Sets of divergence of sequences of linear fractional transformations: the general classification and closure under finite intersections

A set of divergence is a subset $E$ of the Riemann sphere realized by a sequence of linear fractional (Möbius) transformations that diverges at every point of $E$ and converges at every point of its complement. Piranian and Thron (1957) introduced these sets, motivated by the convergence theory of continued fractions, and posed a list of open questions; Erdős and Piranian (1959) settled the case of sets lying on a straight line (exactly the $G_{\delta\sigma}$ sets of the line) and the denumerable case (exactly the sets eliminated by a transfinite geometric derivation), together with the hereditary question for denumerable sets, but left all other divergence sets unclassified. Crane and Short (2007) identified sets of divergence with conical limit sets of subsets of hyperbolic space, generalized the earlier theorems to all dimensions, and proved strict inclusions between the $G_\delta$ and $G_{\delta\sigma}$ classes, quasisymmetric invariance, and non-invariance under arbitrary homeomorphisms, while stating that the 1957 intersection question remains unsolved. The surviving open problem is to characterize the class of sets of divergence beyond the line and denumerable regimes, and in particular to decide whether the intersection of two sets of divergence is again a set of divergence.

Background

A linear fractional transformation (also called a Möbius map) is a map $T(z)=\frac{az+b}{cz+d}$ with $ad-bc\neq 0$; it acts as a homeomorphism of the Riemann sphere $\mathbb{C}_\infty$, the extended complex plane. Such maps are the building blocks of continued fractions: for a continued fraction with tail maps $s_n(z)=\frac{a_n}{b_n+z}$, the classical approximants are the compositions $S_n=s_1\circ s_2\circ\cdots\circ s_n$, and the continued fraction converges exactly when the sequence $(S_n(0))$ converges. In the 1950s George Piranian, Wolfgang Thron, and Paul Erdős, extending an older line of questions about divergence sets of Taylor and trigonometric series (Erdős, Herzog, Piranian 1954), posed the inverse problem: which subsets of the sphere can occur as the exact locus of divergence of a sequence of Möbius maps?

Following the definition of Piranian and Thron (1957), a set $E\subseteq\mathbb{C}_\infty$ is a set of divergence if there exists a sequence $(T_n)$ of linear fractional transformations such that $(T_n(z))$ diverges, in the spherical (chordal) metric, for every $z\in E$ and converges for every $z\notin E$. Elementary descriptive-set-theoretic arguments show every set of divergence is a $G_{\delta\sigma}$ set, meaning a countable union of countable intersections of open sets, because the convergence set of any sequence of continuous functions is an $F_{\sigma\delta}$ set, that is, a countable intersection of countable unions of closed sets; the 1957 and 1959 papers established conversely that every $G_\delta$ set is a set of divergence, and that the class is closed under finite unions.

Erdős and Piranian (1959) settled two classification regimes. First, a set lying on a straight line is a set of divergence if and only if it is a $G_{\delta\sigma}$ subset of that line. Second, a denumerable (countable) set $E$ is a set of divergence if and only if the transfinite derived sequence becomes empty at some ordinal: $E^1=E\cap\mathrm{gd}(E)$, $E^\alpha=E^{\alpha-1}\cap\mathrm{gd}(E^{\alpha-1})$ for successor ordinals, $E^\alpha=\bigcap_{\beta<\alpha}E^\beta$ for limit ordinals, with $E^\alpha=\emptyset$ for some $\alpha$. Here $\mathrm{gd}(E)$ is the set of geometric limit points of $E$: the points $z$ such that for every $\varepsilon>0$ there is a $\delta>0$ such that every $t$ with $|t-z|<\delta$ has some $w\in E$ with $|w-t|<\varepsilon\,|t-z|$; informally, the points that cannot be separated from $E$ by an arbitrarily thin angular sector. As a corollary, every subset of a denumerable set of divergence is again a set of divergence. The paper states no further open question, but it leaves entirely untreated the divergence sets that are neither on a line nor denumerable, and it settles only the second and third of the open questions listed by Piranian and Thron (1957), according to the contemporaneous Zentralblatt review of the paper.

Beardon (2004) determined, in all dimensions, the structure of the convergence side: if a sequence of Möbius maps on $\mathbb{R}^k_\infty$, the one-point compactification of $\mathbb{R}^k$, converges pointwise exactly on a set $C$ to a map $g$, then either $g$ is constant on $C$, or $C$ is the image under a Möbius map of $V\cup\{\infty\}$ for a subspace $V$ and $g$ is the restriction of a Möbius map, or $C$ is the whole sphere and $g$ is constant off a single point. This complements, but does not address, the divergence-set classification.

Crane and Short (2007) recast the entire problem in hyperbolic geometry, meaning the geometry of hyperbolic space $H^{m+1}$, the simply connected complete $(m+1)$-dimensional space of constant negative curvature. Identifying the sphere $S^m$ with the ideal boundary of $H^{m+1}$, a sequence $(z_n)$ in $H^{m+1}$ converges conically to a boundary point $x\in S^m$ if it stays within bounded hyperbolic distance of some geodesic ray landing at $x$; the conical limit set $\Lambda_c(W)$ of a subset $W\subseteq H^{m+1}$ is the set of boundary points reached conically by sequences from $W$. Crane and Short proved that a subset of $S^m$ is a set of divergence of some sequence of orientation-preserving Möbius maps if and only if it is the conical limit set of some subset of $H^{m+1}$; conical limit sets are a notion imported from Kleinian group theory, the theory of discrete groups of Möbius transformations acting on hyperbolic space, where they are normally attached to group orbits rather than to arbitrary subsets. Writing $\mathrm{CL}(m)$ for the class of conical limit sets in $S^m$, they proved for every $m$ that the $G_\delta$ class lies strictly inside $\mathrm{CL}(m)$, which lies strictly inside the $G_{\delta\sigma}$ class; that every $G_{\delta\sigma}$ subset of a codimension-one sphere (a copy of $S^{m-1}$ inside $S^m$) belongs to $\mathrm{CL}(m)$, generalizing the line theorem; that $\mathrm{CL}(m)$ is closed under finite unions and under intersection with open sets, has a local open-cover character, and is closed under locally quasisymmetric homeomorphisms, that is, homeomorphisms of the sphere whose distortion of distance ratios within triples of points is uniformly controlled; that $\mathrm{CL}(m)$ is not invariant under arbitrary homeomorphisms of the sphere; and that the countable members of $\mathrm{CL}(m)$ are exactly the sets eliminated by a transfinite derivation $E^{\chi+1}=E^\chi\setminus\Lambda_c(H^{m+1}\setminus\mathrm{co}(E^\chi))$, where $\mathrm{co}$ denotes hyperbolic convex hull, the geometric form of the Erdős–Piranian criterion. Non-invariance under homeomorphisms means that no purely topological characterization of divergence sets can exist: any complete classification must be geometric.

Despite this progress the classification is incomplete, and one concrete structural question from the 1957 list has never been answered: whether the intersection of two sets of divergence is again a set of divergence. Crane and Short state that this Piranian–Thron question remains unsolved, equivalently that it is not known whether $\mathrm{CL}(m)$ is closed under finite intersections, and they rule out the easy counterexample routes: in any hypothetical counterexample neither factor nor the intersection can be countable, and any countable intersection of conical limit sets that is countable is again a conical limit set.

Problem Statement

Determine the structure of the class of sets of divergence of sequences of linear fractional transformations on the Riemann sphere $\mathbb{C}_\infty$, equivalently, by the Crane–Short identification, of the class $\mathrm{CL}(2)$ of conical limit sets of subsets of hyperbolic space $H^3$, beyond the settled regimes of sets on a line and denumerable sets. The overarching objective is an intrinsic characterization of which subsets of the sphere arise as exact divergence loci of Möbius sequences, together with the closure properties of this class under the basic set operations; any such characterization must be geometric, since the class is invariant under locally quasisymmetric maps but not under arbitrary homeomorphisms.

The concrete named core, open since Piranian and Thron (1957) and unanswered by Erdős–Piranian (1959) and Crane–Short (2007), is the intersection question (Crane and Short's Problem 1): given two sequences of linear fractional transformations whose sets of divergence are $A$ and $B$, must there exist a sequence of linear fractional transformations whose set of divergence is exactly $A\cap B$? Equivalently, is the class $\mathrm{CL}(m)$ closed under finite intersections, for $m=2$ or in general dimension? Crane and Short pose two adjacent open questions of the same shape, intersection with $G_\delta$ sets (their Problem 2) and products of conical limit sets (their Problem 3), which are companions of the intersection question within the same classification objective rather than consequences of it.

References

1. George Piranian and W. J. Thron, Convergence properties of sequences of linear fractional transformations, Michigan Mathematical Journal 4 (1957), no. 2, 129–135. DOI 10.1307/mmj/1028989001, https://doi.org/10.1307/mmj/1028989001.
2. Paul Erdős and George Piranian, Sequences of linear fractional transformations, Michigan Mathematical Journal 6 (1959), no. 3, 205–209. DOI 10.1307/mmj/1028998224, https://doi.org/10.1307/mmj/1028998224.
3. Alan F. Beardon, The pointwise convergence of Möbius maps, Michigan Mathematical Journal 52 (2004), no. 3, 483–489. DOI 10.1307/mmj/1100623408, https://doi.org/10.1307/mmj/1100623408.
4. Edward Crane and Ian Short, Conical limit sets and continued fractions, Conformal Geometry and Dynamics 11 (2007), 224–249. DOI 10.1090/S1088-4173-07-00169-5, arXiv:0708.1730, https://doi.org/10.1090/S1088-4173-07-00169-5.
5. Ian Short, Hausdorff dimension of sets of divergence arising from continued fractions, Proceedings of the American Mathematical Society 140 (2012), no. 4, 1371–1385. DOI 10.1090/S0002-9939-2011-11032-3, https://doi.org/10.1090/S0002-9939-2011-11032-3.
6. Paul Erdős, Fritz Herzog, and George Piranian, Sets of divergence of Taylor series and of trigonometric series, Mathematica Scandinavica 2 (1954), 262–266. DOI 10.7146/math.scand.a-10413, https://doi.org/10.7146/math.scand.a-10413.
7. Matthew Jacques and Ian Short, Semigroups of Isometries of the Hyperbolic Plane, International Mathematics Research Notices 2022 (2022), no. 9, 6403–6463 (first published online 2020). DOI 10.1093/imrn/rnaa291, https://doi.org/10.1093/imrn/rnaa291.