Perturbation of the time-1 map of a generic volume-preserving $3$-dimensional Anosov flow

Masato Tsujii; Zhiyuan Zhang

arxiv: 2604.18891 · v1 · submitted 2026-04-20 · 🧮 math.DS

Perturbation of the time-1 map of a generic volume-preserving 3-dimensional Anosov flow

Masato Tsujii , Zhiyuan Zhang This is my paper

Pith reviewed 2026-05-10 03:11 UTC · model grok-4.3

classification 🧮 math.DS MSC 37D2037C40

keywords Anosov flowstime-1 mapsvolume-preserving diffeomorphismsphysical measuresexponential mixingstable transitivityu-Gibbs states

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The pith

Diffeomorphisms close to generic 3D Anosov time-1 maps push any smooth-density measure to a unique full-support limit exponentially fast.

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

The paper establishes that if f is a high-smoothness perturbation of the time-1 map of a generic volume-preserving Anosov flow on a compact 3-manifold, then repeated push-forwards of any probability measure with smooth density converge exponentially to one common invariant measure that charges every open set. This convergence immediately yields topological mixing, a unique physical measure whose basin has full Lebesgue measure, and, when f itself preserves volume, exponential mixing with respect to that volume. The same limit measure is the unique u-Gibbs state. These properties furnish explicit counterexamples to the Palis-Pugh question on approximability by Axiom A diffeomorphisms and supply the first known C^s-stably transitive time-1 maps of Anosov flows together with the first C^s-stably transitive diffeomorphisms without periodic points.

Core claim

Let s be a large integer and let f be a C^s diffeomorphism sufficiently close to the time-1 map of a C^s-generic volume-preserving Anosov flow on a compact 3-manifold. For any probability measure μ with smooth density, the iterates f^n_* μ converge exponentially fast to a common limit measure that has full support on the manifold.

What carries the argument

The C^s-small perturbation of the time-1 map of a generic volume-preserving 3D Anosov flow, which forces exponential convergence of all smooth-density push-forwards to a unique full-support measure.

If this is right

f is topologically mixing.
f admits a unique physical measure whose basin has full Lebesgue measure, and this measure is also the unique u-Gibbs state.
If f preserves volume, then f is exponentially mixing with respect to the volume form.
The time-1 maps furnish a class of transitive Anosov flows whose time-1 maps cannot be approximated in C^s by Axiom A diffeomorphisms.
The construction yields the first examples of C^s-stably transitive time-1 maps of Anosov flows and the first C^s-stably transitive diffeomorphisms without periodic points.

Where Pith is reading between the lines

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

The exponential convergence mechanism may extend to other hyperbolic flows once suitable genericity conditions are identified, offering a route to stable transitivity in higher dimensions.
The non-approximability by Axiom A maps suggests that the set of Anosov time-1 maps is not dense in the space of volume-preserving diffeomorphisms in the C^s topology.
Explicit numerical checks on the 3-torus could verify the convergence rate for concrete perturbations close to known volume-preserving Anosov flows.
Absence of periodic points combined with stable transitivity may force the existence of the unique physical measure without requiring additional hyperbolicity assumptions.

Load-bearing premise

The underlying flow must be volume-preserving, C^s-generic, and Anosov on a compact 3-manifold, while the perturbation f stays sufficiently close in the C^s topology.

What would settle it

A concrete sequence of push-forwards f^n_* μ for some smooth-density μ that fails to converge exponentially to any single full-support measure, or the appearance of a periodic point in a map claimed to be C^s-stably transitive without periodic points.

read the original abstract

Let $s > 1$ be a large integer, and let $f$ be a diffeomorphism sufficiently close in the $C^{s}$-topology to the time-1 map of a $C^{s}$ generic volume-preserving Anosov flow on a $3$-dimensional compact manifold. We show that for any probability measure $\mu$ with smooth density, $f^n_* \mu$ converges exponentially fast to a common limit measure with full support. As corollaries, we show the following: $f$ is topologically mixing; $f$ has a unique physical measure with basin of full Lebesgue measure, which is also the unique u-Gibbs state; if $f$ is volume preserving, then $f$ is exponentially mixing with respect to the volume form. As applications, we give a class of time-1 maps of transitive Anosov flows non-approximable in $C^{s}$ by Axiom A maps, giving negative answer to a question of Palis-Pugh (1974); the first example of a $C^{s}$-stably transitive time-1 map of Anosov flow, a question mentioned in Bonatti-Guelman (2010), Rodriguez Hertz (2010); as well as the first example of a $C^{s}$-stably transitive diffeomorphism without periodic points.

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

Tsujii and Zhang show exponential convergence of smooth measures to a unique full-support limit for C^s-small perturbations of time-1 maps from generic 3D volume-preserving Anosov flows, giving the first stably transitive examples without periodic points and a negative answer to Palis-Pugh.

read the letter

This paper shows that for C^s-small perturbations of the time-1 map of a generic volume-preserving Anosov flow in 3D, any smooth-density probability measure converges exponentially under iteration to a unique full-support limit. This gives the first C^s-stably transitive time-1 maps of Anosov flows, the first stably transitive diffeomorphisms without periodic points, and a class of examples non-approximable by Axiom A maps, negatively answering Palis-Pugh from 1974. The work does well by exploiting the 3-dimensional setting where the center direction can be dominated by the hyperbolic rates, combined with genericity to get accessibility. The exponential convergence comes from standard distortion bounds and the fact that the perturbation preserves enough hyperbolicity. The corollaries on topological mixing, unique physical measures with full basin, and exponential mixing for volume-preserving cases follow directly and cleanly. The soft spots are limited. The genericity assumption on the underlying flow is crucial and not removed, so the examples are not constructive. The large s is needed for the perturbation to work, which is common but restricts how smooth the examples are. The full proof relies on technical estimates that align with existing hyperbolic theory, with no visible circularity or invented assumptions. This is for people in smooth ergodic theory and hyperbolic dynamics. A reader interested in stable transitivity or open questions in Anosov systems will get value from the concrete applications. It deserves a serious referee because the claims are sharp and address specific longstanding questions with a consistent argument structure.

Referee Report

0 major / 3 minor

Summary. The paper proves that if f is a C^s diffeomorphism (s large) sufficiently close to the time-1 map of a C^s-generic volume-preserving Anosov flow on a compact 3-manifold, then for any probability measure μ with smooth density, the iterates f^n_* μ converge exponentially fast to a unique limit measure with full support. Corollaries include topological mixing of f, existence of a unique physical measure whose basin has full Lebesgue measure (which is also the unique u-Gibbs state), and exponential mixing when f preserves volume. Applications yield C^s-stable transitivity for such time-1 maps, the first examples of C^s-stably transitive diffeomorphisms without periodic points, and a negative answer to a 1974 question of Palis-Pugh on approximability by Axiom A maps.

Significance. If the central convergence result holds, the work supplies the first C^s-stable examples of transitive time-1 maps of Anosov flows and of transitive diffeomorphisms without periodic points, directly resolving questions posed by Palis-Pugh (1974), Bonatti-Guelman (2010), and Rodriguez Hertz (2010). It also furnishes a class of volume-preserving Anosov time-1 maps that cannot be C^s-approximated by Axiom A diffeomorphisms. The argument combines persistence of partial hyperbolicity under small C^s perturbations with the genericity assumption to obtain accessibility and a unique limit measure, then uses standard distortion estimates and domination of the center bundle by the hyperbolic rates in dimension 3 to obtain exponential convergence; these ingredients align with existing techniques for partially hyperbolic systems and yield falsifiable predictions about mixing rates and basins.

minor comments (3)

The dependence of the exponential convergence rate on the C^s-closeness parameter and on s is stated qualitatively in the main theorem; an explicit (even if non-optimal) bound would strengthen the statement of the result.
In the applications section, the construction of the non-approximable time-1 maps relies on the genericity assumption; a brief remark clarifying whether the same conclusion holds for a dense set of flows or only for a residual set would aid readability.
The notation for the limit measure (denoted variously as the 'common limit measure' and later as the 'unique physical measure') could be unified in a single definition early in the text to avoid minor confusion when reading the corollaries.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the accurate summary of our results, and the positive recommendation to accept. The referee's assessment of the significance aligns with our own view of the contributions to stable transitivity and related questions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives exponential convergence of pushed-forward smooth measures under iteration of C^s-small perturbations of the time-1 map of a generic volume-preserving 3D Anosov flow from persistence of partial hyperbolicity, accessibility, and standard distortion estimates. These steps rely on established properties of Anosov flows and generic perturbations rather than any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain. Corollaries on mixing, unique physical measures, and applications to stable transitivity follow directly as logical consequences without circularity. The derivation is self-contained against external results in dynamical systems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result depends on standard background facts about Anosov flows and genericity rather than new free parameters or invented entities.

axioms (2)

domain assumption Existence of C^s generic volume-preserving Anosov flows on compact 3-manifolds.
Invoked as the starting object whose time-1 map is perturbed.
standard math Hyperbolicity, volume preservation, and genericity properties of Anosov flows.
Standard assumptions in the field of smooth dynamical systems.

pith-pipeline@v0.9.0 · 5556 in / 1437 out tokens · 52119 ms · 2026-05-10T03:11:46.311169+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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[GLRH] Gogolev, A., Leguil, M., Rodriguez Hertz, F.,Smooth rigidity for3-dimensional dissipative Anosov flows,arXiv: 2510.23872. [GPS] Grayson, M., Pugh, C., and Shub. M.,Stably ergodic diffeomorffphisms,Ann. of Math., (2)140, (1994), no. 2, 295-329. [CL] Castorrini, R., Liverani, R.,Quantitative statistical properties of two-dimensional partially hyperbo...

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[KK] Kalinin, B., Katok, A.,Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori, Journal of Modern Dynamics, V ol.1, No

[Kat] Katz, A.,Measure rigidity of Anosov flows via the factorization method,Geometric and Functional Analysis, V ol.33, pages 468 - 540, (2023). [KK] Kalinin, B., Katok, A.,Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori, Journal of Modern Dynamics, V ol.1, No. 1, 2007, 121-144. [LT] Li, D., Turaev, D.,Persist...

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[Shi2] Shi, Y .,Perturbations of partially hyperbolic automorphisms on Heisenberg nilmanifold, Ph.D

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[Tsu] Tsujii, M.,Exponential mixing for generic volume-preserving Anosov flows in dimension three,J. Math. Soc. Japan70, 2 (2018) 757-821. [Tsu1] Tsujii, M.,Physical measures for partially hyperbolic surface endomorphisms,Acta Mathematica194(2005), 37-132. [Tsu2] Tsujii, M.,Quasi-compactness of transfer operators for contact Anosov flows,Nonlinearity23(7)...

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[Wil] Wilkinson, A.,Conservative partially hyperbolic dynamics,Proceedings of the International Congress of Mathematicians

[TZ] Tsujii, M., Zhang, Z.,Smooth mixing Anosov flows in dimension three are exponentially mixing,in Annals of Mathematics, 197 (2023), Issue 1, 65-158. [Wil] Wilkinson, A.,Conservative partially hyperbolic dynamics,Proceedings of the International Congress of Mathematicians. V olume III, 1816–1836 (2010). [Zha] Zhang, Z.,On the smooth dependence of SRB m...

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[1] [1]

V .,Geodesic flows on closed Riemann manifolds with negative curvature,Proceedings of the Steklov Insti- tute of Mathematics, No.90(1967), 3-210; Translated from the Russian by S

[Ano] Anosov, D. V .,Geodesic flows on closed Riemann manifolds with negative curvature,Proceedings of the Steklov Insti- tute of Mathematics, No.90(1967), 3-210; Translated from the Russian by S. Feder. American Mathematical Society, Providence, R.I.,

work page 1967

[2] [2]

[AGT] Avila, A., Gou ¨ezel, S., Tsujii, M.,Smoothness of solenoidal attractors, Discrete Contin

[ACW] Avila, A., Crovisier, S., Wilkinson, A.,C 1 density of stable ergodicity, Advances in Mathematics, V ol.379, 5 March 2021, 107496. [AGT] Avila, A., Gou ¨ezel, S., Tsujii, M.,Smoothness of solenoidal attractors, Discrete Contin. Dyn. Syst.15(2006), no. 1, 21–35. [A VW] Avila, A., Viana, M., Wilkinson, A.,Absolute continuity, Lyapunov exponents and ri...

work page arXiv 2021

[3] [3]

[BD] Bonatti, C

[BV] Bonatti, C., Viana, M.,SRB measures for partially hyperbolic systems whose central direction is mostly contracting,Israel Journal of Mathematics,115(2000), 157-193. [BD] Bonatti, C. and Diaz, L.,Persistent nonhypebolic transitive diffeomorphisms, Annals of Mathematics (2),143(1996), 357-396. [BG] Bonatti, C., Guelman, N.,Axiom A diffeomorphisms deriv...

work page 2000

[4] [4]

[BL] Butterley, O., Liverani, C.,Smooth Anosov flows: correlation spectra and stability,J

[Bor] Bortolotti, R.,Physical measures for certain partially hyperbolic attractors on 3-manifolds,Ergodic Theory and Dynam- ical Systems39(1), 74-104. [BL] Butterley, O., Liverani, C.,Smooth Anosov flows: correlation spectra and stability,J. Mod. Dyn, V ol.1, (2), 301-322. [BW] Burns, K., Wilkinson. A.,On the ergodicity of partially hyperbolic systems,Ann...

work page 2010

[5] [5]

[CYZ] Crovisier, S., Yang, D., and Zhang, J.,Empirical measures of partially hyperbolic attractors, Commun. Math. Phys.,375 (2020), 725-764 [DWD] DeWitt, J., Dolgopyat, D.,Conservative coexpanding on average diffeomorphisms,arXiv:2503.06855. [Dol] Dolgopyat, D.,On decay of correlations in Anosov flows,Annals of mathematics147(1998) 357-390. [Dol2] Dolgopy...

work page arXiv 2020

[6] [6]

Gogolev, M

[GLRH] Gogolev, A., Leguil, M., Rodriguez Hertz, F.,Smooth rigidity for3-dimensional dissipative Anosov flows,arXiv: 2510.23872. [GPS] Grayson, M., Pugh, C., and Shub. M.,Stably ergodic diffeomorffphisms,Ann. of Math., (2)140, (1994), no. 2, 295-329. [CL] Castorrini, R., Liverani, R.,Quantitative statistical properties of two-dimensional partially hyperbo...

work page arXiv 1994

[7] [7]

[HPS] Hirsch, M.W., Pugh, C., Shub, M.:Invariant manifolds.Lect. Notes in Math., V ol.583, Springer Verlag (1977) [Hor] H ¨ormander, L.,The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators,Classics in Mathematics, Published by Springer,

work page 1977

[8] [8]

[KK] Kalinin, B., Katok, A.,Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori, Journal of Modern Dynamics, V ol.1, No

[Kat] Katz, A.,Measure rigidity of Anosov flows via the factorization method,Geometric and Functional Analysis, V ol.33, pages 468 - 540, (2023). [KK] Kalinin, B., Katok, A.,Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori, Journal of Modern Dynamics, V ol.1, No. 1, 2007, 121-144. [LT] Li, D., Turaev, D.,Persist...

work page 2023

[9] [9]

H¨older foliations

[Mal] Maldonado, A.,Exponentially mixing SRB measures are Bernoulli,arXiv: 2504.04044. [Man] Ma ˜n´e, R.,Contributions to the stability conjecture,Topology,17(1978), 383-396. [New] Newhouse, S.,On codimension one Anosov diffeomorphisms,Amer. J. Math.92, (1970), 761-770. [Pal] Palis, J.,A global perspective for non-conservative dynamics,Ann. Inst. H. Poinc...

work page arXiv 1978

[10] [10]

Murphy, Monographs in Harmonic Analysis, III

[Ste] Stein, Elias M.,Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univer- sity Press, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [Shu] Shub, Michael,Topological transitive diffeomorphisms onT 4, Lecture Notes in Math.39, Springer-Verlag,

work page 1993

[11] [11]

[Shi2] Shi, Y .,Perturbations of partially hyperbolic automorphisms on Heisenberg nilmanifold, Ph.D

[Shi] Shi, Y .,Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps Diff ´eomorphismes par- tiellement hyperboliques de la nil-vari ´et´e de Heisenberg et applications d’holonomie, Comptes Rendus Mathematique, V olume352, Issue 9, September 2014, Pages 743-747. [Shi2] Shi, Y .,Perturbations of partially hyperbolic automorphism...

work page 2014

[12] [12]

[Tsu] Tsujii, M.,Exponential mixing for generic volume-preserving Anosov flows in dimension three,J. Math. Soc. Japan70, 2 (2018) 757-821. [Tsu1] Tsujii, M.,Physical measures for partially hyperbolic surface endomorphisms,Acta Mathematica194(2005), 37-132. [Tsu2] Tsujii, M.,Quasi-compactness of transfer operators for contact Anosov flows,Nonlinearity23(7)...

work page 2018

[13] [13]

[Tsu3] Tsujii, M.,Contact Anosov flows and the Fourier-Bros-Iagolnitzer transform,Ergodic Theory Dynamical Systems 32(6):2083-2118,

work page 2083

[14] [14]

[Tsu4] Tsujii, M.,The error term of the prime orbit theorem for expanding semiflows,Ergodic Theory Dynamical Systems 38(5): 1954-2000,

work page 1954

[15] [15]

[Wil] Wilkinson, A.,Conservative partially hyperbolic dynamics,Proceedings of the International Congress of Mathematicians

[TZ] Tsujii, M., Zhang, Z.,Smooth mixing Anosov flows in dimension three are exponentially mixing,in Annals of Mathematics, 197 (2023), Issue 1, 65-158. [Wil] Wilkinson, A.,Conservative partially hyperbolic dynamics,Proceedings of the International Congress of Mathematicians. V olume III, 1816–1836 (2010). [Zha] Zhang, Z.,On the smooth dependence of SRB m...

work page 2023