Myrberg Points: Dynamics and Boundaries

• Myrberg points are distinguished boundary points that mark maximal topological and dynamical density of orbit images in various geometric contexts.
• They connect conicality, recurrence, and geodesic flow conservativity through tools like Patterson–Sullivan measures and BMS constructions in settings such as hyperbolic, CAT(0), and projective spaces.
• Recent developments extend their role to uniform visibility manifolds and convergence boundaries while linking Hausdorff dimension and topological freeness in discrete group actions.

Myrberg points, or the Myrberg limit set, are distinguished boundary points for discrete group actions and geodesic flows that encode maximal topological and dynamical density of orbit images of geodesic rays on the boundary. In classical hyperbolic and Kleinian theory, the relevant measure-theoretic and dynamical statements were developed in works of Myrberg, Agard, Tukia, and Stratmann; papers extend the framework to uniform visibility manifolds without conjugate points, convergence boundaries for groups with contracting elements, projective Veronese actions on $\mathbb{C}P^n$, and visual, horofunction, and Roller boundaries of $\operatorname{CAT}(0)$ spaces (Liu, 2024, Yang, 2022, Ucan-Puc et al., 2021, Ma et al., 7 Oct 2025). Across these settings, Myrberg points are closely tied to conicality, Patterson–Sullivan theory, Hopf–Tsuji–Sullivan type dichotomies, Hausdorff dimension, and topological freeness.

1. Definitions across geometric settings

In the setting of a simply connected uniform visibility manifold without conjugate points $X$, with $\Gamma \leq \operatorname{Iso}(X)$ discrete and $M=\Gamma\backslash X$, the Myrberg limit set is defined dynamically by

$$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$

This definition is adapted to visibility manifolds and generalizes the classical hyperbolic definition (Liu, 2024).

For groups with contracting elements acting properly discontinuously by isometries on a proper geodesic metric space $(X,d)$, the paper on convergence boundaries formulates Myrberg points on a boundary $\partial\overline X$ equipped with a $G$-invariant partition $[\cdot]$. A non-pinched $\operatorname{CAT}(0)$0 is Myrberg if for any $\operatorname{CAT}(0)$1 and any two distinct $\operatorname{CAT}(0)$2 in the reduced boundary $\operatorname{CAT}(0)$3, there is a sequence $\operatorname{CAT}(0)$4 with $\operatorname{CAT}(0)$5 and $\operatorname{CAT}(0)$6. Equivalently, the $\operatorname{CAT}(0)$7-translates of the ordered pair $\operatorname{CAT}(0)$8 are dense in the double boundary of distinct pairs (Yang, 2022).

A formally different, but closely related, notion appears for discrete subgroups of $\operatorname{CAT}(0)$9. If $X$0 denotes the set of quasi-projective limits of sequences in $X$1, the Myrberg limit set is

$X$2

In this projective setting, the complement of the equicontinuity region coincides with the Myrberg limit set: $X$3 For Veronese groups this gives an explicit geometric realization in terms of tangent or osculating hyperplanes (Ucan-Puc et al., 2021).

For proper $X$4 spaces, the visual-boundary definition is again phrased through density of pair-orbits. If $X$5 acts properly by isometries on $X$6 and $X$7 is the limit set, then $X$8 is Myrberg if for some, equivalently any, $X$9, the pair-orbit

$\Gamma \leq \operatorname{Iso}(X)$0

is dense in $\Gamma \leq \operatorname{Iso}(X)$1 in the cone topology (Ma et al., 7 Oct 2025).

2. Conicality, recurrence, and orbit-density

A fundamental structural fact is that Myrberg points are conical in the visibility setting: if $\Gamma \leq \operatorname{Iso}(X)$2 is a simply connected uniform visibility manifold without conjugate points, then

$\Gamma \leq \operatorname{Iso}(X)$3

The proof uses uniform visibility, continuity of the endpoint map, and angle estimates to obtain bounded distance between a ray $\Gamma \leq \operatorname{Iso}(X)$4 and suitable orbit points $\Gamma \leq \operatorname{Iso}(X)$5, thereby exhibiting $\Gamma \leq \operatorname{Iso}(X)$6 as conical (Liu, 2024).

In the axiomatized theory of convergence boundaries, Myrberg points admit a sharper recurrence description. For every choice of three mutually independent non-pinched contracting elements $\Gamma \leq \operatorname{Iso}(X)$7, and for every integer $\Gamma \leq \operatorname{Iso}(X)$8 large enough, the Myrberg set is the countable intersection of all corresponding $\Gamma \leq \operatorname{Iso}(X)$9-conical sets. This bridges dynamical density and conical recurrence. At the same time, Myrberg points are not merely synonymous with conical points: fixed points of a contracting or hyperbolic element are conical but never Myrberg, and the Myrberg set is a strict subset of the conical set in general (Yang, 2022).

In proper Gromov hyperbolic and CAT$M=\Gamma\backslash X$0 settings, a loxodromic-axis characterization is available. A limit point $M=\Gamma\backslash X$1 is Myrberg if and only if there exists a geodesic ray $M=\Gamma\backslash X$2 and a sequence $M=\Gamma\backslash X$3 such that the intersection $M=\Gamma\backslash X$4 has diameter tending to infinity for some universal $M=\Gamma\backslash X$5 and for every loxodromic $M=\Gamma\backslash X$6. In this formulation, a Myrberg ray spends longer and longer segments near translates of every loxodromic axis, so Myrberg points correspond to transitive geodesic directions. The same source states that $M=\Gamma\backslash X$7 is contained in the conical set, but disjoint from the uniformly conical set unless $M=\Gamma\backslash X$8 acts convex-cocompactly (Mj et al., 5 Jun 2025).

For non-elementary, non-convex-cocompact Kleinian groups, the nonuniform aspect becomes explicit. Every Myrberg point is nonuniformly conical, and hence

$M=\Gamma\backslash X$9

This isolates Myrberg behavior from uniformly conical recurrence and links it to sublinear, but not uniformly bounded, return of orbit points to the geodesic ray (Choi, 2 Dec 2025).

3. Patterson–Sullivan theory, BMS measures, and HTS dichotomies

In the visibility-without-conjugate-points framework, the measure-theoretic theory begins with the Poincaré series

$$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$0

and the critical exponent

$$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$1

A $$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$2-dimensional Busemann density $$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$3 satisfies

$$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$4

and, under Axiom 2, one constructs the Bowen–Margulis–Sullivan measure from the boundary-pair measure

$$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$5

The central Myrberg-type dichotomy then states: $$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$6 Moreover, if $$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$7 is conservative, then

$$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$8

Thus, in the conservative regime, the Myrberg limit set is a full Patterson–Sullivan measure subset of the conical limit set (Liu, 2024).

The same paper places this result inside a Hopf–Tsuji–Sullivan dichotomy proved in the visibility/no-conjugate-points setting. For a complete uniform visibility manifold without conjugate points satisfying Axiom 2, the following are equivalent: $$L_m(\Gamma)\triangleq \left\{\xi\in X(\infty): \forall \eta\ne\eta'\in L(\Gamma),\ \forall x\in X,\ \exists \{\alpha_n\}\subset\Gamma\ \text{with}\ \alpha_n x\to \eta,\ \alpha_n \xi\to \eta'\right\}.$$9; the geodesic flow is conservative; the geodesic flow is ergodic; the $(X,d)$0-action on $(X,d)$1 is ergodic; and the Poincaré series diverges at the relevant exponent. This refines the classical hyperbolic picture by replacing the conical set with the finer Myrberg set as a detector of conservativity (Liu, 2024).

In the broader convergence-boundary framework, an analogous dichotomy holds for groups with contracting elements. Let $(X,d)$2 be a $(X,d)$3-dimensional $(X,d)$4-quasi-equivariant quasi-conformal density with $(X,d)$5. Then divergence of the Poincaré series at $(X,d)$6, positivity of the Myrberg set, fullness of the Myrberg set inside $(X,d)$7, positivity of the conical set, and fullness of the conical set inside $(X,d)$8 are equivalent. Under divergence type, the reduced Myrberg set supports unique and ergodic Patterson–Sullivan densities up to bounded factors (Yang, 2022).

Historically, this is the measure-theoretic content of the Tukia–Stratmann picture in a much broader setting. In hyperbolic manifolds, Tukia proved that the Myrberg limit set is a full Patterson–Sullivan measure subset of the conical limit set, and the recent visibility result extends that statement from constant or negative curvature to uniform visibility manifolds without conjugate points under Axiom 2 (Liu, 2024).

4. Hausdorff dimension and sublinear recurrence

A major recent development is the dimension theory of Myrberg sets. For a proper Gromov hyperbolic space $(X,d)$9 with visual metric $\partial\overline X$0, the paper “Hausdorff Dimension of non-conical and Myrberg limit sets” proves

$\partial\overline X$1

In CAT$\partial\overline X$2 settings, where one may take $\partial\overline X$3, this becomes

$\partial\overline X$4

The result confirms the Falk–Matsuzaki conjecture and removes the earlier finiteness hypothesis on the Bowen–Margulis–Sullivan measure. The same work also records that, in finitely generated geometrically infinite Kleinian groups, the uniformly conical set, the Myrberg set, and the non-conical set are mutually disjoint, and each has Hausdorff dimension $\partial\overline X$5 (Mj et al., 5 Jun 2025).

For non-elementary, non-convex-cocompact Kleinian groups acting on $\partial\overline X$6, a semigroup-based refinement identifies the Hausdorff dimension of the sublinearly conical Myrberg limit set. If

$\partial\overline X$7

then

$\partial\overline X$8

This yields a different proof of a theorem of M. Mj and W. Yang and does so through divergence-type semigroups and a Patterson–Sullivan theory for semigroups (Choi, 2 Dec 2025).

The semigroup construction is quantitative. For a suitably chosen divergence-type semigroup $\partial\overline X$9, the associated Patterson–Sullivan measure $G$0 satisfies the shadow principle

$G$1

for all $G$2, with explicit $G$3. The argument combines semiconvexity, a semigroup analogue of the Hopf–Tsuji–Sullivan dichotomy, Borel–Cantelli, and a Frostman-type estimate to show that $G$4-almost every point is both $G$5-Myrberg and sublinearly conical (Choi, 2 Dec 2025).

These two 2025 results separate measure from dimension in a precise way. One paper proves maximal Hausdorff dimension of the Myrberg set in full generality for proper Gromov hyperbolic actions; the other isolates a sublinear subclass and still obtains full dimension $G$6 in the non-convex-cocompact Kleinian case (Mj et al., 5 Jun 2025, Choi, 2 Dec 2025).

5. Projective representations and Veronese geometry

For Kleinian groups acting through the irreducible representation

$G$7

Myrberg points admit an explicit projective-geometric description. The action preserves the rational normal curve $G$8, and for $G$9 one has

$[\cdot]$0

where $[\cdot]$1 is the unique osculating or tangent hyperplane to $[\cdot]$2 at $[\cdot]$3. In words, Myrberg points are exactly the points lying on hyperplanes tangent to $[\cdot]$4 at the Veronese image of classical limit points of $[\cdot]$5 (Ucan-Puc et al., 2021).

This description is linked to quasi-projective limits and to the failure of equicontinuity. Any divergent sequence in $[\cdot]$6 admits subsequences converging to quasi-projective maps, and equicontinuity fails precisely along the projectivized kernel of the limit. The Myrberg set is therefore the union of those kernels. For Veronese groups, the kernel hyperplanes are precisely the osculating hyperplanes to the rational normal curve (Ucan-Puc et al., 2021).

When $[\cdot]$7 is convex-cocompact, the projective Myrberg and Kulkarni limit sets coincide: $[\cdot]$8 Equivalently,

$[\cdot]$9

The role of convex-cocompactness is that there are no parabolic elements, all limit points are conical, and every divergent sequence is of loxodromic type, so the $\operatorname{CAT}(0)$00-lemma applies uniformly (Ucan-Puc et al., 2021).

The case $\operatorname{CAT}(0)$01 recovers the classical picture. Then $\operatorname{CAT}(0)$02 is the identity, $\operatorname{CAT}(0)$03, tangent hyperplanes are points, and

$\operatorname{CAT}(0)$04

This identifies the projective definition with the classical one in dimension one (Ucan-Puc et al., 2021).

6. $\operatorname{CAT}(0)$05 boundaries, Coxeter groups, and topological freeness

In proper $\operatorname{CAT}(0)$06 spaces with rank-one elements, Myrberg points admit a recurrence characterization: $\operatorname{CAT}(0)$07 is Myrberg if and only if, for some or any $\operatorname{CAT}(0)$08, the ray $\operatorname{CAT}(0)$09 is recurrent to every rank-one $\operatorname{CAT}(0)$10 with arbitrary accuracy. The same paper proves that every Myrberg point is visible from any other boundary point, that Myrberg points are uncountably many, and that Myrberg points cannot be fixed by rank-one elements (Ma et al., 7 Oct 2025).

These properties feed directly into boundary dynamics. If $\operatorname{CAT}(0)$11 is non-elementary, acts properly by isometries on a proper $\operatorname{CAT}(0)$12 space $\operatorname{CAT}(0)$13 with a rank-one element, the elliptic radical $\operatorname{CAT}(0)$14 is trivial, and either the action is cocompact or $\operatorname{CAT}(0)$15 is geodesically complete, then there exists a Myrberg point $\operatorname{CAT}(0)$16 with $\operatorname{CAT}(0)$17. Consequently, the action $\operatorname{CAT}(0)$18 is topologically free; together with rank-one north–south dynamics and minimality, it is a topologically free strong boundary action (Ma et al., 7 Oct 2025).

For irreducible non-spherical non-affine Coxeter groups, the horofunction boundary carries an analogous theory. Rank-one elements exist and coincide with contracting isometries on the Cayley graph; north–south dynamics on the horofunction boundary has singleton attracting and repelling classes; the action on the horofunction boundary is minimal; and the fixed-point pairs of contracting elements are dense in the boundary square. The Myrberg limit sets in the visual boundary of the Davis complex and in the horofunction boundary of the Cayley graph are canonically $\operatorname{CAT}(0)$19-equivariantly homeomorphic. Under these hypotheses, the horofunction boundary action is minimal, topologically free, topologically amenable, and a strong boundary action (Ma et al., 7 Oct 2025).

For irreducible non-Euclidean $\operatorname{CAT}(0)$20 cube complexes, Myrberg points align with squeezing points in the sense used for Roller boundaries. If $\operatorname{CAT}(0)$21 is a proper irreducible $\operatorname{CAT}(0)$22 cube complex and $\operatorname{CAT}(0)$23 acts properly essentially, contains a rank-one element, and has trivial elliptic radical, then the action on the Roller limit set is topologically free; under the locally finite, essential, irreducible, non-Euclidean, finite-dimensional, proper cocompact hypotheses, the action on the Nevo–Sageev boundary is topologically free, topologically amenable, and a strong boundary action. The corresponding crossed products yield unital Kirchberg algebras satisfying the UCT, and the visual-boundary case gives new examples of $\operatorname{CAT}(0)$24-selfless groups and of exact, purely infinite, simple reduced crossed product $\operatorname{CAT}(0)$25-algebras (Ma et al., 7 Oct 2025).

Several open directions remain explicit in the visibility-manifold literature. They include whether $\operatorname{CAT}(0)$26 holds without the visibility assumption for manifolds without conjugate points, whether the non-wandering set characterization via endpoints in $\operatorname{CAT}(0)$27 remains valid if uniform visibility is dropped, ergodicity with respect to Liouville measure under Axiom 2 for closed surfaces without conjugate points of genus $\operatorname{CAT}(0)$28, and the equivalence of visibility and uniform visibility axioms in higher-dimensional compact manifolds without conjugate points (Liu, 2024).

This suggests that Myrberg points are best viewed not as a single rigid definition, but as a boundary-theoretic locus where strong recurrence, conical approximation, conformal density theory, and topological freeness meet. Across negatively curved, visibility, projective, and $\operatorname{CAT}(0)$29 contexts, they isolate the directions in which orbit dynamics are simultaneously most recurrent and most globally distributed.