Hausdorff dimensions of Julia sets in the Speiser class $\mathcal{S}_2$: is every value in $(1,2]$ attained?

Bergweiler and Cui (2022) proved that for every $d\in(0,2]$ there is a transcendental meromorphic function whose inverse has three singular values and whose Julia set has Hausdorff dimension exactly $d$, and noted that it “seems likely” that the interval $(1,2]$ is also realized in the Speiser class $\mathcal{S}_2$ of functions whose inverse has two singular values. The class $\mathcal{S}_2$ is completely classified: its members are exactly the maps $L(e^{az})$ with $a\ne 0$ and $L$ a fractional linear transformation, having no critical points and two asymptotic values. Within this class the interval $(1/2,1]$ is realized by the tangent family $\lambda\tan z$ (Barański 1995), and the value $2$ is realized by every entire member such as $\lambda e^z$ (McMullen 1987), but no member of $\mathcal{S}_2$ is known to have Julia-set dimension strictly between $1$ and $2$. The problem is to prove or refute that every $d\in(1,2]$ occurs as $\dim J(f)$ for some $f\in\mathcal{S}_2$ — equivalently, that the Möbius-exponential family $L\circ\exp$ realizes every Julia-set dimension in $(1,2)$.

Background

Complex dynamics studies the behaviour of points under repeated application of a holomorphic map. For a transcendental meromorphic function $f$ (meromorphic on the whole plane, so with an essential singularity at $\infty$), the Fatou set $F(f)$ is the largest open set on which the family of iterates $f^n$ is normal, and the Julia set $J(f)=\widehat{\mathbb C}\setminus F(f)$, where $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ is the Riemann sphere, is its complement, the locus of chaotic behaviour. A standard measure of the fractal size of $J(f)$ is its Hausdorff dimension $\dim J(f)$, a number in $[0,2]$.

For transcendental maps the key dynamical data of $f$ are the singular values of its inverse: a point $v\in\widehat{\mathbb C}$ is a critical value if $v=f(z)$ for some $z$ with $f'(z)=0$, and an asymptotic value if $f(z(t))\to v$ along some curve $z(t)\to\infty$. Write $\operatorname{sing}(f^{-1})$ for the set of critical and asymptotic values. The Speiser class $\mathcal S$ consists of the transcendental meromorphic functions for which $\operatorname{sing}(f^{-1})$ is finite, with $\mathcal S_q$ the subclass having exactly $q$ singular values; the Eremenko–Lyubich class $\mathcal B\supset\mathcal S$ requires only that $\operatorname{sing}(f^{-1})\setminus\{\infty\}$ be bounded. Off its singular values, $f$ is a covering map, so small $q$ means high rigidity; the classes $\mathcal S_q$ are the finite-parameter worlds of transcendental dynamics, and the monodromy theorem forces $q\ge 2$.

The possible Hausdorff dimensions of Julia sets have been mapped for the wide class $\mathcal B$: Stallard constructed, for every $d\in(1,2)$, a transcendental entire function with $\dim J(f)=d$, and, for every $d\in(0,1)$, a transcendental meromorphic one; together with the classical facts $J(e^z)=\mathbb C$ and $J(\tan z)=\widehat{\mathbb R}$ (the circle $\mathbb R\cup\{\infty\}$), every $d\in(0,2]$ is realized inside $\mathcal B$. Bergweiler and Cui (2022) then showed that three singular values already suffice in the Speiser class: for every $d\in(0,2]$ there exists $f\in\mathcal S_3$ with $\dim J(f)=d$.

The minimal class $\mathcal S_2$ is completely classified: by a classical observation of Eremenko, restated by Bergweiler and Cui, every $f\in\mathcal S_2$ has the form $f(z)=L(e^{az})$ with $a\ne 0$ and $L$ a fractional linear (Möbius) transformation — a map with no critical points and exactly two asymptotic values, $L(0)$ and $L(\infty)$. The two standard subfamilies are the exponentials $\lambda e^z$ (asymptotic values $0$ and $\infty$) and the tangent maps $\lambda\tan z$ (asymptotic values $\pm i\lambda$). For this class the following is known. First, $\dim J(f)>1/2$ for every $f\in\mathcal S_2$ (Barański in the hyperbolic setting, and Mayer in general, as cited by Bergweiler–Cui). Second, Barański proved that for $f_\lambda(z)=\lambda\tan z$ the function $\lambda\mapsto\dim J(f_\lambda)$ is real-analytic on the disc $0<|\lambda|<1$ and near real $\lambda$ with $|\lambda|>1$, satisfies $\dim J(f_\lambda)=1/2+O(\sqrt{|\lambda|})$ near $\lambda=0$, and is strictly increasing along the real segment $(0,1)$; combined with continuity of the dimension at hyperbolic parameters (Kotus–Urbański; hyperbolic means the closure of the forward orbits of the singular values stays disjoint from the Julia set, giving uniform expansion there), the real tangent segment realizes exactly the interval $(1/2,1]$, while for real $|\lambda|\ge 1$ one has $J(\lambda\tan z)=\widehat{\mathbb R}$, of dimension $1$. Third, $\dim J(\lambda e^z)=2$ for every $\lambda\ne 0$ (McMullen), and in fact every entire member of $\mathcal S_2$ — that is, every map $\alpha e^{az}+\beta$ — has $\dim J=2$, since entire functions of finite order in $\mathcal B$ have $\dim J=2$ (Theorem 9.6 of the Bergweiler–Rempe survey). By contrast, the escaping set $I(f)=\{z: f^n(z)\to\infty\}$ behaves dichotomically in $\mathcal S_2$: Bergweiler–Cui record that $\dim I(f)\in\{1/2,2\}$ for every $f\in\mathcal S_2$, so the Julia-set dimension is the only invariant of the class with room between the endpoints.

The gap is the interval $(1,2)$: no $f\in\mathcal S_2$ is known with $\dim J(f)$ strictly between $1$ and $2$, and Bergweiler and Cui write: "It seems likely that for $d\in(1,2]$ there also exists $f\in\mathcal S_2$ such that $\dim J(f)=d$."

Problem Statement

Let $\mathcal S_2$ be the class of transcendental meromorphic functions $f:\mathbb C\to\widehat{\mathbb C}$ whose inverse has exactly two singular values (critical or asymptotic); equivalently, by the classification of the class, $f(z)=L(e^{az})$ for some $a\in\mathbb C\setminus\{0\}$ and some fractional linear transformation $L$. Prove, or refute: for every $d\in(1,2]$ there exists $f\in\mathcal S_2$ such that the Hausdorff dimension of the Julia set of $f$ equals $d$.

The known boundary of the question: $\dim J(f)>1/2$ for every $f\in\mathcal S_2$; every $d\in(1/2,1]$ is attained within the real tangent family $f_\lambda(z)=\lambda\tan z$, $0<\lambda\le 1$; and $d=2$ is attained, e.g. by $\lambda e^z$ (indeed by every entire member of $\mathcal S_2$). The open core is the interval $(1,2)$: no $f\in\mathcal S_2$ with $\dim J(f)\in(1,2)$ is currently known, so it is open whether even a single such value, let alone all of them, is realized. In view of the classification, the question is exactly whether the Möbius-exponential family $L\circ\exp$ realizes every Julia-set dimension in $(1,2)$ — necessarily through members with two finite asymptotic values, since the members with an infinite asymptotic value are the entire ones, all of dimension $2$.

References

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