Cannon--Thurston maps for Anosov foliations

Abstract: Universal circles, introduced by Thurston and Calegari--Dunfield, are not well understood in general. Recently, the author together with Taylor showed that Anosov foliations with branching admit nonconjugate universal circles. We continue the study of these universal circles and show that for an Anosov foliation with branching on a hyperbolic manifold, the leftmost universal circle admits a Cannon--Thurston-type map to the ideal 2-sphere. This is a new type of construction of a Cannon--Thurston map. As a corollary, we show the fundamental group of the manifold acts on the leftmost universal circle with pseudo-Anosov dynamics.

• The paper constructs a continuous, surjective, π₁(M)-equivariant Cannon–Thurston map from the leftmost universal circle of the weak unstable foliation to S²∞.
• The proof classifies universal-circle sections by their bases and uses quadrant-local extremality, master-set finiteness, and leaf-space geometry to establish well-definedness and continuity.
• The resulting map gives each fundamental-group element finitely many alternating attractor and repellor fixed points after taking a power, supporting the conjecture that universal circles arise from almost pseudo-Anosov flowspace boundaries.

Context and motivation

For a closed fibered hyperbolic 3-manifold $M$ with fiber $\Sigma$, the lift of $\Sigma$ to $\widetilde{M}$ is identified with $\mathbb{H}^2$ and compactified by its ideal circle. Cannon and Thurston showed that the inclusion $\Sigma \hookrightarrow \widetilde{M}$ extends continuously to a surjective, $\pi_1(M)$-equivariant map from this circle to the ideal 2-sphere $S^2_\infty$ [CannonThurston]. Frankel generalized this: for any quasigeodesic flow on a hyperbolic 3-manifold, the flow space boundary $\partial O$ admits such a map, obtained by collapsing master sets [Frankel_thesis].

Universal circles, introduced by Thurston and developed by Calegari–Dunfield, generalize the fiber-boundary picture to taut foliations more broadly, but their behavior is poorly understood in general; in particular, different constructions can yield nonconjugate actions on circles. Buckminster and Taylor recently showed that for the weak stable/unstable foliations of non-$R$-covered Anosov flows, the Calegari–Dunfield and Fenley/Landry–Minsky–Taylor constructions produce genuinely nonconjugate universal circles [buckminster2025universal], making these foliations a natural test case. The paper under review addresses the natural question: when does a universal circle admit a Cannon–Thurston-type map to $\Sigma$0?

Main results

The main theorem states that for a non-$\Sigma$1-covered Anosov flow on a closed hyperbolic 3-manifold $\Sigma$2, with $\Sigma$3 the leftmost universal circle of the weak unstable foliation, there exists a continuous, surjective, $\Sigma$4-equivariant map

$\Sigma$5

This is a new type of construction of a Cannon–Thurston map: since the action on $\Sigma$6 is nonconjugate to the action on $\Sigma$7 by [buckminster2025universal, Theorem B], the map $\Sigma$8 is distinct from the classical map $\Sigma$9 of Frankel and Fenley. The paper also formulates a general definition of a Cannon–Thurston map for a foliation — a minimal universal circle together with an equivariant sphere-filling curve compatible with the continuous extensions of each leaf — and shows that $\Sigma$0 satisfies it.

A corollary concerns dynamics: every element of $\Sigma$1 has some power acting on $\Sigma$2 with a positive, finite number of fixed points alternating between attractors and repellors. This uses forthcoming work of Fenley–Mann–Potrie showing any Cannon–Thurston map is uniformly finite-to-one. Since the same dynamical property holds on $\Sigma$3, the corollary is presented as evidence that $\Sigma$4 should be conjugate to an action on the orbit space boundary of a transverse pseudo-Anosov flow, in line with the Landry–Minsky–Taylor conjecture that all universal circles for non-$\Sigma$5-covered foliations arise as flowspace boundaries of almost pseudo-Anosov flows.

Background framework

The setting is a non-$\Sigma$6-covered (topological) Anosov flow $\Sigma$7 on a closed hyperbolic 3-manifold, with weak unstable foliation $\Sigma$8. Key structural facts used throughout:

• Quasigeodesicity: Fenley proved an Anosov flow is quasigeodesic if and only if it is non-$\Sigma$9-covered [fenley2022nonrcoveredanosovflows], so each lifted unstable leaf $\widetilde{M}$0 has a continuous extension $\widetilde{M}$1.
• Stitching map: By [buckminster2025universal, Theorem C], the stitching map $\widetilde{M}$2 is a $\widetilde{M}$3-equivariant homeomorphism sending stable leaves to markers, reducing the study of the circle bundle at infinity $\widetilde{M}$4 to the orbit space.
• Master sets: For $\widetilde{M}$5, the master set rooted at $\widetilde{M}$6 is the union of preimages under the endpoint maps $\widetilde{M}$7; these are connected unions of leaves of $\widetilde{M}$8 sharing endpoints in $\widetilde{M}$9. A lemma proved here (using Fenley's results on branching leaves and closed orbits) establishes that every master set contains only finitely many leaves — a finiteness fact that is invoked repeatedly later.
• Leaf space structure: The leaf space $\mathbb{H}^2$0 is a simply connected, generally non-Hausdorff 1-manifold whose nonseparated points form cataclysms; pairs of leaves are joined by unique zigzag paths crossing cataclysms at launching/landing leaves.

The maps $\mathbb{H}^2$1 are collated into a single discontinuous function $\mathbb{H}^2$2, where $\mathbb{H}^2$3 extends $\mathbb{H}^2$4 by sending the nonmarker point $\mathbb{H}^2$5 to $\mathbb{H}^2$6. The discontinuity of $\mathbb{H}^2$7 occurs precisely when sequences limit onto nonmarker points of branching leaves; two lemmas show $\mathbb{H}^2$8 is continuous away from this failure mode. The map $\mathbb{H}^2$9 is further extended to the end space $\Sigma \hookrightarrow \widetilde{M}$0 via a map $\Sigma \hookrightarrow \widetilde{M}$1, shown to be a single well-defined point of $\Sigma \hookrightarrow \widetilde{M}$2 independent of the approximating zigzag ray $\Sigma \hookrightarrow \widetilde{M}$3.

Basepoints of sections

Points of $\Sigma \hookrightarrow \widetilde{M}$4 are sections of the circle bundle $\Sigma \hookrightarrow \widetilde{M}$5: special sections (leftmost sections based at a point) or limit sections. The central device is the decomposition of $\Sigma \hookrightarrow \widetilde{M}$6 into the leftmost up region $\Sigma \hookrightarrow \widetilde{M}$7 and rightmost down region $\Sigma \hookrightarrow \widetilde{M}$8 of a section $\Sigma \hookrightarrow \widetilde{M}$9. Three structural lemmas drive everything:

• Every leaf lies in $\pi_1(M)$0;
• If two nonseparated leaves branch from above, at most one lies in $\pi_1(M)$1 (dually for $\pi_1(M)$2);
• If $\pi_1(M)$3 sits on a marker over one leaf, the whole marker's leaf interval lies in the corresponding region.

These force the coloring of $\pi_1(M)$4 to alternate across cataclysms along zigzag paths. The base $\pi_1(M)$5 of a section is then defined as $\pi_1(M)$6 when nonempty, and otherwise as the set of leaves/ends from which every zigzag path is oriented "with the $\pi_1(M)$7-current." Nonemptiness follows from a no-sink argument: flowing against the $\pi_1(M)$8-current from any leaf either terminates at a global source or escapes out an end.

The classification result (Proposition: base) states that the base of any section is either a point, embedded interval, or line in $\pi_1(M)$9 — in which case $S^2_\infty$0 is special — or a single point of $S^2_\infty$1, in which case $S^2_\infty$2 is a limit section. The key exclusion is that type (I) sections (limit sections based at interior leaves) do not exist: the proof iterates through pinching configurations within a cataclysm and terminates because master sets are finite. This dichotomy is what makes the subsequent definition of $S^2_\infty$3 tractable.

Quadrant-local extremality

The Cannon–Thurston map is defined by $S^2_\infty$4 for $S^2_\infty$5, or $S^2_\infty$6 for a base end $S^2_\infty$7. Well-definedness would be immediate if special sections had unique bases, but they do not; the bridging concept is quadrant-local extremality (ql-extremality): a marker is ql-extremal over a zigzag path if it is extremal among markers with the same endpoint in the same quadrant, and a section is ql-extremal over $S^2_\infty$8 if all its markers over $S^2_\infty$9 are, its intersection with the nonmarker section is discrete, and it evaluates to nonmarker points at breakpoints.

Two facts connect this notion to the rest of the machinery. First, a ql-extremal section has $\partial O$0 contained in a single master set, so $\partial O$1 is constant over $\partial O$2 — this gives well-definedness immediately, since every section is ql-extremal over its base. Second, the key limiting proposition: if sections $\partial O$3 converge to $\partial O$4 and a zigzag path $\partial O$5 is oriented with the $\partial O$6-current but against each $\partial O$7-current, then $\partial O$8 is ql-extremal over $\partial O$9. Its proof rules out accumulation of nonmarker points using finiteness of nonseparated leaf sets on hyperbolic manifolds.

Proof of the main theorem

Continuity proceeds by compactness. Given sections $R$0, one extracts a subsequence such that $R$1 converges in the compact closure $R$2 to a point $R$3, which corresponds via a surjectivity/injectivity lemma for $R$4 on $R$5 to a leaf or end $R$6. A technical lemma then shows that $R$7 lies in the limit set of the zigzag rays from the $R$8 to any fixed end — proved case-by-case depending on whether $R$9 lies in $\Sigma$00, $\Sigma$01, or neither. This forces the limit section $\Sigma$02 to be ql-extremal over the zigzag path from $\Sigma$03 to a basepoint of $\Sigma$04, whence $\Sigma$05 and continuity follows. Equivariance comes from equivariance of $\Sigma$06 and $\Sigma$07; surjectivity from minimality of $\Sigma$08; compatibility with leaf extensions gives the Cannon–Thurston property in the sense of the paper's definition.

The dynamical corollary follows cleanly: periodic points of $\Sigma$09 on $\Sigma$10 are exactly $\Sigma$11, finite in number by uniform finite-to-one-ness, with sink/source behavior inherited from $\Sigma$12 via continuity of $\Sigma$13.

Limitations and open questions

Several dependencies should be noted. The uniform finite-to-one property of Cannon–Thurston maps is cited from forthcoming work [FMP] rather than proved here, so the dynamical corollary depends on that result. The main theorem is specific to the leftmost universal circle of the weak unstable foliation of a non-$\Sigma$14-covered Anosov flow; the $\Sigma$15-covered case is handled separately and essentially reduces to known constructions via regulating pseudo-Anosov flows. The proof also relies on hyperbolicity of $\Sigma$16 in an essential way (finiteness of master sets, Gromov-hyperbolicity of $\Sigma$17), and the extension of the framework to other foliations or universal circles is not addressed.

Two questions remain open. First, whether $\Sigma$18 is isomorphic as a universal circle — not merely as a circle with an action — to the flowspace boundary of an almost transverse pseudo-Anosov flow, which would require intertwining the monotone maps of the universal circle structure. Second, the Landry–Minsky–Taylor conjecture that all universal circles for non-$\Sigma$19-covered foliations arise from almost pseudo-Anosov flowspace boundaries; the theorem is offered as evidence but does not settle either question.

Conclusion

The paper constructs the first Cannon–Thurston map from a universal circle that is provably not conjugate to a flowspace boundary, answering Question (main) affirmatively for the leftmost universal circle of a non-$\Sigma$20-covered Anosov foliation. The method — combining the classification of sections by their bases with quadrant-local extremality linking section limits to master sets — provides a template that may apply to other universal circles, and the resulting pseudo-Anosov-type dynamics on $\Sigma$21 strengthens the case for the flowspace-boundary conjecture in this setting.