Ergodic measures of intermediate entropies for $\mathbb{Z}^{d}$-action

Ercai Chen; Xiaoyao Zhou; Yage Liu

arxiv: 2605.21048 · v1 · pith:VMS5SZEJnew · submitted 2026-05-20 · 🧮 math.DS

Ergodic measures of intermediate entropies for mathbb{Z}^(d)-action

Yage Liu , Ercai Chen , Xiaoyao Zhou This is my paper

Pith reviewed 2026-05-21 02:02 UTC · model grok-4.3

classification 🧮 math.DS

keywords ergodic measuresintermediate entropyZ^d actionsKatok conjectureinvariant measuresentropy expansivenessproduct propertygenericity

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The pith

For systems with the approximate Z^d-product property and asymptotic entropy expansiveness, ergodic measures with any given intermediate entropy are generic in natural subspaces of the invariant measures.

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

The paper proves that dynamical systems obeying the approximate Z^d or Z_+^d-product property and asymptotic entropy expansiveness have a highly structured space of invariant measures. In this structure, the ergodic measures that realize any fixed entropy value strictly between the global minimum and maximum form a generic subset inside certain natural subspaces. A sympathetic reader cares because this supplies not only existence but typicality of ergodic measures at every entropy level, directly settling Katok's conjecture for the systems in question and clarifying how orbit complexity distributes across the full entropy range.

Core claim

Under the assumptions of the approximate Z^d or Z_+^d-product property together with asymptotic entropy expansiveness, the space of invariant measures admits a precise description in which, for every intermediate entropy value h, the ergodic measures with entropy exactly h are generic in appropriate natural subspaces. This structural result immediately yields the existence of ergodic measures of every intermediate entropy and thereby confirms Katok's conjecture for all such systems.

What carries the argument

The approximate Z^d-product property combined with asymptotic entropy expansiveness, which together produce a decomposition of the invariant-measure space allowing genericity of fixed-entropy ergodic measures inside natural subspaces.

If this is right

Ergodic measures exist for every possible intermediate entropy value.
The invariant-measure space decomposes into subspaces indexed by entropy in which ergodic measures are dense and typical.
Katok's conjecture on intermediate-entropy ergodic measures holds for all systems obeying the stated hypotheses.
The global structure of the measure space is determined by these genericity properties rather than by isolated examples.

Where Pith is reading between the lines

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

The same genericity statements may extend to other countable amenable group actions once an analogous product property is established.
Thermodynamic formalism for these systems could be refined by using the subspace decomposition to locate equilibrium states at every entropy level.
Explicit constructions in symbolic dynamics or cellular automata might now be guided by the requirement that the product property holds, yielding new families of examples with full intermediate-entropy spectra.

Load-bearing premise

The dynamical systems must satisfy the approximate Z^d or Z_+^d-product property and be asymptotically entropy expansive.

What would settle it

A concrete counterexample would be any explicit Z^2 or Z_+^2 action that satisfies both the approximate product property and asymptotic entropy expansiveness yet has some intermediate entropy value h for which the ergodic measures of entropy h fail to be generic in the corresponding subspace.

read the original abstract

For dynamical systems satisfying the approximate $\mathbb{Z}^{d}$ or $\mathbb{Z}_+^{d}$-product property and asymptotically entropy expansiveness, we establish a precise description of the structure of their space of invariant measures. In particular, we prove that the set of ergodic measures with any given intermediate entropy is generic in certain natural subspaces. As a consequence, this result confirms Katok's conjecture on the existence of ergodic measures with arbitrary intermediate entropy for such systems.

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 confirms Katok's conjecture for Z^d actions meeting the approximate product property and asymptotic entropy expansiveness by showing genericity of intermediate-entropy ergodic measures via standard constructions.

read the letter

The key takeaway from this paper is that it confirms Katok's conjecture on the existence of ergodic measures with arbitrary intermediate entropies for Z^d-actions that satisfy the approximate product property and are asymptotically entropy expansive. It does this by proving that such measures are actually generic in appropriate subspaces. What the authors do well is lay out a structural description of the invariant measures under these conditions. They build on entropy approximation by periodic orbits, extend controls to product systems, and deploy Baire category arguments to show density and genericity in the weak* topology. This leads neatly to the existence result as a byproduct. The approach feels grounded in the existing literature on topological dynamics for higher-rank actions, and the stress-test indicates no hidden circularity or unjustified leaps in the logic. The result is new as a completed confirmation for this class, going beyond earlier partial results on entropy in Z^d settings. It gives a precise picture rather than just existence. On the soft side, the assumptions are quite specific. The approximate product property isn't something every system has, so while the proof works here, it leaves open how much this tells us about systems without it. Also, for higher d, one might wonder if the genericity holds uniformly or if there are dimension-dependent subtleties, though nothing in the outline flags a problem. The paper doesn't seem to introduce major new techniques, which is fine for a conjecture confirmation but means the impact depends on how broadly the assumptions apply. This kind of work is for specialists in ergodic theory and dynamical systems, particularly those interested in entropy, invariant measures, and multidimensional actions. A reader who knows the background on Katok's conjecture would appreciate the closure it provides for this case. I would bring it to a reading group for discussion on the Baire category application. It deserves serious peer review because it addresses an open question with a direct and apparently consistent argument. Referees could check the technical details on the product extensions and genericity.

Referee Report

0 major / 2 minor

Summary. The paper proves that for dynamical systems satisfying the approximate Z^d or Z_+^d-product property together with asymptotic entropy expansiveness, the ergodic measures realizing any prescribed intermediate entropy value form a generic subset in certain natural subspaces of the space of invariant measures. This structure is derived via entropy approximation by periodic measures, control of entropy on product extensions, and Baire-category arguments in the weak* topology, yielding existence of such measures as a corollary and thereby confirming Katok's conjecture for this class of systems.

Significance. If the derivations hold, the result supplies a precise description of the entropy spectrum for invariant measures of Z^d-actions under the stated hypotheses, extending known results from Z-actions to higher dimensions. The approach relies on standard, direct constructions in topological dynamics without ad-hoc parameters or circular reductions, providing a clean confirmation of the conjecture for systems meeting the approximate product and entropy expansiveness conditions. This constitutes a solid contribution to the study of entropy spectra and generic properties in multidimensional dynamics.

minor comments (2)

The introduction would benefit from a brief explicit statement of the precise subspaces in which genericity is claimed, to make the main theorem immediately accessible without cross-referencing later sections.
Notation for the approximate product property is introduced clearly but could include a short comparison table with the classical specification property to aid readers transitioning from Z-action literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our manuscript. The summary accurately captures our main results on the genericity of ergodic measures with prescribed intermediate entropies for systems satisfying the approximate product property and asymptotic entropy expansiveness, and we appreciate the recognition that this confirms Katok's conjecture for the indicated class of Z^d-actions. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives its main theorem on the structure of invariant measures and genericity of ergodic measures with prescribed intermediate entropy directly from the given assumptions of the approximate Z^d or Z_+^d-product property together with asymptotic entropy expansiveness. It employs standard constructions from topological dynamics (entropy approximation by periodic measures, entropy control on product extensions, and Baire-category arguments in the weak* topology) without any reduction of the central claims to fitted parameters, self-definitional equivalences, or load-bearing self-citations. The argument is self-contained and independent of the target result, confirming existence as a corollary without circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are identifiable. The result relies on background assumptions in ergodic theory such as the definitions of product properties and entropy expansiveness, which are standard but not detailed here.

pith-pipeline@v0.9.0 · 5603 in / 1073 out tokens · 23125 ms · 2026-05-21T02:02:54.499579+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

Theorem 1.4: asymptotically entropy expansive system with approximate L-product property implies entropy-generic (Me(X,T,α) residual in Mα(X,T))
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Definition 2.13 and Prop 2.15: approximate L-product property implies entropy-dense

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

26 extracted references · 26 canonical work pages

[1]

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F. B\'eguin, S. Crovisier, F. Le Roux, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique. Ann. Sci. \'Ec. Norm. Sup\'er. 40 (4)(2007)251-308

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[2]

Bowen, Periodic points and measures for Axiom A diffeomorphisms

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms. Transl. Am. Math. Soc. 154 (1971) 377-397

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[3]

Burguet, Topological and almost Borel universality for systems with the weak specification property

D. Burguet, Topological and almost Borel universality for systems with the weak specification property. Ergod. Theory Dyn. Syst. 40 (8) (2020) 2098-2115

work page 2020
[4]

Chandgotia, T

N. Chandgotia, T. Meyerovitch, Borel subsystems and ergodic universality for compact Z ^d -systems via specification and beyond. Proc. Lond. Math. Soc. 123 (2021) 231-312

work page 2021
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Constantine, J

D. Constantine, J. Lafont, D. J. Thompson, The weak specification property for geodesic flows on CAT( -1 ) spaces, Groups Geom. Dyn. 14 (1) (2020) 297-336

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Glasner, B

E. Glasner, B. Weiss, Strictly ergodic, uniform positive entropy models.Bull. Soc. Math. Fr. 122 (3) (1994) 399-412

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L. Guan, P. Sun, W. Wu, Measures of intermediate entropies and homogeneous dynamics.Nonlinearity.30 (2017) 3349-3361

work page 2017
[8]

F. Hahn, Y. Katznelson, On the entropy of uniquely ergodic transformations. Trans. Am. Math. Soc. 126 (1967) 335-360

work page 1967
[9]

Huang, L

W. Huang, L. Xu, S. Xu, Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electron. Res. Arch. 29 (4) (2021) 2819-2827

work page 2021
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Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms

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work page 1980
[11]

Konieczny, M

J. Konieczny, M. Kupsa, D. Kwietniak, Arcwise connectedness of the set of ergodic measures of hereditary shifts. Proc. Am. Math. Soc. 146 (8) (2018) 3425-3438

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Kwietniak, M

D. Kwietniak, M. Lacka, P. Oprocha, A panorama of specification-like properties and their consequences. Contemp. Math. 669 (2016) 155-186

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[13]

J. Li, P. Oprocha, Properties of invariant measures in dynamical systems with the shadowing property. Ergod. Theory Dyn. Syst. 38 (2018) 2257-2294

work page 2018
[14]

Lindenstrauss, G

J. Lindenstrauss, G. Olsen, Y.Sternfeld, The Poulsen simplex. Ann. Inst. Fourier (Grenoble). 28 (1),vi, 91-114 (1978)

work page 1978
[15]

Pfister, W.G

C.-E. Pfister, W.G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the -shifts. Nonlinearity. 18 (2005) 237–261

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Phelps, Lectures on Choquet’s Theorem, second ed., Lecture Notes in Mathematics, vol

R.R. Phelps, Lectures on Choquet’s Theorem, second ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001

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A. Quas, T. Soo, Ergodic universality of some topological dynamical systems. Trans. Am. Math. Soc. 368 (6) (2016) 4137-4170

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X. Ren, W. Sun: Local entropy, metric entropy and topological entropy for countable discrete amenable group actions. Int. J. Bifurc. Chaos. 26 (07) (2016) 1650110

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Sun, Zero-entropy invariant measures for skew product diffeomorphisms.Ergod

P. Sun, Zero-entropy invariant measures for skew product diffeomorphisms.Ergod. Theory Dyn. Syst. 30 (2010) 923-930

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Sun, Measures of intermediate entropies for skew product diffeomorphisms.Discrete Contin

P. Sun, Measures of intermediate entropies for skew product diffeomorphisms.Discrete Contin. Dyn. Syst., Ser.A 27 (3) (2010) 1219-1231

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Sun, Density of metric entropies for linear toral automorphisms

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Sun, Equilibrium states of intermediate entropies

P. Sun, Equilibrium states of intermediate entropies. Dyn. Syst. 36 (1) (2021) 69-78

work page 2021
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Sun, Ergodic measures of intermediate entropies for dynamical systems with approximate product property

P. Sun, Ergodic measures of intermediate entropies for dynamical systems with approximate product property. Adv. Math. 465 (8) (2025) 110159

work page 2025
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Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part

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[25]

Walters, An Introduction to Ergodic Theory

P. Walters, An Introduction to Ergodic Theory. Springer-Verlag, 1982

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D. Yang, J. Zhang, Non-hyperbolic ergodic measures and horseshoes in partially hyperbo lic homoclinic classes. J. Inst. Math. Jussieu. 19 (5) (2020) 1765-1792

work page 2020

[1] [1]

B\'eguin, S

F. B\'eguin, S. Crovisier, F. Le Roux, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique. Ann. Sci. \'Ec. Norm. Sup\'er. 40 (4)(2007)251-308

work page 2007

[2] [2]

Bowen, Periodic points and measures for Axiom A diffeomorphisms

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms. Transl. Am. Math. Soc. 154 (1971) 377-397

work page 1971

[3] [3]

Burguet, Topological and almost Borel universality for systems with the weak specification property

D. Burguet, Topological and almost Borel universality for systems with the weak specification property. Ergod. Theory Dyn. Syst. 40 (8) (2020) 2098-2115

work page 2020

[4] [4]

Chandgotia, T

N. Chandgotia, T. Meyerovitch, Borel subsystems and ergodic universality for compact Z ^d -systems via specification and beyond. Proc. Lond. Math. Soc. 123 (2021) 231-312

work page 2021

[5] [5]

Constantine, J

D. Constantine, J. Lafont, D. J. Thompson, The weak specification property for geodesic flows on CAT( -1 ) spaces, Groups Geom. Dyn. 14 (1) (2020) 297-336

work page 2020

[6] [6]

Glasner, B

E. Glasner, B. Weiss, Strictly ergodic, uniform positive entropy models.Bull. Soc. Math. Fr. 122 (3) (1994) 399-412

work page 1994

[7] [7]

L. Guan, P. Sun, W. Wu, Measures of intermediate entropies and homogeneous dynamics.Nonlinearity.30 (2017) 3349-3361

work page 2017

[8] [8]

F. Hahn, Y. Katznelson, On the entropy of uniquely ergodic transformations. Trans. Am. Math. Soc. 126 (1967) 335-360

work page 1967

[9] [9]

Huang, L

W. Huang, L. Xu, S. Xu, Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electron. Res. Arch. 29 (4) (2021) 2819-2827

work page 2021

[10] [10]

Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. IHES 51 (1980) 137-173

work page 1980

[11] [11]

Konieczny, M

J. Konieczny, M. Kupsa, D. Kwietniak, Arcwise connectedness of the set of ergodic measures of hereditary shifts. Proc. Am. Math. Soc. 146 (8) (2018) 3425-3438

work page 2018

[12] [12]

Kwietniak, M

D. Kwietniak, M. Lacka, P. Oprocha, A panorama of specification-like properties and their consequences. Contemp. Math. 669 (2016) 155-186

work page 2016

[13] [13]

J. Li, P. Oprocha, Properties of invariant measures in dynamical systems with the shadowing property. Ergod. Theory Dyn. Syst. 38 (2018) 2257-2294

work page 2018

[14] [14]

Lindenstrauss, G

J. Lindenstrauss, G. Olsen, Y.Sternfeld, The Poulsen simplex. Ann. Inst. Fourier (Grenoble). 28 (1),vi, 91-114 (1978)

work page 1978

[15] [15]

Pfister, W.G

C.-E. Pfister, W.G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the -shifts. Nonlinearity. 18 (2005) 237–261

work page 2005

[16] [16]

Phelps, Lectures on Choquet’s Theorem, second ed., Lecture Notes in Mathematics, vol

R.R. Phelps, Lectures on Choquet’s Theorem, second ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001

work page 2001

[17] [17]

A. Quas, T. Soo, Ergodic universality of some topological dynamical systems. Trans. Am. Math. Soc. 368 (6) (2016) 4137-4170

work page 2016

[18] [18]

X. Ren, W. Sun: Local entropy, metric entropy and topological entropy for countable discrete amenable group actions. Int. J. Bifurc. Chaos. 26 (07) (2016) 1650110

work page 2016

[19] [19]

Sun, Zero-entropy invariant measures for skew product diffeomorphisms.Ergod

P. Sun, Zero-entropy invariant measures for skew product diffeomorphisms.Ergod. Theory Dyn. Syst. 30 (2010) 923-930

work page 2010

[20] [20]

Sun, Measures of intermediate entropies for skew product diffeomorphisms.Discrete Contin

P. Sun, Measures of intermediate entropies for skew product diffeomorphisms.Discrete Contin. Dyn. Syst., Ser.A 27 (3) (2010) 1219-1231

work page 2010

[21] [21]

Sun, Density of metric entropies for linear toral automorphisms

P. Sun, Density of metric entropies for linear toral automorphisms. Dyn. Syst. 27 (2) (2012) 197-204

work page 2012

[22] [22]

Sun, Equilibrium states of intermediate entropies

P. Sun, Equilibrium states of intermediate entropies. Dyn. Syst. 36 (1) (2021) 69-78

work page 2021

[23] [23]

Sun, Ergodic measures of intermediate entropies for dynamical systems with approximate product property

P. Sun, Ergodic measures of intermediate entropies for dynamical systems with approximate product property. Adv. Math. 465 (8) (2025) 110159

work page 2025

[24] [24]

Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part

R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part. Proc. Am. Math. Soc.140 (6) (2012) 1973-1985

work page 2012

[25] [25]

Walters, An Introduction to Ergodic Theory

P. Walters, An Introduction to Ergodic Theory. Springer-Verlag, 1982

work page 1982

[26] [26]

D. Yang, J. Zhang, Non-hyperbolic ergodic measures and horseshoes in partially hyperbo lic homoclinic classes. J. Inst. Math. Jussieu. 19 (5) (2020) 1765-1792

work page 2020