations of an infinitely regularly Hénon-like renormalizable map converge super-exponentially to one-dimensional systems and their one-dimensional profiles converge exponentially to the 1D renormalization attractor (Theorem A); finite-time checkability — if a Hénon-like map $F$ with return times $\{R_n\}$ of $\mathbf{b}$-bounded type has $n_1$ nested $(L,\varepsilon_0,\lambda)$-regular Hénon-like returns, and an exponent $\varepsilon_0\in(0,1)$ and depths $n_0\le n_1$ are chosen so that

$\mathbf{b}\varepsilon_0^{d}<1,\qquad C\lambda^{\varepsilon_0 R_{n_0}}<1,\qquad K\lambda^{\varepsilon_0 R_{n_1}}<1,$

with $d\in(0,1)$ a uniform constant independent of $F$, $C\ge 1$ depending on $L$, $\lambda$, $\lambda^{1-\varepsilon_0}\|DF^{-1}\|$, $\|F\|_{C^3}$ and $\mathbf{b}$, and $K\ge 1$ depending on $C$, the $C^6$-norm of the depth-$n_0$ return map restricted to its renormalization domain $\mathcal{B}^{n_0}$, and the valuable curvature $\kappa_{n_0}$ (the curvature, measured in the straightening chart, of the image of a horizontal segment — a quantitative measure of the nonlinearity of the fold), then regularity persists automatically at all deeper depths (Theorem B); realization — in a $C^1$ one-parameter family whose renormalized one-dimensional profiles form a full family (a family of unimodal maps containing representatives of every combinatorial type), every bounded-type combinatorics is realized by an infinitely regularly renormalizable map (Theorem C); and regular unicriticality — the dynamics on the renormalization limit set has a unique orbit of quadratic tangencies between strong-stable and center manifolds, outside a slow-exponentially shrinking neighborhood of which it is uniformly partially hyperbolic (Theorem E).

The quantitative content, however, is left implicit: throughout the source papers, constants are absorbed into a notation ($\bar\eta$ denotes any number between $\eta$ and $C\eta^D$ for uniform unspecified $C,D$) that deliberately hides all explicit values. Remark 3.1 of the source paper states that explicit estimates of $d$, $C$, $K$ — 'while it is not done in this paper' — are possible, and Examples 3.2–3.3 work out the consequence: for the one-parameter Jacobian slices $\mathcal{H}_\lambda=\{F_{a,\lambda}\}_{a\in\mathbb{R}}$ (with $L=1$) the constant $K$ depends only on $\lambda$, so explicit estimates would immediately yield a definite lower bound $\lambda_0\in(0,1)$ on the admissible Jacobian — at depth $n_1=0$ it is characterized by the inequality $C\lambda_0^{\varepsilon_0}<1$ — and a finite-depth numerical protocol that could in principle certify infinitely renormalizable Hénon maps with Jacobians arbitrarily close to 1. No such explicit constants existed in the literature as of August 2026.

Problem Statement

Make the finite-time checkability criteria of regular Hénon-like renormalization numerically effective for the Hénon family $\mathcal{H}=\{F_{a,b}(x,y)=(x^2+a-by,\,x)\}$. Concretely, following Remark 3.1 and Examples 3.2–3.3 of Yang's 'On Regular Hénon-like Renormalization':

1. Derive explicit, computable estimates for the constants $d$, $C$ and $K$ in the finite-time criteria of Theorem B, namely

$\mathbf{b}\varepsilon_0^{d}<1,\qquad C\lambda^{\varepsilon_0 R_{n_0}}<1,\qquad K\lambda^{\varepsilon_0 R_{n_1}}<1,$

with $d$ a universal constant, $C$ an explicit function of $L$, $\lambda$, $\lambda^{1-\varepsilon_0}\|DF^{-1}\|$, $\|F\|_{C^3}$ and the combinatorial bound $\mathbf{b}$, and $K$ an explicit function of $C$, $\|F^{R_{n_0}}|_{\mathcal{B}^{n_0}}\|_{C^6}$ (the $C^6$-norm of the depth-$n_0$ return map on its renormalization domain $\mathcal{B}^{n_0}$) and the valuable curvature $\kappa_{n_0}$ — with every use of the source papers' absorption notation replaced by a quantitative inequality and no hidden uniform constant remaining.

2. Applying these estimates to the one-parameter Jacobian slices $\mathcal{H}_\lambda=\{F_{a,\lambda}\}_{a\in\mathbb{R}}$ (for which the source observes, setting $L=1$, that $K$ depends only on $\lambda$; here the contraction base $\lambda$ of the regularity condition coincides with the Jacobian), produce an explicit $\lambda_0\in(0,1)$ — defined, as in Example 3.2, at depth $n_1=0$, where it is characterized by the inequality $C\lambda_0^{\varepsilon_0}<1$ — such that, for every Jacobian $\lambda\in(0,\lambda_0)$ and every $\mathbf{b}$-bounded renormalization type, the criterion of Theorems B and C certifies the existence of an infinitely regularly Hénon-like renormalizable Hénon map with Jacobian $\lambda$ realizing that combinatorics — a definite certified lower bound on the admissible Jacobian, obtained by the non-perturbative theory.

3. Following Example 3.3 of the source, specify the corresponding finite-depth verification protocol: the finite, explicitly enumerated computations (enclosures of the $C^6$-norms and valuable curvatures of the first $N\ge n_1$ return maps, verification of the $(L,\varepsilon_0,\lambda)$-regularity of these returns, and fullness of the renormalized one-dimensional profiles, over a parameter interval) whose rigorous execution certifies infinitely regularly Hénon-like renormalizable Hénon maps at Jacobians beyond $\lambda_0$, potentially arbitrarily close to 1.

Determining sharper (larger) certified thresholds — the largest $\lambda_0$ the criteria can reach — is the natural refinement of this problem; no arbitrary numerical threshold is imposed.

References

1. J. Yang, 'On Regular Hénon-like Renormalization', preprint, arXiv:2411.08317 (2024). DOI: 10.48550/arxiv.2411.08317, https://arxiv.org/abs/2411.08317
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