Density of Hyperbolicity for Complex Polynomials

Density of Hyperbolicity for Complex Polynomials

Definitions and mathematical statement

Let $\operatorname{Poly}_d=\{a_dz^d+\cdots+a_0:a_d\ne0\}$ with its coefficient topology. Regard a polynomial as a rational map of the Riemann sphere; infinity is its superattracting fixed point. A polynomial is hyperbolic when every critical orbit converges to an attracting periodic cycle, allowing escape to infinity in this sphere convention. The assertion is \[ \overline{\{f\in\operatorname{Poly}_d:f\text{ hyperbolic}\}}=\operatorname{Poly}_d \quad\text{for each }d\ge2. \] Equivalently, every coefficient neighborhood of every degree-$d$ polynomial contains such a map of the same degree. Requiring convergence only to finite attracting cycles would wrongly exclude escaping critical points and change the statement.

Formulation and concept references: [S1].

Short English statement

Can every complex polynomial be approximated in its coefficients by one whose critical orbits all tend to attracting cycles, including the cycle at infinity?

Research attempt

Current frontier.

The target is complex, fixed-degree density, including escape to infinity [S1], Problem 11. The theorem of Kozlovski--Shen--van Strien [E1], Theorem 1, concerns hyperbolicity of real polynomial dynamics on the real line; its definition does not impose attraction of every nonreal critical point. It must not be substituted for the present complex statement. Dudko's December 2025 manuscript for ICM 2026 reports bounded-type quadratic renormalization and local-connectivity results, and explicitly isolates remaining unbounded satellite and interpolation problems [E2], Sections 3.8--3.9 and 4.3--4.4. These results suggest using quantitative critical relations, but do not supply a perturbation theorem for all complex polynomials of every degree. All estimates below are deductions made in this attempt; no global closing lemma is being attributed to those sources.

Attack route.

Preserve every critical orbit that already has a finite certificate of attraction or escape, and try to close only the remaining critical orbits. A simultaneous quantitative implicit-function argument could make those critical points periodic and hence superattracting. The key issue is not merely a return close to a critical point: the return error must be small relative to the inverse parameter derivative and its variation. A second route permits previously uncaptured critical points to escape. Two exact quadratic examples below show why neither closing every critical point nor assuming recurrence of every bad critical point is a valid universal plan.

Attempt.

1. Robust finite capture certificates. For an attracting periodic point $z_*$ of period $q$, sufficiently small closed disks $D$ about $z_*$ satisfy \[ f^q(D)\subset\operatorname{int}D, \qquad \sup_{z\in D}|(f^q)'(z)|<\theta<1. \tag{215.1} \] A critical orbit attracted to this cycle eventually enters the interior of $D$. The finitely many iterates up to entry, the strict inclusion, and the derivative inequality persist on a sufficiently small coefficient neighborhood. The contraction theorem then supplies an attracting periodic point for every perturbed map in that neighborhood. For escape, a uniform escape radius $R$ exists on a small coefficient neighborhood with the leading coefficient bounded away from zero. Entry into $|z|>R$ is again a finite open condition. These arguments prove persistence without taking an unjustified infinite intersection of neighborhoods.

It is enough for density to handle polynomials with simple critical points: the discriminant of $f'$ is a nonzero polynomial in the coefficients, so its nonvanishing locus is dense. First approximate a multiple-critical-point polynomial within half the requested tolerance, then solve the approximation problem for that simple-critical polynomial within the remaining half. This reduction does not make a further parameter Jacobian automatically invertible.

2. A simultaneous closing lemma with explicit constants. Let $a\mapsto f_a$ be a holomorphic $k$-parameter family of polynomials of the same degree, defined on a neighborhood of the closed ball $\|a\|\le r$ in ${\mathbb{C}}^k$. Assume its remaining $k$ critical points have holomorphic markings $c_i(a)$ there; all other critical points have the persistent capture certificates above. For chosen positive integers $n_i$, put \[ F_i(a)=f_a^{n_i}(c_i(a))-c_i(a),\qquad J=DF(0). \] Suppose $J$ is invertible, $L=\|J^{-1}\|$, and \[ \|J^{-1}F(0)\|\le r/2, \qquad \sup_{\|a\|\le r}\|DF(a)-J\|\le(2L)^{-1}. \tag{215.2} \] Then a parameter $a_*$ with $\|a_*\|\le r$ satisfies $F(a_*)=0$. Indeed $T(a)=a-J^{-1}F(a)$ has derivative norm at most $1/2$ throughout the convex ball, and \[ \|T(a)\|\le\|J^{-1}F(0)\|+\tfrac12\|a\|\le r. \] Thus $T$ is a self-map contraction on a complete space. Its fixed point also satisfies $\|a_*\|\le2\|J^{-1}F(0)\|$.

Every newly periodic critical point lies on a superattracting cycle, because the multiplier contains a factor $f_{a_*}'(c_i(a_*))=0$. The other critical points remain attracted or escaping. Consequently $f_{a_*}$ is hyperbolic in precisely the sphere convention of the target. This is a proved finite-parameter lemma. Applying it arbitrarily close to every simple-critical polynomial requires the hypotheses $215.2$ at arbitrarily small parameter scales; that requirement remains unproved. When there are fewer bad critical points than available parameters, a $k$-dimensional parameter slice with invertible restricted Jacobian would suffice, but its existence is also a hypothesis, not a genericity shortcut.

3. The derivative includes critical-point motion. The size and rank of $J$ can be tested rather than guessed. Write $f_a=f+ah+O(a^2)$ along a parameter direction and let $c$ be a simple critical point of $f$. Differentiating $f_a'(c(a))=0$ gives \[ Dc=-\frac{h'(c)}{f''(c)}. \] For $n\ge1$, the chain rule gives the exact identity \[ \begin{split} D\big(f_a^n(c(a))-c(a)\big)\big|_{a=0} ={}&\sum_{j=0}^{n-1} (f^{n-1-j})'(f^{j+1}(c))\,h(f^j(c))\\ &+\frac{h'(c)}{f''(c)}. \end{split}\tag{215.3} \] The initial-point contribution to the derivative of the iterate is zero because $(f^n)'(c)=0$; the last term comes from subtracting the moving critical point. Omitting it can give the wrong rank or condition number. For a concrete check take $f_b(z)=z^3+bz$, $3c^2+b=0$, $c\ne0$, and $n=1$. Since $F=-2c^3-c$ and $dc/db=-1/(6c)$, direct differentiation gives $dF/db=c+1/(6c)$, exactly as $215.3$ with $h(z)=z$ predicts. The reproduction script verifies this identity symbolically.

Long iterates create products of derivatives in $215.3$. A tiny orbit-return error can coexist with an almost singular $J$ or enormous variation of $DF$ on the proposed parameter ball. Thus recurrence alone is not enough to invoke a qualitative implicit-function theorem at a point where $F(0)\ne0$. Conditions $215.2$ retain precisely the quantitative control that such an invocation would hide.

4. A false strengthening fails on an open hyperbolic set. For $f_c(z)=z^2+c$, consider \[ |c-1/10|<1/100,\qquad D=\{z:|z|\le1/5\}. \] For $z\in D$, \[ |f_c(z)|\le\frac1{25}+\frac{11}{100}=\frac3{20}<\frac15, \qquad |f_c'(z)|\le\frac25. \] The map is a contraction on $D$ and its only finite critical point $0$ is attracted to its fixed point there. These polynomials are hyperbolic. If $0$ were periodic, contraction would force it to be that fixed point, which would require $c=0$, excluded by the chosen parameter disk. Hence even an open set of already hyperbolic maps need not contain a map whose finite critical point is periodic. Density of critical-periodic centers in the whole coefficient space is a false replacement for the question. This is why the proposed strategy preserves captured critical points instead of closing all of them.

5. A bad critical orbit need not return near its critical point. At $c=-2$, the critical orbit is $0,-2,2,2,\ldots$, so $|f_c^n(0)-0|=2$ for all $n\ge1$. There is no small return residual for the closing lemma at this parameter. Nevertheless, for real $c=-a$ with $a>2$, one has \[ f_c^2(0)=a^2-a>a. \] For $x\ge a$, $x^2-a-x\ge a(a-2)>0$, so successive iterates grow without bound. Thus $c<-2$ gives nearby hyperbolic maps by escape. The example disproves an assumed universal recurrence input, not density. A complete approach needs a rigorous alternative between controlled critical closing, controlled escape, and other critical-relation perturbations, including the neutral and infinitely renormalizable cases.

Stress test.

The closing lemma is simultaneous: independently solving the $k$ scalar equations and then combining their parameters is invalid. Its ball must stay inside a common holomorphic marking domain and the capture neighborhood for every already good critical point. No compactness principle used here supplies those radii uniformly over long iterates. The simple-critical reduction resolves multiplicities for a density argument, but does not resolve singular Jacobians, nonrecurrence, parabolic implosion, or recurrent critical orbits with badly controlled scales. Polynomial perturbations also cannot independently alter one orbit location by a localized smooth bump while keeping the degree fixed; the holomorphic coefficient constraint is essential.

The role of renormalization is therefore specific: it would need to supply quantitative parameter-scale control or a valid alternative to $215.2$, not merely assert that a dynamical return exists. The unfinished regimes isolated in [E2] are relevant warnings against assuming such bounds. Hyperbolicity of a renormalization operator in a restricted class is not itself the requested density of hyperbolic polynomials in all coefficient neighborhoods.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt proves a robust-capture lemma and an explicit simultaneous critical-closing criterion, with the full parameter derivative and two exact tests rejecting stronger but false shortcuts. These are local tools, not a solution of the density conjecture. The missing result is a fixed-degree perturbation theorem guaranteeing controlled capture or escape of every remaining critical orbit arbitrarily close to every simple-critical polynomial. No argument here supplies that theorem, and no counterexample to the original density statement has been constructed.

Sources

• S1 Steve Smale, Mathematical Problems for the Next Century, 1998. https://www.fim.uni-passau.de/fileadmin/dokumente/fakultaeten/fim/lehrstuhl/muller/SmaleProblems1998.pdf. Catalogue formulation source.
• E1 O. Kozlovski, W. Shen and S. van Strien, "Density of hyperbolicity in dimension one," Annals of Mathematics 166 $2007$, 145--182, Theorem 1 and its definition of real-line hyperbolicity. https://annals.math.princeton.edu/wp-content/uploads/annals-v166-n1-p04.pdf.
• E2 D. Dudko, "On the MLC Conjecture and the Renormalization Theory in Complex Dynamics," arXiv:2512.24171 (30 December 2025), ICM 2026 manuscript, Sections 3.8--3.9 and 4.3--4.4, Theorems 3.3 and 4.1. The full MLC problem is not asserted solved there. https://arxiv.org/abs/2512.24171.