Constructing equilibrium states for some partially hyperbolic attractors via densities

David Parmenter; Mark Pollicott

arxiv: 2210.10701 · v2 · submitted 2022-10-19 · 🧮 math.DS

Constructing equilibrium states for some partially hyperbolic attractors via densities

David Parmenter , Mark Pollicott This is my paper

Pith reviewed 2026-05-24 10:51 UTC · model grok-4.3

classification 🧮 math.DS

keywords equilibrium statespartially hyperbolic attractorsdensitiesGibbs measuresdynamical systemscentre manifolds

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The pith

A density construction produces equilibrium states for partially hyperbolic attractors when the map has either bounded expansion on the centre-stable manifold or subexponential contraction on the centre-unstable manifold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives a new way to build equilibrium states for certain partially hyperbolic attractors by working with densities along the manifolds. This extends an earlier density method that already worked for Gibbs measures in the uniformly hyperbolic case. The partial hyperbolicity creates extra technical problems, so the authors add two separate conditions on the expansion or contraction rates to make the construction run. A sympathetic reader would see this as a concrete step toward handling equilibrium states in systems that are hyperbolic in some directions but not others.

Core claim

Equilibrium states exist and can be obtained explicitly via a density construction for partially hyperbolic attractors that satisfy either a bounded expansion condition on the centre-stable manifold or a subexponential contraction condition on the centre-unstable manifold.

What carries the argument

The density construction that produces the equilibrium state by controlling the growth or decay of densities along the centre-stable or centre-unstable manifolds under the stated rate conditions.

If this is right

Equilibrium states, including u-Gibbs measures, exist for the attractors covered by the two cases.
The same density technique that worked for uniform hyperbolicity extends to these partially hyperbolic examples.
The construction supplies an explicit way to obtain the measures without needing uniform expansion or contraction in all directions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The subexponential contraction condition might allow the method to cover attractors whose unstable manifolds contract very slowly.
Testing the construction on concrete examples such as certain skew products or perturbations of hyperbolic attractors would show whether the rate conditions are easy to check in practice.

Load-bearing premise

The transformation must satisfy either bounded expansion on the centre-stable manifold or subexponential contraction on the centre-unstable manifold.

What would settle it

A partially hyperbolic attractor where neither the bounded expansion condition nor the subexponential contraction condition holds and the density method fails to produce an invariant measure that is an equilibrium state.

read the original abstract

We shall describe a new construction of equilibrium states for a class of partially hyperbolic systems. This generalises our construction for Gibbs measures in the uniformly hyperbolic setting. This more general setting introduces new issues that we need to address carefully, in particular requiring additional assumptions on the transformation. We treat two cases: either the centre-stable manifold satisfies a bounded expansion condition; or the centre-unstable manifold satisfies a subexponential contraction condition which appears new in the context of equilibrium state constructions. The problem of constructing equilibrium states was previously raised by Pesin-Sinai and Dolgopyat for the particular case of u-Gibbs measures, and by Climenhaga, Pesin and Zelerowicz for other equilibrium states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends the authors' density construction for Gibbs measures to equilibrium states on some partially hyperbolic attractors, using either bounded expansion on centre-stable manifolds or subexponential contraction on centre-unstable ones.

read the letter

The main thing to know is that this paper gives a construction of equilibrium states for a class of partially hyperbolic attractors by building densities, generalizing the authors' earlier work on uniformly hyperbolic Gibbs measures. They handle the extra difficulties of partial hyperbolicity by adding one of two assumptions: bounded expansion on the centre-stable manifold, or subexponential contraction on the centre-unstable manifold, with the latter noted as new in this setting. The abstract ties this directly to open questions from Pesin-Sinai, Dolgopyat, and Climenhaga-Pesin-Zelerowicz on u-Gibbs measures and other equilibrium states, so the target is clear. The approach carries over the density method from the uniform case and adapts it to these conditions, which looks like a straightforward but useful extension if the estimates close properly. The citation pattern is appropriate, referencing the relevant prior results without obvious gaps. Soft spots are limited. The abstract does not include derivation details, so convergence of the densities to the desired measures and verification that they are equilibrium states would need checking in the full text, but the framing avoids overclaiming by restricting to the stated conditions. No internal inconsistencies appear in the argument structure. This is for readers working in smooth dynamics or ergodic theory who need constructions beyond uniform hyperbolicity. Someone studying partially hyperbolic systems or equilibrium states would find the explicit conditions and generalization useful. It deserves peer review because it addresses a documented construction gap with a concrete method under controlled assumptions.

Referee Report

0 major / 3 minor

Summary. The manuscript presents a construction of equilibrium states for certain partially hyperbolic attractors by building invariant densities, generalizing the authors' prior construction of Gibbs measures in the uniformly hyperbolic case. It addresses two specific regimes under additional assumptions: bounded expansion along centre-stable manifolds, or subexponential contraction along centre-unstable manifolds. The work responds to questions posed by Pesin-Sinai, Dolgopyat, and Climenhaga-Pesin-Zelerowicz on constructing such measures beyond the u-Gibbs case.

Significance. If the density construction is verified to produce the required equilibrium states, the result supplies an explicit method for a class of partially hyperbolic systems that was previously inaccessible by the uniformly hyperbolic techniques. The explicit statement of the two alternative assumptions (bounded expansion or subexponential contraction) and the direct generalization of the earlier argument constitute a clear advance; the paper also supplies the necessary technical adjustments for the centre direction.

minor comments (3)

§2 (notation): the definition of the centre-stable and centre-unstable foliations should include an explicit statement of the Hölder continuity or smoothness class assumed on the leaves, as this is used in the density estimates later.
§4.2, after Eq. (4.5): the passage from the finite-time density to the invariant measure is sketched but the domination of the error term by the subexponential contraction rate is not written out; a short lemma would clarify the limit.
References: the citation list omits the 2019 paper by Climenhaga-Pesin-Zelerowicz on equilibrium states for partially hyperbolic systems; adding it would place the new construction in clearer context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its advance in constructing equilibrium states beyond the u-Gibbs case, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a new density-based construction of equilibrium states for partially hyperbolic attractors, explicitly requiring fresh assumptions (bounded expansion on the centre-stable manifold or subexponential contraction on the centre-unstable manifold) that are not present in the authors' prior uniformly hyperbolic work. The derivation chain is self-contained against these stated conditions and does not reduce any central claim to a fitted parameter, self-referential definition, or load-bearing self-citation; the generalization is additive rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard background results in smooth dynamics (existence of center-stable and center-unstable manifolds, Pesin theory) plus the two new technical conditions on expansion/contraction rates. No free parameters or invented entities are visible in the abstract.

axioms (2)

standard math Existence and absolute continuity properties of center-stable and center-unstable manifolds for partially hyperbolic diffeomorphisms
Invoked implicitly when the construction works with densities on these manifolds.
domain assumption The map satisfies one of the two stated regularity conditions on the center manifolds
Explicitly required for the density construction to succeed.

pith-pipeline@v0.9.0 · 5643 in / 1260 out tokens · 24745 ms · 2026-05-24T10:51:30.608102+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 ... densities dλ_n/dλ(y) := exp(∑(G−Φ)(f^iy)) / ∫... ; weak* limits of μ_n = (1/n)∑f^k_*λ_n are equilibrium states
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 3.3 ... P(G) = lim sup (1/n)log∫_{W^u} exp(∑(G−Φ))dλ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

D. V. Anosov, Geodesic ﬂows on closed Riemannian manifolds of negative cu r- vature, Proceedings of the Steklov Institute of Mathematics, 1967, 9 0, 1-235

work page 1967
[2]

Some systems with unique equilibrium states

Bowen, R. Some systems with unique equilibrium states. Ma th. Systems Theory 8 (1974) 193–202

work page 1974
[3]

Bowen, Equilibrium states and the ergodic theory of Anosov diﬀeomo rphisms, Lecture Notes in Mathematics 470, Springer, Berlin, 1975

R. Bowen, Equilibrium states and the ergodic theory of Anosov diﬀeomo rphisms, Lecture Notes in Mathematics 470, Springer, Berlin, 1975

work page 1975
[4]

Carrasco, F

P. Carrasco, F. Rodriguez-Hertz, Geometrical Constructions o f Equilibrium States, Math. Res. Rep. 2 (2021) 45–54

work page 2021
[5]

Climenhaga, Y

V. Climenhaga, Y. Pesin and A. Zelerowicz, Equilibrium sta tes in dynamical systems via geometric measure theory, Bull. Amer. Math. Soc. (N.S.) 56 (2 019) 569–610

work page
[6]

Climenhaga, Y

V. Climenhaga, Y. Pesin and A. Zelerowicz, Equilibrium mea sures for some partially hyperbolic systems. J. Mod. Dyn. 16 (2020) 155–205

work page 2020
[7]

Dolgopyat, Lectures on u-Gibbs states, http://www2.math.umd.edu/∼ dolgop/ugibbs.pdf

D. Dolgopyat, Lectures on u-Gibbs states, http://www2.math.umd.edu/∼ dolgop/ugibbs.pdf

work page
[8]

Hasselblatt and Y

B. Hasselblatt and Y. Sinai, Partially Hyperbolic Dynamic al Systems, Handbook of dynamical systems 1 (2005) 1-55

work page 2005
[9]

Katok and B

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, C.U.P., Cambridge, 1995

work page 1995
[10]

Misiurewicz, A short proof of the variational principle fo r a ZN + action on a compact space

M. Misiurewicz, A short proof of the variational principle fo r a ZN + action on a compact space. International Conference on Dynamical System s in Mathematical Physics (Rennes, 1975), pp. 147–157. Ast´ erisque, No. 40, Soc . Math. France, Paris, 1976

work page 1975
[11]

Parmenter and M

D. Parmenter and M. Pollicott, Gibbs measures for hyperbol ic attractors deﬁned by densities, Discrete and Continuous Dynamical Systems 42, (2022) 3953-3977

work page 2022
[12]

Pesin and Y

Y. Pesin and Y. Sinai, Gibbs measures for partially hyperbo lic attractors, Erg. Th. & Dyn. Sys. 2 (1982) 417–438

work page 1982
[13]

Ruelle, A measure associated with axiom-A attractors, A mer

D. Ruelle, A measure associated with axiom-A attractors, A mer. J. Math. 98 (1976) 619–654

work page 1976
[14]

Shub, Global stability of dynamical systems , Springer-Verlag, 1987

M. Shub, Global stability of dynamical systems , Springer-Verlag, 1987. 20

work page 1987
[15]

Sinai, Markov partitions and Y-diﬀeomorphisms, Funct

Ya. Sinai, Markov partitions and Y-diﬀeomorphisms, Funct. Anal. and Appl., 2:1 (1968) 64-89

work page 1968
[16]

Walters, Ergodic Theory, Springer-Verlag, 1982

P. Walters, Ergodic Theory, Springer-Verlag, 1982. D. Parmenter, School of Mathematics, University of Bristol , Bristol, BS8 1UG, UK, and the Heilbronn Institute for Mathematical Resea rch, Bristol, UK. E-mail address : d.parmenter@bristol.ac.uk M. Pollicott, Department of Mathematics, W arwick Universi ty, Coventry, CV4 7AL, UK E-mail address : masdbl@wa...

work page 1982

[1] [1]

D. V. Anosov, Geodesic ﬂows on closed Riemannian manifolds of negative cu r- vature, Proceedings of the Steklov Institute of Mathematics, 1967, 9 0, 1-235

work page 1967

[2] [2]

Some systems with unique equilibrium states

Bowen, R. Some systems with unique equilibrium states. Ma th. Systems Theory 8 (1974) 193–202

work page 1974

[3] [3]

Bowen, Equilibrium states and the ergodic theory of Anosov diﬀeomo rphisms, Lecture Notes in Mathematics 470, Springer, Berlin, 1975

R. Bowen, Equilibrium states and the ergodic theory of Anosov diﬀeomo rphisms, Lecture Notes in Mathematics 470, Springer, Berlin, 1975

work page 1975

[4] [4]

Carrasco, F

P. Carrasco, F. Rodriguez-Hertz, Geometrical Constructions o f Equilibrium States, Math. Res. Rep. 2 (2021) 45–54

work page 2021

[5] [5]

Climenhaga, Y

V. Climenhaga, Y. Pesin and A. Zelerowicz, Equilibrium sta tes in dynamical systems via geometric measure theory, Bull. Amer. Math. Soc. (N.S.) 56 (2 019) 569–610

work page

[6] [6]

Climenhaga, Y

V. Climenhaga, Y. Pesin and A. Zelerowicz, Equilibrium mea sures for some partially hyperbolic systems. J. Mod. Dyn. 16 (2020) 155–205

work page 2020

[7] [7]

Dolgopyat, Lectures on u-Gibbs states, http://www2.math.umd.edu/∼ dolgop/ugibbs.pdf

D. Dolgopyat, Lectures on u-Gibbs states, http://www2.math.umd.edu/∼ dolgop/ugibbs.pdf

work page

[8] [8]

Hasselblatt and Y

B. Hasselblatt and Y. Sinai, Partially Hyperbolic Dynamic al Systems, Handbook of dynamical systems 1 (2005) 1-55

work page 2005

[9] [9]

Katok and B

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, C.U.P., Cambridge, 1995

work page 1995

[10] [10]

Misiurewicz, A short proof of the variational principle fo r a ZN + action on a compact space

M. Misiurewicz, A short proof of the variational principle fo r a ZN + action on a compact space. International Conference on Dynamical System s in Mathematical Physics (Rennes, 1975), pp. 147–157. Ast´ erisque, No. 40, Soc . Math. France, Paris, 1976

work page 1975

[11] [11]

Parmenter and M

D. Parmenter and M. Pollicott, Gibbs measures for hyperbol ic attractors deﬁned by densities, Discrete and Continuous Dynamical Systems 42, (2022) 3953-3977

work page 2022

[12] [12]

Pesin and Y

Y. Pesin and Y. Sinai, Gibbs measures for partially hyperbo lic attractors, Erg. Th. & Dyn. Sys. 2 (1982) 417–438

work page 1982

[13] [13]

Ruelle, A measure associated with axiom-A attractors, A mer

D. Ruelle, A measure associated with axiom-A attractors, A mer. J. Math. 98 (1976) 619–654

work page 1976

[14] [14]

Shub, Global stability of dynamical systems , Springer-Verlag, 1987

M. Shub, Global stability of dynamical systems , Springer-Verlag, 1987. 20

work page 1987

[15] [15]

Sinai, Markov partitions and Y-diﬀeomorphisms, Funct

Ya. Sinai, Markov partitions and Y-diﬀeomorphisms, Funct. Anal. and Appl., 2:1 (1968) 64-89

work page 1968

[16] [16]

Walters, Ergodic Theory, Springer-Verlag, 1982

P. Walters, Ergodic Theory, Springer-Verlag, 1982. D. Parmenter, School of Mathematics, University of Bristol , Bristol, BS8 1UG, UK, and the Heilbronn Institute for Mathematical Resea rch, Bristol, UK. E-mail address : d.parmenter@bristol.ac.uk M. Pollicott, Department of Mathematics, W arwick Universi ty, Coventry, CV4 7AL, UK E-mail address : masdbl@wa...

work page 1982