Unboundedness of the parameter derivative of the critical orbit at non-hyperbolic parameters of the unicritical family (Avila's Problem 1)

In 'Infinitesimal perturbations of rational maps' (Nonlinearity 15 (2002), no. 3, 695-704), Artur Avila poses in the Appendix, as Problem 1: for a parameter $c$ in the generalized Mandelbrot set $M_d$ of the unicritical polynomial family $p_c(z)=z^d+c$ such that $p_c$ is non-hyperbolic, is the sequence $D p_c^{\,n-1}(c)\sum_{k=0}^{n-1}\frac{1}{D p_c^k(c)}$ unbounded as $n\to\infty$? By the chain rule this sequence equals $\frac{d}{dc}p_c^{\,n}(0)$, the derivative of the critical orbit with respect to the parameter, so the question is whether the critical orbit moves with unbounded infinitesimal speed under parameter perturbation at every non-hyperbolic parameter. The summable case is settled inside the same paper (Corollary A.4, with the key non-vanishing of the critical-orbit series independently obtained by Genadi Levin), and subsequent work extended instability criteria to weakly expanding critical orbits (Levin), Abel-Nörlund-Voronoi summability (Cabrera-Makienko-Poirier), and automorphic measures (Makienko-Cabrera), but always under additional hypotheses; the non-summable regimes - infinitely renormalizable (Feigenbaum) parameters, Siegel and Cremer parameters, parabolic parameters, and the hypothetical non-hyperbolic interior components of $M_d$ - remain uncovered. An affirmative answer in full generality would imply the Fatou conjecture (density of hyperbolicity) for these families.

Background

One-dimensional holomorphic dynamics studies the iteration of rational maps on the Riemann sphere. The Fatou set $F(R)$ of a rational map $R$ is the largest open set on which the family of iterates $\{R^n\}$ is normal (equicontinuous in the spherical metric); its complement, the Julia set $J(R)$, is the closure of the repelling periodic orbits and the locus of unstable behavior. A map is hyperbolic if every critical point (a point where the derivative vanishes) has an orbit converging to an attracting or superattracting cycle, equivalently if $R$ is uniformly expanding on $J(R)$; hyperbolic maps are structurally stable, meaning all sufficiently close maps are topologically conjugate to them. In the 1920s Pierre Fatou conjectured that hyperbolic maps are dense in the space of rational maps of any given degree. Even the restriction to the quadratic polynomial family $p_c(z)=z^2+c$ is open: writing $M_2=\{c\in\mathbb{C}:\{p_c^n(0)\}\text{ is bounded}\}$ for the Mandelbrot set (the locus where the critical orbit does not escape to infinity), the conjecture is equivalent to the statement that every component of the interior of $M_2$ consists of parameters with an attracting cycle ('no queer components'; a queer component is an interior component containing no hyperbolic parameter). The unicritical family $p_c(z)=z^d+c$, $d\ge2$, whose single finite critical point is $z=0$ with critical value $c=p_c(0)$, has the same structure with the generalized Mandelbrot set $M_d$; the case $d=2$ is the classical one.

Avila's paper develops an infinitesimal, perturbative approach to this conjecture, following the stability theory of Mañé-Sad-Sullivan. For a rational map $R$ and a holomorphic vector field $v$ near $J(R)$ (an infinitesimal perturbation direction, $R+\lambda v$), the response of a critical point $c$ is

$$\frac{d}{d\lambda}(R+\lambda v)^n(c)\Big|_{\lambda=0}=DR^{\,n-1}(R(c))\sum_{k=0}^{n-1}\frac{v(R^k(c))}{DR^k(R(c))},$$

and Corollary 2.9 of the paper shows this sequence is uniformly bounded in $n$ whenever $R$ is structurally stable (via an identity with a quasiconformal vector field - a vector field arising as the derivative at $\lambda=0$ of a family of quasiconformal conjugacies - or by Cauchy estimates for the holomorphic motion of the critical orbit: on a structurally stable parameter region the critical orbit depends holomorphically on the parameter over a definite neighborhood, so its parameter derivatives are bounded). Unboundedness of this sequence is thus the infinitesimal signature of bifurcation.

For polynomials the natural perturbation theory is Douady-Hubbard's polynomial-like framework: a polynomial-like map is a holomorphic degree-$d$ covering $f:U\to U'$ between Jordan disks with $U$ relatively compact in $U'$; two polynomial-like maps with connected filled Julia sets (filled Julia set = the set of non-escaping points) are hybrid equivalent when a quasiconformal conjugacy (a homeomorphism with bounded infinitesimal distortion) between them is conformal on the filled Julia set. The hybrid classes laminate the space of quadratic-like germs, and Lyubich (1999), in his work on Feigenbaum universality and Milnor's hairiness conjecture, gave this lamination a complex-analytic structure and proved that the quadratic family $c\mapsto z^2+c$ is transverse to it (meets every hybrid class non-tangentially; Theorem 4.11 of that paper, cited and used in Avila's Appendix A).

A critical point $c$ in the Julia set is called summable if $\sum_{k\ge0}1/|DR^k(R(c))|<\infty$ - derivative growth along the critical orbit, as holds for Collet-Eckmann maps (exponential growth of derivatives along the critical orbit) and Misiurewicz maps (critical orbit eventually lands on a repelling cycle). Under summability, the series $\mu_{R,c}(v)=\sum_{k\ge0}\frac{v(R^k(c))}{DR^k(R(c))}$ converges absolutely and defines a complex measure supported on the critical orbit (an eigenmeasure of the Ruelle transfer operator, the operator that averages an observable over inverse branches of the map). Theorem A.3 and Corollary A.4 of the paper identify the tangent space to the hybrid class of a polynomial with a summable critical point as the kernel of the related functional $\mu_{p_c,0}$, and then use Lyubich's transversality of the constant vector field to conclude $\sum_{k\ge0}\frac{1}{D p_c^k(c)}\neq0$; since summability forces $|D p_c^k(c)|\to\infty$, the sequence of Problem 1 is unbounded in the summable case (Avila notes the same non-vanishing was independently obtained by Levin, and the work of Avila-Lyubich-de Melo indicates the affirmative answer should also hold for real quadratic maps whose postcritical set - the closure of the forward orbit of the critical point - is non-minimal, i.e. properly contains a nonempty closed invariant subset). By the chain rule, since the parameter $c$ enters $z^d+c$ additively,

$$D p_c^{\,n-1}(c)\sum_{k=0}^{n-1}\frac{1}{D p_c^k(c)}=\frac{d}{dc}\,p_c^{\,n}(0),$$

so Problem 1 asks whether the critical orbit moves with unbounded infinitesimal speed under parameter perturbation at every non-hyperbolic parameter. An affirmative answer would, through Corollary 2.9, exclude non-hyperbolic structurally stable parameter regions of the connectedness locus - that is, would prove the Fatou conjecture for the family $z^d+c$. The genuinely open regimes are the non-summable ones: infinitely renormalizable (Feigenbaum) parameters, where the map is self-similar under return maps of the critical point at infinitely many scales; Siegel and Cremer parameters, whose critical orbit accumulates on a cycle with multiplier of modulus 1 that is not a root of unity (an irrationally indifferent cycle) - when the dynamics is conjugate to an irrational rotation on a disk around such a cycle (a Siegel disk; this holds whenever the rotation angle $\theta$ is Diophantine, i.e. badly approximable by rationals) the parameter is a Siegel parameter, and when the cycle is not linearizable it is a Cremer parameter; parabolic parameters, with a periodic cycle of multiplier a root of unity (for Problem 1 the relevant case is a critical orbit converging to the parabolic cycle); and slowly recurrent critical orbits (recurrent critical orbits - returning indefinitely close to the critical point - at a slow rate).

Problem Statement

Let $d\ge2$ and $p_c(z)=z^d+c$ for $c\in\mathbb{C}$, with connectedness locus $M_d=\{c\in\mathbb{C}:\{p_c^n(0)\}_{n\ge0}\text{ is bounded}\}$ (the generalized Mandelbrot set - equivalently, the set of parameters for which the Julia set of $p_c$ is connected, since for a polynomial the Julia set is connected exactly when the critical orbit is bounded; $M_2$ is the classical Mandelbrot set). For every $c\in M_d$ such that $p_c$ is non-hyperbolic (the orbit of the critical point 0 does not converge to an attracting or superattracting cycle), is the sequence

$$D p_c^{\,n-1}(c)\sum_{k=0}^{n-1}\frac{1}{D p_c^k(c)}$$

unbounded as $n\to\infty$? (The terms are well defined under the non-hyperbolicity hypothesis: $D p_c^k(c)=0$ for some $k$ would mean $p_c^{m}(0)=0$ for some $1\le m\le k$, i.e. the critical point itself is periodic, which is the superattracting - hence hyperbolic - case.) Equivalently: is $\frac{d}{dc}p_c^{\,n}(0)$, the parameter derivative of the critical orbit, unbounded for every non-hyperbolic $c\in M_d$? This is Problem 1 of Avila (2002), posed in this full generality (the source's Remark A.2 extends the framework to unicritical polynomial-like maps of degree $d$ hybrid-conjugate to $z^d+c$, in which $M_d$ is a global transversal of the hybrid-class lamination, though one hybrid class may meet $M_d$ in more than one point). The summable case is already settled (Corollary A.4 of the source and independent work of Levin and Makienko), and the remaining uncovered classes include the infinitely renormalizable (Feigenbaum), Siegel, Cremer, and parabolic parameters, slowly recurrent critical orbits, and the hypothetical non-hyperbolic interior components of $M_d$.

References

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