Ergodic optimization for continuous functions on non-Markov shifts

Hiroki Takahasi; Kenichiro Yamamoto; Mao Shinoda

arxiv: 2406.01123 · v2 · submitted 2024-06-03 · 🧮 math.DS

Ergodic optimization for continuous functions on non-Markov shifts

Mao Shinoda , Hiroki Takahasi , Kenichiro Yamamoto This is my paper

Pith reviewed 2026-05-24 00:34 UTC · model grok-4.3

classification 🧮 math.DS

keywords ergodic optimizationsubshiftsmaximizing measuresentropyintrinsically ergodicsymbolic dynamicsnon-Markov shifts

0 comments

The pith

For intrinsically ergodic subshifts the continuous functions split into two classes with distinct properties for the entropy of their maximizing measures.

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

The paper shows that for a wide class of intrinsically ergodic subshifts over finite alphabets the space of continuous functions splits into two subsets. One is a dense Gδ set in which every maximizing measure has relatively small entropy. The other lies inside the closure of the functions that possess uncountably many fully supported ergodic maximizing measures of relatively large entropy. This unifies earlier findings and covers systems such as transitive piecewise monotonic maps, certain coded shifts and multidimensional β-transformations. A sympathetic reader cares because the split describes what happens for typical functions versus a dense collection of exceptional ones in ergodic optimization.

Core claim

For a wide class of intrinsically ergodic subshifts over a finite alphabet, the space of continuous functions on the shift space splits into two subsets: one is a Gδ dense set for which all maximizing measures have relatively small entropy; the other is contained in the closure of the set of functions having uncountably many, fully supported ergodic measures with relatively large entropy.

What carries the argument

The dichotomy that partitions the space of continuous functions into a dense Gδ set whose maximizing measures all have relatively small entropy and a complementary set whose closure contains functions with uncountably many large-entropy fully supported ergodic maximizers.

If this is right

The splitting holds for any transitive piecewise monotonic interval map.
The splitting holds for some coded shifts and multidimensional β-transformations.
The splitting holds for systems without Bowen's specification property.
There exist intrinsically ergodic subshifts that have positive obstruction entropy to specification.

Where Pith is reading between the lines

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

The same partition may appear in optimization problems on other classes of dynamical systems once the intrinsic ergodicity condition is met.
One could test the boundary case by building a concrete subshift with controlled obstruction entropy and checking the entropy values numerically for sample functions.
The result supplies a template for classifying how many distinct ergodic measures can arise as maximizers when the underlying shift lacks specification.

Load-bearing premise

The subshifts under study are intrinsically ergodic over a finite alphabet.

What would settle it

Exhibit one intrinsically ergodic subshift over a finite alphabet together with a continuous function whose maximizing measures fall into neither of the two described classes.

read the original abstract

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space splits into two subsets: one is a $G_\delta$ dense set for which all maximizing measures have `relatively small' entropy; the other is contained in the closure of the set of functions having uncountably many, fully supported ergodic measures with `relatively large' entropy. This result considerably generalizes and unifies the results of Morris (2010) and Shinoda (2018), and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without Bowen's specification property, including any transitive piecewise monotonic interval map, some coded shifts and multidimensional $\beta$-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification.

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

Generalizes the dichotomy to non-Markov shifts and adds a useful example, but the interval-map application rests on an assumption that does not always hold.

read the letter

The main thing here is a clean extension of the Morris-Shinoda dichotomy to intrinsically ergodic subshifts that lack Bowen's specification. The space of continuous functions splits into a dense Gδ set where all maximizing measures have relatively small entropy and a second set whose closure contains functions with uncountably many fully supported ergodic measures of relatively large entropy. They also give an example of an intrinsically ergodic shift with positive obstruction entropy to specification, which is new and shows the result is not limited to the specification case. The applications to coded shifts and multidimensional β-transformations look like straightforward corollaries once the main theorem is in place. That part is solid and worth having in the literature. The soft spot is the claim that the result covers any transitive piecewise monotonic interval map. Not every such map is intrinsically ergodic; some have multiple ergodic measures of maximal entropy. The main theorem requires intrinsic ergodicity, so either the paper shows that all transitive piecewise monotonic maps satisfy the hypothesis or the stated generality is too broad. The stress-test concern lands. The definitions of relatively small and large entropy are only sketched in the abstract, so the precise statements and the Gδ-density argument need the full proofs to judge. The citation pattern to Morris and Shinoda is appropriate and the work stays within standard symbolic dynamics. This is for people who already work on ergodic optimization or symbolic models of interval maps. A serious referee should see it because the core generalization and the new example are substantive even if the applications section needs tightening. I would send it to review with a request to clarify the interval-map claim.

Referee Report

2 major / 0 minor

Summary. The paper claims that for a wide class of intrinsically ergodic subshifts over a finite alphabet (including those without Bowen's specification property), the space of continuous functions splits into a Gδ-dense subset where all maximizing measures have 'relatively small' entropy, and a second subset contained in the closure of functions admitting uncountably many fully supported ergodic maximizing measures with 'relatively large' entropy. The result generalizes Morris (2010) and Shinoda (2018) and is applied to transitive piecewise monotonic interval maps, coded shifts, and multidimensional β-transformations; an example of an intrinsically ergodic subshift with positive obstruction entropy to specification is also given.

Significance. If the main dichotomy holds with the stated generality, the result would unify and extend the literature on generic properties of maximizing measures in ergodic optimization beyond Markov shifts and specification systems, providing a topological dichotomy in C(X) for a broad class of intrinsically ergodic symbolic systems. The applications to interval maps and other concrete examples would be a notable strength if the hypotheses are verified.

major comments (2)

[Abstract / applications] Abstract and applications paragraph: the assertion that the result applies to 'any transitive piecewise monotonic interval map' is not supported, because there exist transitive piecewise monotonic maps that admit multiple distinct ergodic measures of maximal entropy and hence fail to be intrinsically ergodic; the main theorem's hypothesis is therefore not satisfied for those examples, so the claimed generality of the application does not hold.
[Main theorem / Gδ-density argument] Main theorem statement and § on the Gδ-density argument: the definitions of 'relatively small' and 'relatively large' entropy are not supplied in the abstract and the verification of the Gδ-density construction is unavailable in the provided text; these are load-bearing for the claimed dichotomy and must be explicitly stated and proved before the result can be assessed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying these issues. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / applications] Abstract and applications paragraph: the assertion that the result applies to 'any transitive piecewise monotonic interval map' is not supported, because there exist transitive piecewise monotonic maps that admit multiple distinct ergodic measures of maximal entropy and hence fail to be intrinsically ergodic; the main theorem's hypothesis is therefore not satisfied for those examples, so the claimed generality of the application does not hold.

Authors: We agree that the claim as stated is incorrect. Not every transitive piecewise monotonic interval map is intrinsically ergodic. We will revise the abstract and the applications paragraph to read 'any intrinsically ergodic transitive piecewise monotonic interval map' (and similarly qualify the statements for coded shifts and multidimensional β-transformations). revision: yes
Referee: [Main theorem / Gδ-density argument] Main theorem statement and § on the Gδ-density argument: the definitions of 'relatively small' and 'relatively large' entropy are not supplied in the abstract and the verification of the Gδ-density construction is unavailable in the provided text; these are load-bearing for the claimed dichotomy and must be explicitly stated and proved before the result can be assessed.

Authors: The notions of 'relatively small' and 'relatively large' entropy are defined immediately after the statement of the main theorem (Theorem 2.1) as entropy strictly less than, respectively strictly greater than, the entropy of the unique measure of maximal entropy. The Gδ-density construction and its verification appear in full in Section 3 (proof of Theorem 3.1). We will add a short parenthetical gloss of the two entropy notions to the abstract and ensure the complete proofs are visible in the version sent to the referee. revision: yes

Circularity Check

0 steps flagged

No circularity: pure existence theorem on intrinsically ergodic subshifts

full rationale

The paper is a self-contained existence result in ergodic theory. It proves a Gδ-density dichotomy for continuous functions on a class of subshifts defined by the external hypothesis of intrinsic ergodicity. The derivation generalizes Morris (2010) and Shinoda (2018) but does not reduce any claimed prediction or dichotomy to a fitted parameter, self-definition, or load-bearing self-citation chain. No equations or steps equate outputs to inputs by construction. The application statements invoke the hypothesis explicitly and do not manufacture circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the domain assumption that the subshifts are intrinsically ergodic; no free parameters or invented entities are introduced.

axioms (1)

domain assumption The subshifts are intrinsically ergodic over a finite alphabet
Explicitly stated as the setting for the dichotomy in the abstract.

pith-pipeline@v0.9.0 · 5700 in / 1153 out tokens · 21039 ms · 2026-05-24T00:34:18.868343+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

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Hofbauer and P

F. Hofbauer and P. Raith. Density of periodic orbit measures fo r transformations on the interval with two monotonic pieces. Dedicated to the memory of Wies /suppress law Szlenk.Fund. Math. 157 (1998), 221–234

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H. Hu, X. Li and Y. Yu. A note on ( − β)-shifts with the speciﬁcation property. Publ. Math. Debr. 91 (2017), 123–131

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O. Jenkinson. Ergodic optimization in dynamical systems. Ergod. Th. & Dynam. Sys. 39 (2019), 2593–2618

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W. Krieger. On the uniqueness of the equilibrium state. Math. Syst. Theory 8 (1974/75), 97–104

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T. Kucherenko, M. Schmoll and C. Wolf. Ergodic theory on code d shift spaces. arXiv:2310.18855

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D. Kwietniak, M. /suppress L¸ acka and P. Oprocha. A panorama of speciﬁcation-like properties and their consequences. Dynamics and numbers , 155–186, Contemp. Math. 669 (2016), Amer. Math. Soc., Providence, RI

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Lind and B

D. Lind and B. Marcus. An introduction to symbolic dynamics and coding . Second edition. Cambridge Mathematical Library. Cambridge University Press, Cam bridge, 2021

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I. D. Morris. The Ma˜ n´ e-Conze-Guivarc’h lemma for intermittent maps of the circle. Ergod. Th. & Dynam. Sys. 29 (2009), 1603–1611. ERGODIC OPTIMIZATION FOR CONTINUOUS FUNCTIONS 27

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I. D. Morris. Ergodic optimization for generic continuous funct ions. Discrete Contin. Dyn. Syst. 27 (2010), 383-388

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Oguchi and M

M. Oguchi and M. Shinoda. Hausdorﬀ dimension of the paramete rs for (α, β)-transformations with the speciﬁcation property. arXiv:2403.14230

work page arXiv
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Schmeling

J. Schmeling. Symbolic dynamics for β-shifts and self-normal numbers. Ergod. Th. & Dynam. Sys. 17 (1997), 675–694

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M. Shinoda. Uncountably many maximizing measures for a dense s et of continuous functions. Nonlinearity 31 (2018), 2192–2200

work page 2018
[36]

Shinoda and H

M. Shinoda and H. Takahasi. Lyapunov optimization for non-gen eric one-dimensional ex- panding Markov maps. Ergod. Th. & Dynam. Sys. 40 (2020), 2571–2592

work page 2020
[37]

Shinoda and K

M. Shinoda and K. Yamamoto. Density of periodic measures and la rge deviation principle for generalised mod one transformations. Nonlinearity 37 (2023), 025003

work page 2023
[38]

K. Sigmund. Generic properties of invariant measures for Axiom A diﬀeomorphisms. Invent. math. 11 (1970), 99–109

work page 1970
[39]

K. Sigmund. On the distribution of periodic points for β-shifts. Monatsh. Math. 82 (1976), 247–252

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[40]

K. Sigmund. On the connectedness of ergodic systems. Manuscripta Math . 22 (1977), 27–32

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[41]

P. Walters. An Introduction to Ergodic Theory (Graduate Texts in Mathem atics, 79) . Springer, New York, 1982

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[42]

Yamamoto

K. Yamamoto. On the density of periodic measures for piecewise monotonic maps and their coding spaces. Tsukuba J. Math. 44 (2020), 309–324

work page 2020
[43]

Yuan and B

G. Yuan and B. Hunt. Optimal orbits of hyperbolic systems. Nonlinearity 12 (1999), 1207– 1224. Department of Mathematics, Ochanomizu University, 2-1-1 O tsuka, Bunkyo-ku, Tokyo, 112-8610, JAPAN Email address : shinoda.mao@ocha.ac.jp Department of Mathematics, Keio University, Yokohama, 223 -8522, JAPAN Email address : hiroki@math.keio.ac.jp Department of ...

work page 1999

[1] [1]

A. T. Baraviera, R. Leplaideur and A. O. Lopes. Ergodic optimization, zero temperature limits and the max-plus algebra. IMPA Mathematical Publications, 29th Brazilian Mathematics Colloquium, Instituto Nacional de Matem´ atica Pura e Aplicada (IMPA ), Rio de Janeiro,

work page

[2] [2]

J. Bochi. Ergodic optimization of Birkhoﬀ averages and Lyapunov exponents. Proceedings of the International Congress of Mathematicians , Rio de Janeiro, 2018. Vol. III. Invited lectures, 1825–1846, World Sci. Publ., Hackensack, NJ, 2018

work page 2018

[3] [3]

T. Bousch. Le poisson n’a pas d’arˆ etes. Ann. Inst. Poincar´ e Probab. Statist.36 (2000), 489– 508

work page 2000

[4] [4]

T. Bousch. La condition de Walters, Ann. Sci. de l’ ´Ecole normale sup. 34 (2001), 5988–6017. 26 MAO SHINODA, HIROKI TAKAHASI, KENICHIRO YAMAMOTO

work page 2001

[5] [5]

Bousch and O

T. Bousch and O. Jenkinson. Cohomology class of dynamically non- negative Ck functions. Invent. Math. 148 (2002), 207–217

work page 2002

[6] [6]

R. Bowen. Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414

work page 1971

[7] [7]

Br´ emont

J. Br´ emont. Entropy and maximizing measures of generic contin uous functions. C. R. Math. Acad. Sci. 346 (2008), 199–201

work page 2008

[8] [8]

J. Buzzi. Speciﬁcation on the interval. Trans. Amer. Math. Soc. 349 (1997), no.7, 2737–2754

work page 1997

[9] [9]

J. Buzzi. Subshifts of quasi-ﬁnite type. Invent. Math. 159 (2005), 369–406

work page 2005

[10] [10]

Climenhaga

V. Climenhaga. Speciﬁcation and towers in shift spaces. Commun. Math. Phys. 364 (2018), 441–504

work page 2018

[11] [11]

Climenhaga and D

V. Climenhaga and D. J. Thompson. Intrinsic ergodicity beyond s peciﬁcation: β-shifts, S-gap shifts, and their factors. Israel J. Math. 192 (2012), 785–817

work page 2012

[12] [12]

Climenhaga and D

V. Climenhaga and D. J. Thompson. Equilibrium states beyond spe ciﬁcation and the Bowen property. J. Lond. Math. Soc. 87 (2013), 401–427

work page 2013

[13] [13]

Climenhaga and D

V. Climenhaga and D. J. Thompson. Intrinsic ergodicity via obstr uction entropies. Ergodic Theory Dyn. Syst. 34 (2014), 1816–1831

work page 2014

[14] [14]

Climenhaga, D

V. Climenhaga, D. J. Thompson and K. Yamamoto. Large deviatio ns for systems with non- uniform structure. Trans. Amer. Math. Soc. 369 (2017), 4167–4192

work page 2017

[15] [15]

Contreras, A

G. Contreras, A. O. Lopes and P. Thieullen. Lyapunov minimizing m easures for expanding maps of the circle. Ergod. Th. & Dynam. Sys. 21 (2001), 1379–1409

work page 2001

[16] [16]

Hofbauer

F. Hofbauer. On intrinsic ergodicity of piecewise monotonic tran sformations with positive entropy II. Israel J. Math. 38 (1981), 107–115

work page 1981

[17] [17]

Hofbauer

F. Hofbauer. Piecewise invertible dynamical systems. Probab. Theory Related Fields 72 (1986), 359–386

work page 1986

[18] [18]

Hofbauer

F. Hofbauer. Generic properties of invariant measures for sim ple piecewise monotonic trans- formations. Israel J. Math. 59 (1987), 64–80

work page 1987

[19] [19]

Hofbauer

F. Hofbauer. Generic properties of invariant measures for co ntinuous piecewise monotonic transformations. Monatsh. Math. 106 (1988), 301–312

work page 1988

[20] [20]

Hofbauer and P

F. Hofbauer and P. Raith. Density of periodic orbit measures fo r transformations on the interval with two monotonic pieces. Dedicated to the memory of Wies /suppress law Szlenk.Fund. Math. 157 (1998), 221–234

work page 1998

[21] [21]

H. Hu, X. Li and Y. Yu. A note on ( − β)-shifts with the speciﬁcation property. Publ. Math. Debr. 91 (2017), 123–131

work page 2017

[22] [22]

Hunt and E

B. Hunt and E. Ott. Optimal periodic orbits of chaotic systems. Phys. Rev. Lett. 76 2254

work page

[23] [23]

Hunt and E

B. Hunt and E. Ott. Optimal periodic orbits of chaotic systems o ccur at low period. Phys. Rev. E 54 328

work page

[24] [24]

R. B. Israel. Convexity in the theory of lattice gases (Princeton Series i n Physics) Princeton, NJ: Princeton University Press (1979)

work page 1979

[25] [25]

Jenkinson

O. Jenkinson. Ergodic optimization. Discrete Continuous Dyn. Syst. A 15 (2006), 197–224

work page 2006

[26] [26]

Jenkinson

O. Jenkinson. Ergodic optimization in dynamical systems. Ergod. Th. & Dynam. Sys. 39 (2019), 2593–2618

work page 2019

[27] [27]

W. Krieger. On the uniqueness of the equilibrium state. Math. Syst. Theory 8 (1974/75), 97–104

work page 1974

[28] [28]

Kucherenko, M

T. Kucherenko, M. Schmoll and C. Wolf. Ergodic theory on code d shift spaces. arXiv:2310.18855

work page arXiv

[29] [29]

Kwietniak, M

D. Kwietniak, M. /suppress L¸ acka and P. Oprocha. A panorama of speciﬁcation-like properties and their consequences. Dynamics and numbers , 155–186, Contemp. Math. 669 (2016), Amer. Math. Soc., Providence, RI

work page 2016

[30] [30]

Lind and B

D. Lind and B. Marcus. An introduction to symbolic dynamics and coding . Second edition. Cambridge Mathematical Library. Cambridge University Press, Cam bridge, 2021

work page 2021

[31] [31]

I. D. Morris. The Ma˜ n´ e-Conze-Guivarc’h lemma for intermittent maps of the circle. Ergod. Th. & Dynam. Sys. 29 (2009), 1603–1611. ERGODIC OPTIMIZATION FOR CONTINUOUS FUNCTIONS 27

work page 2009

[32] [32]

I. D. Morris. Ergodic optimization for generic continuous funct ions. Discrete Contin. Dyn. Syst. 27 (2010), 383-388

work page 2010

[33] [33]

Oguchi and M

M. Oguchi and M. Shinoda. Hausdorﬀ dimension of the paramete rs for (α, β)-transformations with the speciﬁcation property. arXiv:2403.14230

work page arXiv

[34] [34]

Schmeling

J. Schmeling. Symbolic dynamics for β-shifts and self-normal numbers. Ergod. Th. & Dynam. Sys. 17 (1997), 675–694

work page 1997

[35] [35]

M. Shinoda. Uncountably many maximizing measures for a dense s et of continuous functions. Nonlinearity 31 (2018), 2192–2200

work page 2018

[36] [36]

Shinoda and H

M. Shinoda and H. Takahasi. Lyapunov optimization for non-gen eric one-dimensional ex- panding Markov maps. Ergod. Th. & Dynam. Sys. 40 (2020), 2571–2592

work page 2020

[37] [37]

Shinoda and K

M. Shinoda and K. Yamamoto. Density of periodic measures and la rge deviation principle for generalised mod one transformations. Nonlinearity 37 (2023), 025003

work page 2023

[38] [38]

K. Sigmund. Generic properties of invariant measures for Axiom A diﬀeomorphisms. Invent. math. 11 (1970), 99–109

work page 1970

[39] [39]

K. Sigmund. On the distribution of periodic points for β-shifts. Monatsh. Math. 82 (1976), 247–252

work page 1976

[40] [40]

K. Sigmund. On the connectedness of ergodic systems. Manuscripta Math . 22 (1977), 27–32

work page 1977

[41] [41]

P. Walters. An Introduction to Ergodic Theory (Graduate Texts in Mathem atics, 79) . Springer, New York, 1982

work page 1982

[42] [42]

Yamamoto

K. Yamamoto. On the density of periodic measures for piecewise monotonic maps and their coding spaces. Tsukuba J. Math. 44 (2020), 309–324

work page 2020

[43] [43]

Yuan and B

G. Yuan and B. Hunt. Optimal orbits of hyperbolic systems. Nonlinearity 12 (1999), 1207– 1224. Department of Mathematics, Ochanomizu University, 2-1-1 O tsuka, Bunkyo-ku, Tokyo, 112-8610, JAPAN Email address : shinoda.mao@ocha.ac.jp Department of Mathematics, Keio University, Yokohama, 223 -8522, JAPAN Email address : hiroki@math.keio.ac.jp Department of ...

work page 1999