Gibbs u-states in Dynamical Systems

• Gibbs u-states are invariant measures whose conditional measures along unstable manifolds are absolutely continuous, serving as the unstable analogue of SRB measures.
• They are characterized by unstable Jacobians, entropy, and pressure, linking thermodynamic formalism with partially hyperbolic and non-invertible dynamics.
• Recent extensions generalize these measures to include Haar-type invariance on subresonant orbits, employing factorization methods to establish rigidity and uniqueness.

Gibbs u-states, also called u-Gibbs measures, are invariant probability measures for dynamical systems with unstable directions whose conditional measures along unstable manifolds are absolutely continuous with respect to the corresponding leafwise Riemannian measure. In partially hyperbolic dynamics they provide the unstable-direction analogue of SRB statistics, and in several settings they are characterized by unstable Jacobians, unstable entropy, or unstable pressure. The notion appears in invertible partially hyperbolic diffeomorphisms, non-invertible partially hyperbolic endomorphisms, foliated geodesic flows, and more recent rigidity frameworks based on subresonant normal forms and factorization methods (Hu et al., 2017, Cantarino et al., 2024, Alvarez, 2013, Brown et al., 19 Feb 2025).

1. Definition and foundational formalism

For a $C^1$ partially hyperbolic diffeomorphism $f \colon M \to M$ with invariant splitting

$TM = E^s \oplus E^c \oplus E^u,$

the unstable distribution integrates to an unstable foliation $W^u$. If $\eta$ is a measurable partition subordinate to unstable leaves, a Gibbs u-state is an $f$-invariant probability measure $\mu$ such that, for $\mu$-almost every $x$, the Rokhlin conditional measure $\mu_x^\eta$ is absolutely continuous with respect to the Riemannian leaf volume on $f \colon M \to M$0 (Hu et al., 2017).

In continuous-time settings the same principle is expressed through the unstable Jacobian. For the foliated geodesic flow $f \colon M \to M$1 on the unit tangent bundle of a foliation with negatively curved leaves, a Gibbs u-state is a $f \colon M \to M$2-invariant probability measure whose disintegration along local unstable leaves has densities equivalent to Lebesgue and determined by unstable Jacobian limits. Concretely, on a local unstable leaf,

$f \colon M \to M$3

The same work defines Gibbs su-states by requiring Lebesgue disintegration along both unstable and stable leaves (Alvarez, 2013).

The density formula along unstable leaves also survives in non-invertible partially hyperbolic endomorphisms, but only after passing to the inverse limit space. For a lift $f \colon M \to M$4 of $f \colon M \to M$5 and a $f \colon M \to M$6-subordinate partition $f \colon M \to M$7, the condition is

$f \colon M \to M$8

and the conditional density is given by

$f \colon M \to M$9

where $TM = E^s \oplus E^c \oplus E^u,$0 lies on the local unstable curve through $TM = E^s \oplus E^c \oplus E^u,$1 (Cantarino et al., 2024).

These formulations share a common structure: the relevant measure is invariant, leafwise conditional measures are controlled by unstable Jacobians, and the geometry is organized by measurable partitions subordinate to unstable manifolds.

2. Unstable entropy, unstable pressure, and u-equilibrium states

A thermodynamic formalism for unstable directions was developed for partially hyperbolic diffeomorphisms through unstable pressure $TM = E^s \oplus E^c \oplus E^u,$2. It is defined using $TM = E^s \oplus E^c \oplus E^u,$3 $TM = E^s \oplus E^c \oplus E^u,$4-separated subsets of local unstable leaves and the weighted sums

$TM = E^s \oplus E^c \oplus E^u,$5

where $TM = E^s \oplus E^c \oplus E^u,$6. The resulting unstable topological pressure satisfies the variational principle

$TM = E^s \oplus E^c \oplus E^u,$7

and equivalently the supremum may be taken over ergodic invariant measures (Hu et al., 2017).

Within this framework, a u-equilibrium state for $TM = E^s \oplus E^c \oplus E^u,$8 is an invariant measure attaining the supremum. The connection with Gibbs u-states is exact for the unstable Jacobian potential

$TM = E^s \oplus E^c \oplus E^u,$9

For $W^u$0,

$W^u$1

with equality if and only if $W^u$2 is a Gibbs u-state. Equivalently, Gibbs u-states are precisely the u-equilibrium states for $W^u$3, and

$W^u$4

This recasts the Gibbs u-property as saturation of an entropy-Jacobian inequality rather than only as a leafwise absolute continuity statement (Hu et al., 2017).

The same source develops a differentiability theory for unstable pressure. U-equilibrium states coincide with the $W^u$5-tangent functionals of $W^u$6, Gateaux differentiability at a potential is equivalent to uniqueness of the u-equilibrium state, and Fréchet differentiability yields a stronger stability statement excluding competing ergodic measures with comparable value of $W^u$7 near the distinguished equilibrium (Hu et al., 2017).

This thermodynamic interpretation is especially significant because it isolates the unstable contribution to statistical behavior while allowing center dynamics to remain nontrivial.

3. Non-invertible partially hyperbolic endomorphisms

For partially hyperbolic endomorphisms on $W^u$8, the main geometric complication is that unstable directions need not be globally defined on the base manifold. The standard resolution is to pass to the inverse limit

$W^u$9

equipped with the shift $\eta$0 and projection $\eta$1. On this natural extension there is a continuous $\eta$2-invariant splitting

$\eta$3

with one-dimensional bundles, $\eta$4, and $\eta$5. The center bundle is deterministic in the sense that it depends only on $\eta$6, whereas the unstable bundle may depend on the full past, that is, on the fiber $\eta$7 (Cantarino et al., 2024).

This past dependence leads to the notion of a special map: $\eta$8 is special if $\eta$9 whenever $f$0. Non-speciality is a $f$1-open, $f$2-dense condition in partially hyperbolic endomorphisms. The distinction is dynamical rather than cosmetic, because different backward histories can produce different unstable directions at the same base point (Cantarino et al., 2024).

The principal rigidity statement is Theorem A. Let $f$3 be partially hyperbolic and strongly transitive. If $f$4 is an ergodic u-Gibbs measure with positive center Lyapunov exponent

$f$5

and full support, and if $f$6 is non-special, then $f$7 is the unique absolutely continuous invariant measure. Equivalently, under these hypotheses one has a dichotomy: either the map is special, or the fully supported positive-center u-Gibbs measure is uniquely Lebesgue-equivalent on the base (Cantarino et al., 2024).

The same work derives concrete corollaries. If unstable leaves are dynamically minimal, then every u-Gibbs measure has full support, so any u-Gibbs measure with $f$8 is the unique ACIM for a non-special, strongly transitive map. For non-special perturbations of irreducible linear expanding toral endomorphisms with simple spectrum, uniqueness of the u-Gibbs measure follows, and this measure is also the unique ACIM and is physical. By contrast, the reducible linear map

$f$9

admits infinitely many u-Gibbs measures of the form $\mu$0, and a non-special perturbation of this reducible situation may still admit infinitely many u-Gibbs measures because compact unstable leaves may fail to have dense orbits (Cantarino et al., 2024).

4. Foliated geodesic flows and $\mu$1-harmonic measures

For a compact smooth manifold endowed with a smooth foliation $\mu$2 whose leaves have uniformly pinched negative curvature, the unit tangent bundle $\mu$3 carries a foliated geodesic flow $\mu$4. Inside each leaf, $\mu$5 has a foliated hyperbolic splitting

$\mu$6

together with stable and unstable horospheres described by the Busemann cocycle $\mu$7 (Alvarez, 2013).

In this context Gibbs u-states have leafwise Lebesgue disintegration along unstable horospheres, while Gibbs su-states have such disintegration along both stable and unstable horospheres. A central rigidity theorem states that if $\mu$8 admits a Gibbs su-state $\mu$9, then $\mu$0 is totally invariant: locally it is the product of a transverse invariant measure and the Liouville measures on the leaves of the lifted foliation. In particular, the existence of a Gibbs su-state forces existence of a transverse invariant measure for the foliation (Alvarez, 2013).

The same paper establishes a canonical bijection between Gibbs u-states for $\mu$1 and $\mu$2-harmonic measures on the base foliation. The relevant kernel is the Gibbs kernel

$\mu$3

and local plaque densities admit the Poisson-type representation

$\mu$4

When all leaves have constant curvature $\mu$5, one has $\mu$6 and the Gibbs kernel reduces to the classical Poisson kernel $\mu$7 (Alvarez, 2013).

This setting shows that Gibbs u-states are not limited to partially hyperbolic maps of compact manifolds. They also encode the statistical geometry of flows tangent to negatively curved foliations, where unstable Jacobians interact with boundary-at-infinity structures and transverse measure theory.

5. Normal forms, factorization, and generalized u-Gibbs states

Recent rigidity theory enlarges the classical notion of Gibbs u-state by allowing unstable conditional measures to be Haar on orbits of subgroups of strictly subresonant maps rather than absolutely continuous on the entire unstable manifold. In that framework, a measurable family

$\mu$8

is called compatible if it satisfies $\mu$9-equivariance, partition compatibility, holonomy consistency, and maximality; a measure is a generalized u-Gibbs state when its unstable conditional measures are proportional to Haar measure on the orbits $x$0 (Brown et al., 19 Feb 2025).

The crucial hypothesis is Quantitative Non-Integrability (QNI), which supplies transverse lower bounds between unstable objects and realized center-stable manifolds. Under QNI, the main extra-invariance theorem produces a strictly larger compatible family $x$1 for almost every $x$2. Iterating this enlargement yields measure-classification results for stationary measures of random walks generated by diffeomorphisms and for smooth $x$3-actions (Brown et al., 19 Feb 2025).

The technical engine is the factorization method: one passes to coset spaces $x$4, linearizes via a cocycle on a bundle $x$5, and analyzes carefully constructed Y-configurations and interpolation maps. This produces “friends” of leafwise conditional measures under strictly subresonant maps not contained in $x$6, from which extra invariance follows (Brown et al., 19 Feb 2025).

A closely related but more specialized factorization mechanism appears in the surface endomorphism setting. There the dynamics on center-unstable leaves are linearized by one- and two-dimensional normal forms, and one constructs leaf-wise quotient measures $x$7 on $x$8. These measures satisfy affine covariance rules under forward iteration, movement along center leaves, and movement along unstable leaves. Once sufficiently many affine invariances are obtained, a Kalinin–Katok criterion implies that $x$9 is equivalent to Lebesgue measure, which yields the SRB property and absolute continuity of the original u-Gibbs measure (Cantarino et al., 2024).

These developments indicate that the classical absolute continuity formulation of Gibbs u-states is part of a broader rigidity paradigm organized by unstable holonomies, normal forms, and leafwise symmetry.

6. Relations to SRB measures, uniqueness, and open directions

Across the cited settings, Gibbs u-states are tightly linked to SRB, absolutely continuous, and physical measures, but the precise relation depends on the ambient category. For partially hyperbolic diffeomorphisms, the unstable Jacobian potential $\mu_x^\eta$0 identifies Gibbs u-states with u-equilibrium states, thereby placing SRB-type measures inside unstable thermodynamic formalism (Hu et al., 2017). For surface endomorphisms, SRB on the inverse limit implies absolute continuity on the base, and under the hypotheses of Theorem A the u-Gibbs measure becomes the unique ACIM; in the uniformly expanding corollary this ACIM is also physical (Cantarino et al., 2024). For foliated geodesic flows, ergodic Gibbs u-states are described as the physical/SRB measures, while Gibbs su-states are rigid enough to force total invariance (Alvarez, 2013).

Several limitations are explicit. In the surface endomorphism theory, $\mu_x^\eta$1 regularity is used for normal forms and distortion control, strong transitivity and full support are used to eliminate bad fiberwise sets, and positivity of the center exponent is essential for Lyapunov norms and expansion estimates. Zero center exponent cases remain open, and the paper conjectures positive answers for mostly expanding behavior in the sense of Alves–Bonatti–Viana and Avila–Viana–Santamaria, as well as possible extensions to higher dimensions (Cantarino et al., 2024).

In the generalized setting, the theory assumes measurably good dynamics, holonomies, subresonant normal forms, and in many results finite support of the random walk measure, with removal of finite support deferred to later work. Open directions include broader group actions and sharper stationary-measure classification in symplectic settings (Brown et al., 19 Feb 2025). In unstable thermodynamic formalism, the differentiability criteria suggest routes to uniqueness of u-equilibrium states under additional hypotheses, but global uniqueness results of that kind are not claimed (Hu et al., 2017).

A plausible implication of these parallel developments is that Gibbs u-states now serve less as a single definition than as a family of closely related leafwise regularity conditions, ranging from full absolute continuity on unstable manifolds to Haar-type invariance on subresonant unstable orbits. What remains stable across these variants is the role of unstable geometry as the organizing principle for entropy, rigidity, and statistical behavior.