Sharp $2p/(p+1)$ threshold for the Hausdorff dimension of Julia sets under horocyclic convergence of geometrically finite rational maps

For a geometrically finite rational map $f$ (every critical point in the Julia set is preperiodic; parabolic cycles allowed), McMullen (Comment. Math. Helv. 75 (2000)) proved that algebraic convergence $f_n\to f$ preserving critical relations and horocyclic at every parabolic point makes $J(f_n)\to J(f)$, and that the Hausdorff dimensions converge to $\mathrm{H.dim}\,J(f)$ whenever the convergence is radial or $\mathrm{H.dim}\,J(f)>2p(f)/(p(f)+1)$, where $p(f)$ is the petal number; below the threshold the dimension can jump, and the examples of §14 jump to exactly $2p(f)/(p(f)+1)$. The general lower bound along a horocycle (Theorem 12.1) carries an unproved loss: $\liminf \mathrm{H.dim}\,J(f_n)\ge 2p/(p+1)-\epsilon(\eta,f)$ with $\epsilon\to 0$ as $\eta\to 0$, and McMullen remarks without proof that a direct analysis would give $\epsilon=0$ for sufficiently small $\eta$. The problem is the sharp threshold law: prove or refute the zero-loss bound in the remark's form (petals $p$ at $c$ alone, $2p/(p+1)$, equivalently $\delta(f,g)=2p/(p+1)$ for the implosion pair, for $\eta$ sufficiently small). The radial, super-threshold, borderline, and crude-cap parts are already derivable from the source; the open content is the zero-loss lower bound of the remark. Natural related questions the paper does not pose — an amplified count through preparabolic critical points, a uniform extension to all genuinely horocyclic (non-radial) convergence, and the exact dimension limits in the mixed radial/horocyclic regime — are described below as context, not as part of the objective. No resolution of this bound is known in the literature to date.

Background

A rational map is a holomorphic self-map of the Riemann sphere of degree at least two, given as a ratio of polynomials. Its iterates split the sphere into the Fatou set, where the family of iterates is normal, and the complement, the Julia set $J(f)$, a compact invariant fractal on which the dynamics is chaotic. A basic quantitative invariant of this chaos is the Hausdorff dimension $\mathrm{H.dim}\,J(f)$, which for rational maps can take any value in $(0,2]$.

McMullen (2000) determined the dimension theory of geometrically finite rational maps, the class in which every critical point lying in $J(f)$ is preperiodic (its orbit is eventually periodic), so there are no recurrent critical points on the Julia set, but parabolic cycles, whose multipliers are roots of unity, are allowed. This is the exact counterpart, under the Sullivan dictionary between rational maps and Kleinian groups (discrete groups of Möbius transformations, the other side of the dictionary), of geometrically finite Kleinian groups. The main instruments are: $f$-invariant conformal densities, i.e. finite positive measures $\mu$ on $J(f)$ transforming by $\mu(f(A))=\int_A |f'|^{\alpha}\,d\mu$ on sets $A$ where $f$ is injective, with exponent $\alpha$; the critical dimension $\alpha(f)$, the infimum of exponents admitting a non-atomic such density; and the Poincaré series $P_s(f,x)=\sum_{f^n(y)=x}|(f^n)'(y)|^{-s}$, whose critical exponent is denoted $\delta(f)$. For every rational map $\alpha(f)$ equals the hyperbolic dimension (the supremum of the Hausdorff dimensions of invariant uniformly expanding subsets of $J(f)$), and for geometrically finite maps $\delta(f)=\alpha(f)=\mathrm{H.dim}\,J(f)$, with a unique normalized invariant density (the canonical density).

Parabolic points and petals. A fixed point $c$ of $f$ is parabolic when $f'(c)=1$ (after passing to an iterate); if $f(z)=z+az^{p+1}+O(z^{p+2})$ with $a\ne 0$ near $c$, the Leau–Fatou theorem provides $p$ attracting petals, simply connected domains tangent to $c$ on which all iterates converge to $c$. The petal number $p(f)$ is the largest number of petals at a parabolic periodic point of $f$ or at one of its preimages; a critical point of local degree $d$ whose orbit lands exactly on a $p$-petal parabolic point (a preparabolic critical point; one whose orbit merely converges to the parabolic point along its attracting petal does not count) contributes $dp$ petals. For a single parabolic periodic point $c$, write $q_c$ for the analogous local count: the maximum of the number of petals at $c$ itself and the $dp$ petals contributed by each preparabolic critical point of local degree $d$ whose orbit lands on $c$; thus $p(f)=\max_c q_c$. Always $\alpha(f)>p(f)/(p(f)+1)$.

Convergence modes at parabolic points. Let $f_n\to f$ algebraically (equal degrees, convergent coefficients), preserving critical relations: every critical point $b\in J(f)$ with a relation $f^i(b)=f^j(b)$ persists as critical points $b_n\to b$ of $f_n$, of the same multiplicity, satisfying the same relation. Near a parabolic fixed point $c$ with perturbed fixed points $c_n\to c$ and multipliers $\lambda_n=f_n'(c_n)=\exp(L_n+i\theta_n)\to 1$, one distinguishes: radial convergence, $\theta_n=O(|L_n|)$, i.e. approach to $1$ non-tangentially; horocyclic convergence, $\theta_n^{2}/L_n\to 0$, i.e. approach tangential enough to eventually enter every horocycle tangent to the unit circle at $1$ (radial implies horocyclic); and convergence along the $\eta$-horocycle, $\theta_n^{2}/L_n\to\eta\ne 0$, i.e. escape to $1$ parallel to a single fixed horocycle tangent at $1$, with $\eta$ of either sign (the fixed points $c_n$ may be attracting or repelling, but not indifferent, i.e. not of multiplier modulus one). The convergence at a parabolic point is genuinely horocyclic when it is horocyclic but not radial, i.e. $|\theta_n|/|L_n|\to\infty$; the $\eta$-horocycle regime sits between radial and genuinely horocyclic convergence and is itself neither radial nor horocyclic in the senses above. The convergence $f_n\to f$ is radial (respectively horocyclic) when it is so at every parabolic periodic point of $f$; it is called genuinely horocyclic when it is genuinely horocyclic at at least one parabolic periodic point of $f$, and nowhere radial when it is genuinely horocyclic at every such point. The convergence $(f_n,c_n)\to(f,c)$ is dominant when a non-degeneracy condition on the lower-order coefficients at $c_n$ holds (automatic when $p=1$); it guarantees the perturbed fixed points have controlled principal parts.

Parabolic implosion. Under horocyclic but non-radial convergence, suitable large iterates $f_n^{k(n)}$ converge near $c$ to a new map $g$ commuting with $f$ (a translation in the Fatou coordinates, the coordinates on a petal in which $f$ acts as a translation), so the geometric limit of the dynamical systems contains the semigroup $\langle f,g\rangle$: the exact analogue of a rank-one parabolic cusp of a Kleinian group degenerating to a rank-two cusp, generated in suitable coordinates by the commuting pair $S(z)=(z^{p}+1)^{1/p}$ and $T(z)=(z^{p}+\tau)^{1/p}$ with $\operatorname{Im}\tau\ne 0$.

The results delineating continuity. (i) Theorem 1.3: horocyclic convergence preserving critical relations implies $J(f_n)\to J(f)$ in the Hausdorff metric, and $f_n$ is geometrically finite for all large $n$. (ii) Theorem 11.2 (dynamic convergence): if in addition the convergence is radial, or horocyclic with $\liminf \mathrm{H.dim}\,J(f_n)>2p(f)/(p(f)+1)$, then the canonical densities converge weakly and $\mathrm{H.dim}\,J(f_n)\to \mathrm{H.dim}\,J(f)$; by condition (b') it suffices that $\mathrm{H.dim}\,J(f)>2p(f)/(p(f)+1)$. (iii) Proposition 11.3 (semicontinuity): $\alpha(f)\le\liminf \alpha(f_n)$ for any algebraic convergence. (iv) Theorem 12.1 (dimension along horocycles): if $f$ has $p$ petals at $c$, $(f_n,c_n)\to(f,c)$ dominantly, and $\lambda_n\to 1$ along the $\eta$-horocycle, then $\liminf \alpha(f_n)\ge 2p/(p+1)-\epsilon(\eta,f)$ with $\epsilon(\eta,f)\to 0$ as $\eta\to 0$. The loss $\epsilon$ arises only in the last step of the proof: the implosion pair $\langle f,g\rangle$ is $(1+\epsilon)$-quasiconformally conjugate to the model pair $\langle S,T\rangle$, whose Poincaré-series critical exponent is exactly $2p/(p+1)$ (Proposition 12.3), and quasiconformal distortion degrades the exponent. McMullen remarks, without proof, that a direct analysis "along the lines of Proposition 3.2" would show $\delta(f,g)=2p/(p+1)$, so that in Theorem 12.1 one can actually take $\epsilon=0$ for $\eta$ sufficiently small. (v) Theorem 1.5 (§14): the threshold in (b') cannot be lowered — for every $\epsilon\in(0,1/2)$ there are geometrically finite maps with $p(f)=1$, $\mathrm{H.dim}\,J(f)=1/2+\epsilon$, and $f_n\to f$ horocyclically preserving critical relations with $\mathrm{H.dim}\,J(f_n)\to 1=2p(f)/(p(f)+1)$; the Hausdorff measure concentrates on spiral arms shrinking to the parabolic point. The same numerology underlies Shishikura's theorem that Julia sets at parabolic parameters and the boundary of the Mandelbrot set (the connectedness locus of $z^2+c$) have Hausdorff dimension $2$.

What remains undetermined is the sharp form of the threshold law. The source's own unproved assertion is the exact lower bound along horocycles for $\eta$ sufficiently small: is the loss $\epsilon$ really zero? Beyond the paper lie natural related questions it does not pose: whether an amplified count through preparabolic critical points holds, whether a zero-loss bound extends uniformly to all genuinely horocyclic sequences, and the sharp values in the mixed regime — where the convergence is radial at some parabolic points and genuinely horocyclic at others and only the crude cap $\max(\mathrm{H.dim}\,J(f),\,2p(f)/(p(f)+1))$ follows from Theorem 11.2(b) by the subsequence argument of §14. (The borderline case $\mathrm{H.dim}\,J(f)=2p(f)/(p(f)+1)$, excluded from both Theorem 11.2(b') and the §14 construction, is in fact a formal consequence of Theorem 11.2(b) and Proposition 11.3 by that same subsequence argument.)

Problem Statement

Let $f$ be a geometrically finite rational map and let $c$ be a parabolic periodic point of $f$ at which $f$ has $p$ petals (all terms as defined in the Background). Recall that, near $c$, perturbed fixed points $c_n\to c$ carry multipliers $\lambda_n=f_n'(c_n)=\exp(L_n+i\theta_n)\to 1$; that convergence along the $\eta$-horocycle means $\theta_n^{2}/L_n\to\eta\ne 0$, of either sign, with the perturbed fixed points not indifferent; that $q_c$ denotes the local petal number at $c$ (the maximum of the number of petals at $c$ itself and the $dp$ petals contributed by each preparabolic critical point of local degree $d$ whose orbit lands on $c$), with global petal number $p(f)=\max_c q_c$; and that the convergence $(f_n,c_n)\to(f,c)$ is dominant as defined in the Background (automatic for $p=1$). For the global convergence modes, recall that the convergence $f_n\to f$ (algebraic, preserving critical relations) is radial or horocyclic when it is so at every parabolic periodic point of $f$, is genuinely horocyclic when it is horocyclic but not radial, i.e. $|\theta_n|/|L_n|\to\infty$, at at least one parabolic periodic point, and is nowhere radial when this holds at every one.

The objective is the sharp threshold law for the Hausdorff dimension of Julia sets under parabolic bifurcation: the zero-loss bound asserted, without proof, in McMullen's remark following the proof of Theorem 12.1. Prove or refute the following statement.

Sharp dimension along horocycles (Theorem 12.1 with zero loss). If $(f_n,c_n)\to(f,c)$ dominantly and $\lambda_n=f_n'(c_n)\to 1$ along the $\eta$-horocycle ($\theta_n^{2}/L_n\to\eta\ne 0$), then

$$\liminf_{n\to\infty}\,\mathrm{H.dim}\,J(f_n)\;\ge\;\frac{2p}{p+1},$$

with no $\epsilon(\eta,f)$ loss, for all geometrically finite rational maps $f$ — all degrees, all petal numbers, both signs of $\eta$, and arbitrary configurations of parabolic points and preparabolic critical points — for all sufficiently small $|\eta|$, as asserted in McMullen's remark. Equivalently: the Poincaré-series critical exponent of the implosion pair satisfies $\delta(f,g)=2p/(p+1)$ exactly, with no quasiconformal-distortion loss, so that in Theorem 12.1 one can take $\epsilon=0$ for $\eta$ sufficiently small.

Related questions beyond the source's assertion, recorded as context and not part of the objective: whether the amplified bound $2q_c/(q_c+1)$, counting the petals contributed by preparabolic critical points over $c$, holds — a conjectural transfer of the mechanism of the source's Theorem 3.1 ($\alpha(f)>p(f)/(p(f)+1)$ with the amplified petal count), which concerns $f$ itself rather than implosion pairs, and not a published claim; whether a zero-loss bound extends, with estimates uniform in the horocyclic parameter, to every genuinely horocyclic convergence (sequences diving deeper than any fixed horocycle, $\theta_n^{2}/L_n\to 0$ with $|\theta_n|/|L_n|\to\infty$), which would give $\liminf\,\mathrm{H.dim}\,J(f_n)\ge\max(\mathrm{H.dim}\,J(f),\,2p(f)/(p(f)+1))$ in the nowhere-radial case — the paper's own passage to genuinely horocyclic sequences (Theorems 1.8/13.2) is existential, selecting $\lambda_n$ on the $\eta=1/n$ horocycle and using $\epsilon(\eta,f)\to 0$ for that constructed sequence only; and the exact subsequential limits in the mixed radial/horocyclic regime, where the published arguments yield only the crude cap $\max(\mathrm{H.dim}\,J(f),\,2p(f)/(p(f)+1))$.

Logical structure (what is and is not published). The following parts follow from the source: $\mathrm{H.dim}\,J(f)\le\liminf\,\mathrm{H.dim}\,J(f_n)$ for algebraic convergence preserving critical relations (Proposition 11.3); the crude cap — every subsequential limit is at most $\max(\mathrm{H.dim}\,J(f),\,2p(f)/(p(f)+1))$ (Theorem 11.2(b) via the subsequence argument of §14); continuity along radial sequences and strictly above the threshold (Theorem 11.2(a), (b')); and hence the borderline continuity statement when $\mathrm{H.dim}\,J(f)=2p(f)/(p(f)+1)$. The open content is precisely the zero-loss lower bound of McMullen's remark: $\delta(f,g)=2p/(p+1)$, equivalently $\epsilon=0$ in Theorem 12.1, for $\eta$ sufficiently small.

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