Patterson–Sullivan Measures Overview

• Patterson–Sullivan measures are boundary measures derived from renormalized orbit sums at the critical exponent, forming quasi-conformal densities across various geometric settings.
• They quantify orbit growth and ergodic behavior for discrete group actions in settings ranging from hyperbolic and relatively hyperbolic spaces to higher-rank flag varieties.
• Extensions include weighted versions, coarse cocycle formulations, and applications to entropy, geodesic counting, harmonic analysis, and rigidity theorems.

Patterson–Sullivan measures are boundary measures obtained by renormalizing orbit sums at the critical exponent of a discrete group action. In the classical rank-one setting they form conformal densities on visual or Gromov boundaries, with Radon–Nikodym derivative governed by the Busemann cocycle; in relatively hyperbolic settings they live naturally on the Bowditch boundary; in higher rank they are defined on partial flag manifolds, Furstenberg boundaries, and vector-valued horofunction compactifications via partial or full Iwasawa cocycles; and in abstract form they can be built from expanding coarse cocycles for convergence actions (Yang, 2013, Canary et al., 2023, Blayac et al., 2024, Kim et al., 30 Mar 2026).

1. Classical construction and conformal density

For a discrete group $\Gamma$ acting by isometries on a proper Gromov-hyperbolic or negatively curved space $X$, with basepoint $o\in X$, the Poincaré series is

$P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$

and the critical exponent is

$$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$

Patterson’s construction takes weak-* limits of normalized orbit measures as $s\downarrow \delta$, producing a family $\{\mu_x\}_{x\in X}$ on the boundary. In the rank-one formulation, this family is $\Gamma$-equivariant and conformal: $$\frac{d\mu_y}{d\mu_x}(\xi)=e^{-\delta\,\beta_\xi(y,x)},$$ where $\beta_\xi$ is the Busemann cocycle (Yang, 2013, Canary et al., 2023).

The measure is supported on the limit set, and the shadow lemma gives the basic comparison between boundary mass and orbit growth: for suitable $X$0,

$X$1

In continuous-cocycle settings this quasi-conformal relation becomes an exact equality, while in coarse settings it is replaced by uniform multiplicative bounds (Blayac et al., 2024). These estimates are the core mechanism behind orbit counting, entropy formulas, and boundary ergodicity.

Weighted versions replace distance by a potential-weighted cocycle. For a Hölder potential $X$2 on a negatively curved geodesic flow, the weighted Poincaré series

$X$3

has critical exponent $X$4, and the associated weighted Patterson–Sullivan density satisfies

$X$5

with $X$6 the Gibbs cocycle (Feng et al., 2018).

2. Relatively hyperbolic groups and cusp geometry

For a relatively hyperbolic group $X$7 with peripheral family $X$8, the natural boundary is the Bowditch boundary $X$9, independent of the chosen cusp-uniform hyperbolic model. It consists of conical limit points and bounded parabolic points. The associated Patterson–Sullivan family $o\in X$0 is a $o\in X$1-dimensional quasi-conformal density on $o\in X$2; in the Cayley graph formulation,

$o\in X$3

where $o\in X$4 is the horofunction cocycle at a conical point $o\in X$5 (Yang, 2013).

In the divergent case, these measures have no atoms, are unique in their quasi-conformal class, and are ergodic under the $o\in X$6-action. The same conclusions hold for cusp-uniform actions under divergent type or under the parabolic gap property, which enforces $o\in X$7 for every peripheral subgroup $o\in X$8 (Yang, 2013).

A central refinement is the Partial Shadow Lemma. Besides ordinary shadows

$o\in X$9

one introduces partial shadows $P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$0, which require a nearby $P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$1-transition point, i.e. a point not deeply trapped in a peripheral neighborhood. The resulting estimate

$P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$2

filters out geodesics that linger in cusps and yields uniform control adapted to relatively hyperbolic geometry (Yang, 2013).

These shadow estimates imply purely exponential ball growth and cone growth, support the construction of large geodesic trees with uniformly spaced transition points, and feed into small-cancellation quotients. One outcome is the existence of proper relatively hyperbolic quotients $P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$3 with $P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$4. In the cusp-uniform setting with parabolic gap property, there are proper relatively hyperbolic quotients with

$P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$5

For torsion-free geometrically finite Kleinian groups this gives

$P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$6

answering the question of Dal’bo–Peigné–Picaud–Sambusetti (Yang, 2013).

3. Higher-rank flag varieties, Anosov theory, and new compactifications

In higher rank, the boundary object is typically a partial flag variety $P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$7, not a visual boundary. For a discrete subgroup $P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$8 of a semisimple Lie group and a linear functional $P(s)=\sum_{g\in \Gamma} e^{-s\, d(o,go)},$9, the basic series is

$$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$0

where $$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$1 is the partial Cartan projection. A $$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$2-Patterson–Sullivan measure on $$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$3 satisfies

$$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$4

for the partial Iwasawa cocycle $$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$5 (Canary et al., 2023, Canary et al., 2023).

For transverse subgroups, this higher-rank theory includes rank-one groups, Anosov groups, and relatively Anosov groups. In the divergent case, the conformal measure at critical dimension is unique, the conical limit set has full measure, and the actions on the limit set and on the product boundary are ergodic; in the convergent case, the conical limit set has measure zero and the action is non-ergodic. The same framework proves a higher-rank analogue of the Hopf–Tsuji–Sullivan dichotomy and a variant of Burger’s Manhattan curve theorem, as well as entropy-gap results for proper subgroups (Canary et al., 2023).

For relatively Anosov groups, one obtains divergence at the critical exponent, uniqueness and ergodicity of $$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$6-Patterson–Sullivan measures, strict peripheral entropy gaps $$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$7, and strict concavity of $$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$8 along affine lines, except in the length-rigidity case (Canary et al., 2023).

Recent work enlarges the boundary picture. One direction constructs Patterson–Sullivan measures on vector-valued horofunction boundaries $$\delta=\inf\{ s>0:\,P(s)<\infty\} =\limsup_{R\to\infty}\frac{1}{R}\log\#\{g\in\Gamma:\,d(o,go)\le R\}.$$9 of a symmetric space, where existence is obtained directly from Patterson’s series, and then proves a shadow lemma, uniqueness, and ergodicity for transverse groups (Kim et al., 30 Mar 2026). Another direction constructs exact Patterson–Sullivan measures on the full Furstenberg boundary $s\downarrow \delta$0 for arbitrary Zariski dense $s\downarrow \delta$1-Anosov groups, including non-Borel Anosov groups, with exact conformality

$s\downarrow \delta$2

together with uniqueness, ergodicity, Bowen–Margulis–Sullivan measures on $s\downarrow \delta$3, and strict convexity results for critical exponents (Kim et al., 30 Mar 2026). For relatively Morse groups, a global shadow lemma uniform over centers deep in the cuspidal part gives local doubling estimates and criteria for comparing Patterson–Sullivan measures with Hausdorff measures of visual quasi-metrics (Kim et al., 18 Jun 2026).

4. Coarse cocycles and abstract Patterson–Sullivan systems

A broad abstraction replaces geometric Busemann or Iwasawa cocycles by a $s\downarrow \delta$4-coarse cocycle $s\downarrow \delta$5 for a non-elementary convergence action $s\downarrow \delta$6. If $s\downarrow \delta$7 is expanding, one associates a magnitude $s\downarrow \delta$8, a Poincaré series

$s\downarrow \delta$9

and Patterson measures obtained by a weighted Patterson $\{\mu_x\}_{x\in X}$0-trick. Any weak-* limit at the critical exponent is supported on the limit set and satisfies the coarse quasi-conformal bounds

$\{\mu_x\}_{x\in X}$1

The associated shadows $\{\mu_x\}_{x\in X}$2 obey

$\{\mu_x\}_{x\in X}$3

and in the divergent case the measure is ergodic, coarsely unique, and gives full mass to the conical limit set (Blayac et al., 2024).

This formalism unifies the classical hyperbolic Busemann cocycle, Cartan–Iwasawa cocycles on flag varieties, coarsely additive potentials, Green-metric cocycles from random walks, and visibility-space examples. In the continuous case, it also produces a canonical Bowen–Margulis–Sullivan measure on a flow space and yields the usual conservative and ergodic dichotomy (Blayac et al., 2024).

A further abstraction is the notion of a Patterson–Sullivan system: a group action on a compact metrizable space equipped with a quasi-invariant measure, a cocycle, a magnitude function, and a family of shadows satisfying differentiation and irreducibility axioms. Within this framework one proves a generalized Tukia measurable rigidity theorem: if two PS-systems are measurably intertwined and the pushed-forward measure is not singular, then the magnitudes satisfy a uniform bounded-distortion relation. This gives systematic rigidity and singularity criteria for random walks versus Patterson–Sullivan measures on hyperbolic boundaries, Teichmüller spaces, and higher-rank flag varieties (Kim et al., 22 May 2025).

5. Dynamical consequences: Bowen–Margulis measures, entropy, and counting

Patterson–Sullivan measures are the boundary input for Bowen–Margulis–Sullivan measures. In the transverse higher-rank setting, one defines a Gromov product $\{\mu_x\}_{x\in X}$4 and obtains an invariant measure on the flow space from

$\{\mu_x\}_{x\in X}$5

This measure is used to prove ergodicity and the Hopf–Tsuji–Sullivan dichotomy, and it underlies counting and rigidity arguments (Canary et al., 2023). In the Furstenberg-boundary theory for non-Borel Anosov groups, the analogous construction on $\{\mu_x\}_{x\in X}$6 yields a right $\{\mu_x\}_{x\in X}$7-invariant Bowen–Margulis–Sullivan measure on $\{\mu_x\}_{x\in X}$8, whose ergodicity is equivalent to divergence of the relevant Poincaré series (Kim et al., 30 Mar 2026).

In rank-one manifolds without focal points, Patterson–Sullivan measures on the ideal boundary produce the unique invariant measure of maximal entropy for the geodesic flow. The same construction yields exponential sphere growth,

$\{\mu_x\}_{x\in X}$9

and Margulis-type growth for primitive regular closed geodesics (Liu et al., 2018). In closed rank-one manifolds of nonpositive curvature, the weighted Patterson–Sullivan construction for Hölder potentials produces the unique equilibrium state under the pressure gap condition $\Gamma$0, and this explicit construction implies the Bernoulli property and weighted counting of free homotopy classes (Wu, 12 Sep 2025).

The weighted theory replaces the Busemann cocycle by the Gibbs cocycle and the ordinary critical exponent by $\Gamma$1. The resulting equilibrium measure is written in quasi-product form using the two boundary measures and the cocycle $\Gamma$2, and weighted closed geodesics equidistribute toward it (Wu, 12 Sep 2025). This places equilibrium states, entropy, and geodesic counting within the same boundary-measure framework as the classical maximal-entropy theory.

6. Fine regularity, harmonic analysis, probability, and rigidity

Several recent directions study the fine structure of Patterson–Sullivan measures rather than only their existence. For Zariski dense Anosov groups, the limit set is Ahlfors regular for intrinsic conformal premetrics

$\Gamma$3

and a Patterson–Sullivan measure coincides with the Hausdorff measure if and only if the defining linear form is symmetric under the opposition involution (Dey et al., 2024). By contrast, for geometrically finite Kleinian groups with cusp ranks $\Gamma$4, the Patterson–Sullivan measure is not proportional to the Hausdorff or packing measure of any gauge function, so mixed cusp ranks obstruct any single Hausdorff- or packing-gauge interpretation (Simmons, 2014).

At the level of metric regularity, the regularity dimensions of the Patterson–Sullivan measure of a geometrically finite Kleinian group are

$\Gamma$5

while the Assouad and lower dimensions of the limit set are

$\Gamma$6

The Assouad spectrum refines this further and detects simultaneous contributions of horoballs of different ranks, a phenomenon invisible to Hausdorff, box, and full Assouad dimensions (Fraser, 2017, Fraser et al., 2022).

On the analytic side, the Fourier transform of Patterson–Sullivan measures of convex cocompact real hyperbolic groups decays polynomially: $\Gamma$7 The proof combines an $\Gamma$8-flattening theorem with dynamical self-similarity, and the result has spectral-gap consequences through the fractal uncertainty principle (Khalil, 2024).

Probabilistic applications are equally strong. For stationary symmetric $\Gamma$9-stable random fields indexed by groups acting geometrically on CAT$$\frac{d\mu_y}{d\mu_x}(\xi)=e^{-\delta\,\beta_\xi(y,x)},$$0 spaces, symmetric spaces, or Teichmüller space, the extremal cocycle growth built from the Patterson–Sullivan cocycle controls the asymptotic behavior of partial maxima; non-vanishing extremal cocycle growth is equivalent to finiteness of the Bowen–Margulis measure under a non-arithmetic length-spectrum assumption (Athreya et al., 2018). Twisted Patterson–Sullivan measures, obtained by inserting a positive unitary representation into the Poincaré series, characterize co-amenability by equality of critical exponents under strong positive recurrence: $$\frac{d\mu_y}{d\mu_x}(\xi)=e^{-\delta\,\beta_\xi(y,x)},$$1 (Coulon et al., 2018). In the abstract PS-system framework, analogous rigidity shows that stationary measures of random walks are singular to Patterson–Sullivan measures except in precisely constrained geometric situations, including hyperbolic, Teichmüller, and higher-rank settings (Kim et al., 22 May 2025).

Across these developments, the persistent structure is the same: a boundary measure tied to orbit growth through a cocycle, a shadow principle translating geometry into measure estimates, and an ergodic or rigidity mechanism converting those estimates into asymptotic information. In that sense, Patterson–Sullivan measures remain a unifying instrument for growth, entropy, boundary dynamics, harmonic analysis, and rigidity far beyond their original rank-one formulation.