Avila and Lyubich (Publications Mathematiques de l'IHES 114 (2011) 171-223) proved that the renormalization operator R for real infinitely renormalizable unicritical polynomial-like germs of any even degree d has an invariant horseshoe, conjugate to the two-sided shift over all real combinatorics, on which R is exponentially contracting along the hybrid classes - the stable half of the hyperbolicity picture, valid for all combinatorics and degrees at once. The complementary transverse half - constructing and controlling the unstable direction, which would make R fully hyperbolic and complete the generalization of Lyubich's quadratic-case theorem to all even degrees - was explicitly deferred by the authors to a forthcoming paper that has never appeared. The problem is to establish this transverse unstable direction: prove uniform expansion of R transverse to the hybrid lamination on the horseshoe for arbitrary even degree. No published resolution is known.

Background

The Renormalization Conjecture arose in the mid-1970s from the numerical discovery by Feigenbaum, and independently by Coullet and Tresser, that period-doubling cascades in one-parameter families of interval maps exhibit universal scaling constants. For a unimodal map (a smooth map of an interval with a single turning point; the model families are the unicritical polynomials pc(x)=xd+cp_c(x)=x^d+c with d even (unicritical meaning that the origin is the only critical point)), the renormalization RfRf of an infinitely renormalizable f is the suitably rescaled first return of some iterate fpf^p to a neighborhood of the critical point, which is again a unimodal map of the same type; iterating R walks through ever smaller scales around the critical point. The conjecture asserts that R is hyperbolic on a suitable space of analytic maps, with a horseshoe, i.e. an invariant set on which R acts as the shift over the admissible return patterns (called combinatorics); this would explain universality and control the asymptotic geometry of all maps with the same combinatorics.

The complex point of view made the conjecture tractable. A polynomial-like map is a holomorphic branched covering f:U→Vf:U\to V between topological disks that behaves near its filled Julia set K(f)K(f) (the set of non-escaping points) like a polynomial; it is unicritical of degree d if it has a single critical point, normalized so that f(z)=zd+c+O(zd+1)f(z)=z^d+c+O(z^{d+1}). Two such germs are hybrid equivalent if they have representatives conjugate by a quasiconformal homeomorphism (a homeomorphism whose infinitesimal distortion is uniformly bounded) that is conformal almost everywhere on K(f)K(f); the hybrid classes partition the space of germs and, by the Douady-Hubbard straightening theorem (which assigns to each hybrid class the parameter cc of the polynomial z↦zd+cz\mapsto z^d+c that represents it), each hybrid class meets the Multibrot set MdM_d (the connectedness locus of z↦zd+cz\mapsto z^d+c) in one orbit of the rotation group Z/(d-1)Z. Hybrid classes are the stable leaves of renormalization. The decisive analytic input is beau bounds (a priori bounds): a uniform positive lower bound on the moduli (the conformal invariant measuring thickness) of the annuli between the renormalization domains UnU_n and ranges VnV_n at all sufficiently deep levels - equivalently, precompactness of renormalization orbits in the space of germs. Beau bounds are known for real maps of all degrees (Levin-van Strien; Lyubich-Yampolsky), but in the complex direction only for restricted combinatorial classes (Kahn-Lyubich).

In the quadratic case (d = 2) the conjecture was settled: first for bounded combinatorics (the geometric tower theory of McMullen, and Lyubich's proof that the Feigenbaum renormalization fixed point is hyperbolic with one-dimensional unstable manifold), and then, for all real combinatorics simultaneously, by Lyubich: the renormalization operator admits a full horseshoe on which it is uniformly hyperbolic with a one-dimensional unstable direction transverse to the hybrid lamination, from which he deduced that almost every real quadratic map is either regular (has an attracting cycle) or stochastic (has an absolutely continuous invariant measure).

Avila and Lyubich (2011) generalized the stable half of this picture to every even degree d≥2d\ge 2 and all combinatorics at once, replacing quadratic-specific geometry by an abstract argument. They endowed the hybrid leaves with a path-holomorphic structure (a space parametrized by holomorphic paths) and the induced Caratheodory metric (the weakest metric making all holomorphic functionals on the leaf contractive), proved that the leaves are Caratheodory-hyperbolic, and showed that beau bounds force some iterate of R to carry a neighborhood of the horseshoe strictly inside itself, so that the Schwarz lemma yields uniform exponential contraction along the hybrid lamination; an almost-periodicity argument for cocycles (precompactness of the cyclic semigroup generated by R, whose omega-limit contains a retraction) handles non-stationary combinatorics. The outcome is an R-invariant horseshoe A, conjugate to the two-sided shift over all real combinatorics, which attracts every real infinitely renormalizable unicritical germ of any even degree exponentially fast along hybrid classes. A conditional complex version (beau bounds plus rigidity of the combinatorial family imply the horseshoe) is also proved.

What the paper does not do is control the transverse direction. The authors state that they deal with 'the transverse unstable direction' in a forthcoming paper, which together with the analysis of non-renormalizable maps (Avila-Lyubich-Shen) would prove the Regular or Stochastic Dichotomy in every unicritical family x↦xd+cx\mapsto x^d+c with d even. Since each hybrid class meets the parameter locus in only finitely many points, the transverse direction is expected to be one-dimensional, governed by how R moves the straightening (parameter) coordinate; establishing its uniform expansion is exactly what turned the quadratic horseshoe into a full hyperbolicity statement, and it is this half that has never been published for d≥4d\ge 4.

Problem Statement

Prove that the renormalization operator R is uniformly hyperbolic in the direction transverse to the hybrid lamination on the renormalization horseshoe constructed by Avila and Lyubich for real infinitely renormalizable unicritical polynomial-like germs of even degree. Precisely: for arbitrary even degree d≥2d\ge 2 (the degrees d≥4d\ge 4 beyond the already-settled quadratic case), in the space of normalized degree-d unicritical polynomial-like germs hybrid equivalent to real ones, construct the transverse unstable direction for R on the R-invariant horseshoe A on which R is conjugate to the two-sided shift over all real renormalization combinatorics - the direction is expected to be one-dimensional, corresponding to variation of the straightening parameter across hybrid classes - and prove that R expands uniformly along it, so that the restriction of R to A is uniformly hyperbolic with the hybrid lamination as its stable lamination. The statement is sought for all real combinatorics and arbitrary even degree, within the source paper's setting of germs satisfying beau bounds. This is precisely the content the authors announced for a 'forthcoming paper' that never appeared; completing it finishes the generalization of Lyubich's quadratic-case theorem (the full renormalization horseshoe with one-dimensional unstable direction, from which 'almost every real quadratic map is either regular or stochastic' follows) to arbitrary even degree.

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