Open core of the Avila–Lyubich Feigenbaum Julia set problems: finite invariant geometric measures, the balanced Fibonacci regime, and exact dimension of Feigenbaum Julia sets

In their 2008 Journal of the AMS paper, Avila and Lyubich posed six outstanding problems on the measure and dimension of Julia sets of Feigenbaum (infinitely renormalizable) maps. Later literature has resolved part of the list: positive-area Feigenbaum Julia sets exist (Avila–Lyubich 2022), zero-area dimension-2 examples follow by combining that construction with the 2008 horseshoe interpolation theorem, and the Julia set of the classical period-doubling Feigenbaum polynomial has zero area and Hausdorff dimension strictly less than 2 (Dudko–Sutherland 2020). The surviving open core comprises: (i) whether any Feigenbaum map admits a finite invariant geometric measure (an invariant measure in the class of its critical-exponent conformal measure), possible only in the conservative regime; (ii) the balanced regime of the generalized Fibonacci family — uniqueness of the renormalization cycle for every real criticality and uniqueness of the balanced criticality, with recent computer-assisted progress by Dudko–Gaidashev (2024); (iii) the exact Hausdorff dimension of the classical period-doubling Feigenbaum Julia set; and (iv) the area/dimension dichotomy for the unicritical doubling Feigenbaum maps $F_d:z\mapsto z^d+c_d$ of even degree for big $d$.

Background

For a polynomial (or rational map) $f$, the Julia set $J(f)$ is the boundary of the set of points with bounded forward orbit; it is the locus of chaotic dynamics and carries the map's invariant measures. A quadratic-like map is a holomorphic map that is a two-to-one covering of a topological disk onto another, so that it dynamically resembles a quadratic polynomial on its filled-in set. Such a map is renormalizable if some iterate $f^q$, restricted to a neighborhood of the critical point, is again quadratic-like after rescaling; it is infinitely renormalizable when this can be repeated indefinitely with nesting little Julia sets. A Feigenbaum map is an infinitely renormalizable quadratic-like map with stationary, self-similar combinatorics — the situation of the period-doubling recurrence discovered numerically by Feigenbaum and by Coullet and Tresser, and of the Fibonacci recurrence. The renormalization operator, which sends a renormalizable map to the rescaled restriction, is the central object of the universality theory: its fixed points and periodic orbits organize whole families of maps.

Two basic geometric quantities of a Julia set are its two-dimensional Lebesgue measure (its area) and its Hausdorff dimension (HD), which can take any value in $[0,2]$ and equals 2 whenever the area is positive. A Borel probability measure $\mu$ on $J(f)$ is called $\delta$-conformal if $\mu(f(A))=\int_A |f'|^{\delta}\,d\mu$ for every Borel set $A$ on which $f$ is injective; the minimal such exponent is the critical exponent $\delta_{\mathrm{cr}}$ of the Poincaré series. Avila and Lyubich (2008) proved for Feigenbaum maps that $\delta_{\mathrm{cr}}$ coincides with the hyperbolic dimension $\mathrm{HD}(J_{\mathrm{hyp}})$, the dimension of the hyperbolic part $J_{\mathrm{hyp}}$ (the closure of the repelling periodic points away from the postcritical set); that both coincide with the full dimension $\mathrm{HD}(J)$ whenever the area of $J$ is zero; and that a $\delta_{\mathrm{cr}}$-conformal measure exists and is unique up to scale. They further established a geometric trichotomy for Feigenbaum Julia sets: the 'lean' case $\mathrm{HD}(J)<2$; the 'balanced' boundary case $\mathrm{HD}(J)=2$ with $\mathrm{area}(J)=0$; and the 'black-hole' case $\mathrm{area}(J)>0$, for which they gave a finitely checkable criterion — a ratio $\eta_n/\xi_n$ of return to escape probabilities for the pre-renormalization domains that, when sufficiently large (resp. small) at some level, forces the black-hole (resp. lean) case under a priori geometric bounds. Their Theorem 10.2 constructs, for every Feigenbaum map, an invariant measure equivalent to the $\delta_{\mathrm{cr}}$-conformal measure — the invariant geometric measure — and shows it is $\sigma$-finite. §10.1 of that paper lists five numbered outstanding problems (Problems 1–5), and §10.7 raises a sixth (Problem 6) and studies real generalized Fibonacci maps — unimodal maps with one turning point whose critical orbit returns according to the Fibonacci recurrence, introduced by Lyubich and Milnor (1993) — for which Bruin, Keller, Nowicki and van Strien (1996) proved that sufficiently high criticality (the local degree of the critical point) produces a wild attractor: a Cantor-set attractor of zero Lebesgue measure whose basin nevertheless has positive Lebesgue measure while failing to be a residual subset of the interval — so the attracting set is neither a finite union of intervals nor negligible. In the complex category (even integer criticality), the Fibonacci renormalization has a unique 2-cycle whose geometry was studied by Buff (2000) and whose hyperbolicity was proved by Smania (2007); in the real generalized family with continuous criticality $l>2$, Avila–Lyubich formulated Conjecture 10.7 (a unique renormalization cycle for every $l$) and a conditional trichotomy (Theorem 10.8) whose 'Balanced' boundary case raises Problem 6: is it attained at a unique parameter value, and can it occur at integer criticality (the authors doubt it, in footnote 13)?

Problem Statement

The common scientific objective is to complete the measure-theoretic and dimensional fine structure of the lean/balanced/black-hole trichotomy for Feigenbaum Julia sets that Avila–Lyubich (2008) left open. Four concrete open questions constitute the surviving core:

(i) Finite invariant geometric measure (2008 Problem 5). Does there exist a Feigenbaum map whose invariant geometric measure — the invariant measure equivalent to its $\delta_{\mathrm{cr}}$-conformal measure — is finite? By Theorem 10.2 of the source, such an invariant measure always exists and is $\sigma$-finite; finiteness could occur only in the conservative case. Prove existence or non-existence.

(ii) The balanced regime of the real generalized Fibonacci family (2008 §10.7, Conjecture 10.7 and Problem 6). For the generalized Fibonacci renormalization operator acting on maps with real criticality $l>2$: (a) for every real criticality $l>2$, does there exist a unique renormalization cycle with the Fibonacci combinatorics (Conjecture 10.7, restated by Dudko–Gaidashev 2024 as Conjecture 1.1: for every criticality $d>2$ there is a unique period-two periodic point of the generalized Fibonacci renormalization)? For even integer criticality, uniqueness and hyperbolicity of the complex-category 2-cycle are already known (Buff 2000; Smania 2007), so the open content of (a) concerns odd and non-integer real criticalities. (b) In the conditional trichotomy for the family $f_l$ (Theorem 10.8), is the Balanced boundary case attained at exactly one parameter value $l_0$? (c) Can the Balanced case occur at integer criticality, or is the authors' doubt in footnote 13 justified? The verified instances $d=3.8$ (lean) and $d=5.1$ (wild attractor) of Dudko–Gaidashev bracket these questions.

(iii) Exact dimension of the classical period-doubling Feigenbaum polynomial (2008 Problem 3, remnant). Determine the Hausdorff dimension of the Julia set $J(F_2)$ of the classical doubling Feigenbaum polynomial — the fixed point of quadratic renormalization, at parameter $c\approx-1.401155$; that its area is zero and its dimension is $<2$ is now known (Dudko–Sutherland 2020), but the dimension itself is not.

(iv) Area/dimension dichotomy for unicritical Feigenbaum maps of large even degree (2008 Problem 4). Let $J_d$ be the Julia set of the doubling Feigenbaum map $F_d:z\mapsto z^d+c_d$, where $d\ge 2$ is an even integer: what is $\mathrm{area}(J_d)$ for big $d$, and in case $\mathrm{area}(J_d)=0$, whether $\mathrm{HD}(J_d)=2$ for big $d$ — beyond the asymptotic results $\mathrm{HD}_{\mathrm{hyp}}(J_d)\to 2$ (Levin–Świątek 2005) and $\mathrm{area}(J_d)\to 0$ (Levin–Świątek 2010).

These four items together constitute the open part of the §10 problem list of the source paper, and they are independent sub-goals of that common objective, each resolvable on its own: resolving any single one of them (by settling the question or by refuting its existential form) resolves this task on its own and does not require resolving the others. The same holds separately for each of the sub-parts (ii)(a)–(c) and for either branch of (iv): any one of them is an independently answerable sub-goal whose resolution stands as a contribution in its own right.

References

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