Joint typical periodic optimization: systems with stable hyperbolicity

Oliver Jenkinson; Yinying Huang; Zelai Hao; Zhiqiang Li

arxiv: 2605.01550 · v2 · pith:D7SU7X6Qnew · submitted 2026-05-02 · 🧮 math.DS

Joint typical periodic optimization: systems with stable hyperbolicity

Zelai Hao , Yinying Huang , Oliver Jenkinson , Zhiqiang Li This is my paper

Pith reviewed 2026-05-21 00:47 UTC · model grok-4.3

classification 🧮 math.DS

keywords joint typical periodic optimizationstable hyperbolicityAxiom A diffeomorphismsno-cycle propertyhyperbolic rational mapsperiodic orbitsthermodynamic formalism

0 comments

The pith

An axiomatic joint perturbation framework shows optimizing periodic orbits persist under simultaneous changes to the map and potential for Axiom A diffeomorphisms with the no-cycle property and other stably hyperbolic systems.

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

The paper broadens the theory of joint typical periodic optimization by creating an axiomatic framework that handles simultaneous perturbations to both the dynamical system and the potential function. This applies to stably hyperbolic systems and yields results for Axiom A diffeomorphisms with the no-cycle property, hyperbolic rational maps on the Riemann sphere, real quadratic polynomials, and C^r maps in one dimension. A sympathetic reader would care because the framework demonstrates that certain optimization behaviors remain typical and robust across these important classes of systems even when parameters vary together.

Core claim

The framework of joint typical periodic optimization is extended via an axiomatic joint perturbation framework that accommodates a wider class of stably hyperbolic systems, establishing that optimizing periodic orbits persist under simultaneous perturbation for Axiom A diffeomorphisms with the no-cycle property, hyperbolic rational maps on the Riemann sphere, real quadratic polynomials, and C^r maps in one dimension.

What carries the argument

The axiomatic joint perturbation framework, which guarantees persistence of optimizing periodic orbits when both the map and potential vary simultaneously within classes possessing stable hyperbolicity or the no-cycle property.

If this is right

Optimizing periodic orbits persist under joint perturbations for Axiom A diffeomorphisms with the no-cycle property.
The same persistence holds for hyperbolic rational maps on the Riemann sphere.
Real quadratic polynomials admit joint typical periodic optimization.
C^r maps in one dimension satisfy the persistence property under the framework.

Where Pith is reading between the lines

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

The results suggest joint locking sets are open and dense in the product space of maps and potentials for these systems.
The axiomatic approach may allow application to other families with stable hyperbolicity beyond those explicitly treated.
Connections to thermodynamic formalism could be explored by viewing the potential as a observable in these optimized orbits.

Load-bearing premise

The systems must belong to classes with stable hyperbolicity or the no-cycle property to ensure optimizing periodic orbits persist under joint perturbations.

What would settle it

A specific Axiom A diffeomorphism with the no-cycle property together with a potential where a small simultaneous perturbation of map and potential makes the optimizing periodic orbit disappear or cease to be optimal would falsify the persistence claim.

read the original abstract

The framework of joint typical periodic optimization, in which both the dynamical system and the potential function are allowed to vary simultaneously, was introduced in [HHJL25], in a direction motivated by the work of Yang, Hunt & Ott [YHO00]. For certain classes of hyperbolic systems, it was shown there that optimizing periodic orbits persist under simultaneous perturbation, yielding joint locking sets that contain open dense subsets of the relevant product spaces. In the present article we broaden the scope of this theory, by developing an axiomatic joint perturbation framework that accommodates a wider class of stably hyperbolic systems, and by establishing new joint typical periodic optimization results for several natural and important families: Axiom A diffeomorphisms with the no-cycle property, hyperbolic rational maps on the Riemann sphere, real quadratic polynomials, and $C^r$ maps in one dimension.

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 extends the joint typical periodic optimization framework from HHJL25 to more families of stably hyperbolic systems via a new axiomatic setup, with solid but incremental applications.

read the letter

This paper takes the joint typical periodic optimization idea from HHJL25 and widens it by introducing an axiomatic joint perturbation framework. The core move is to set up conditions under which optimizing periodic orbits persist when both the map and the potential are varied at once, producing joint locking sets that contain open dense subsets of the product space. They then apply the framework to Axiom A diffeomorphisms with the no-cycle property, hyperbolic rational maps on the Riemann sphere, real quadratic polynomials, and C^r maps in one dimension. These are natural families that sit outside the scope of the earlier work, so the extension is concrete rather than purely formal. The axioms appear chosen to capture the stable hyperbolicity and persistence properties already known piecemeal for these classes, which is a reasonable organizing step. The no-cycle condition for Axiom A is a standard hypothesis in this area and is used explicitly to support the persistence claims, so it does not look like an artificial restriction. The main soft spot is that the abstract and summary do not show the explicit verification that each listed family satisfies the joint perturbation axioms. A referee would want to see those checks written out clearly to confirm no hidden assumptions are required for the quadratics or the 1D maps. Nothing in the stated scope points to a contradiction or circularity, and the construction relies on new axioms rather than self-referential fitting. This is for readers already working in ergodic optimization or hyperbolic dynamics who want to see how persistence results can be bundled together. It is worth sending to peer review because the applications are to standard families and the framework is presented as reusable.

Referee Report

0 major / 2 minor

Summary. The manuscript extends the joint typical periodic optimization framework from [HHJL25] by introducing an axiomatic joint perturbation framework for stably hyperbolic systems. It establishes persistence of optimizing periodic orbits under simultaneous perturbations of the map and potential, yielding joint locking sets containing open dense subsets, for Axiom A diffeomorphisms with the no-cycle property, hyperbolic rational maps on the Riemann sphere, real quadratic polynomials, and C^r maps in one dimension.

Significance. If the axiomatic setup and persistence arguments are correct, the work meaningfully broadens the applicability of joint typical periodic optimization to several standard classes of systems with stable hyperbolicity. The concrete treatment of Axiom A systems, rational maps, and one-dimensional maps connects the abstract theory to well-studied families and may support further applications in ergodic optimization.

minor comments (2)

Abstract and introduction: expand the brief mention of the axiomatic framework with a short statement of the key axioms and how they ensure persistence for the listed families.
References: ensure the full bibliographic details for [HHJL25] and [YHO00] appear in the bibliography section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in extending the joint typical periodic optimization framework, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; axiomatic extension is self-contained

full rationale

The paper introduces an axiomatic joint perturbation framework to extend joint typical periodic optimization results to stably hyperbolic systems, including Axiom A diffeomorphisms with the no-cycle property, hyperbolic rational maps, real quadratic polynomials, and C^r maps in one dimension. The derivation relies on persistence of optimizing periodic orbits under simultaneous map-potential perturbations, with the no-cycle property and stable hyperbolicity invoked explicitly as standard hypotheses to support these statements. The reference to the prior framework in [HHJL25] serves as background rather than a load-bearing reduction of the new claims; the central results consist of independent applications and broadenings that do not reduce by construction to fitted inputs, self-definitions, or unverified self-citations within this manuscript. The work remains self-contained against external benchmarks in dynamical systems theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; ledger populated from stated assumptions in abstract. No free parameters visible. Axioms center on stable hyperbolicity and no-cycle property as domain assumptions for persistence.

axioms (2)

domain assumption Systems possess stable hyperbolicity allowing persistence of optimizing periodic orbits under joint perturbations
Invoked to guarantee open dense joint locking sets for the listed families.
domain assumption Axiom A diffeomorphisms satisfy the no-cycle property
Required for the new results on this family.

pith-pipeline@v0.9.0 · 5671 in / 1373 out tokens · 32790 ms · 2026-05-21T00:47:48.988260+00:00 · methodology

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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 2.3 (Joint perturbation for Hölder potentials) … M_max(g, ϕ − 2C |ϕ|_α d_g^{α/2} d(·,O_g)^α + ξ)

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

45 extracted references · 45 canonical work pages · 1 internal anchor

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Quas, A. and Siefken, J. , Ergodic optimization of supercontinuous functions on shift spaces. Ergodic Theory Dynam.\ Systems 32 (2012), 2071--2082

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, The topology of spaces of rational functions

Segal, G. , The topology of spaces of rational functions. Acta Math. 143 (1979), 39--72

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, Differentiable dynamical systems

Smale, S. , Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747--817

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, and Ott, E

Yang , Tsung-Hsun, Hunt, B.R. , and Ott, E. , Optimal periodic orbits of continuous time chaotic systems. Phys.\ Rev.\ E 62 (2000), 1950--1959

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[1] [1]

Iteration of Rational Functions, Springer, New York, 1991

Beardon, A.F. Iteration of Rational Functions, Springer, New York, 1991

work page 1991

[2] [2]

, Ergodic optimization of Birkhoff averages and Lyapunov exponents

Bochi, J. , Ergodic optimization of Birkhoff averages and Lyapunov exponents. In Proc.\ Internat.\ Congr.\ Math.\ (Rio de Janeiro 2018), Volume III, World Sci.\ Publ., Singapore, 2018, pp. 1825--1846

work page 2018

[3] [3]

, La condition de Walters

Bousch, T. , La condition de Walters. Ann.\ Sci.\ \'Ec.\ Norm.\ Sup\'er.\ (4) 34 (2001), 287--311

work page 2001

[4] [4]

, Nouvelle preuve d'un th\'eor\`eme de Yuan et Hunt

Bousch, T. , Nouvelle preuve d'un th\'eor\`eme de Yuan et Hunt. Bull.\ Soc.\ Math.\ France 126 (2008), 227--242

work page 2008

[5] [5]

, Le lemme de Ma\ n\'e-Conze-Guivarc’h pour les syst\`emes amphidynamiques rectifiables

Bousch, T. , Le lemme de Ma\ n\'e-Conze-Guivarc’h pour les syst\`emes amphidynamiques rectifiables. Ann.\ Fac.\ Sci.\ Toulouse Math.\ (6) 20 (2011), 1--14

work page 2011

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and Quas, A

Bressaud, X. and Quas, A. , Rate of approximation of minimizing measures. Nonlinearity 20 (2007), 845--853

work page 2007

[7] [7]

, Ground states are generically a periodic orbit

Contreras, G. , Ground states are generically a periodic orbit. Invent.\ Math. 205 (2016), 383--412

work page 2016

[8] [8]

, Lopes, A.O

Contreras, G. , Lopes, A.O. , and Thieullen, Ph. , Lyapunov minimizing measures for expanding maps of the circle. Ergodic Theory Dynam.\ System 21 (2001), 1379--1409

work page 2001

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and \'Swiatek, G , Generic hyperbolicity in the logistic family

Graczyk, J. and \'Swiatek, G , Generic hyperbolicity in the logistic family. Ann.\ Math.\ (2) 146 (1997), 1--52

work page 1997

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Joint typical periodic optimization

Hao , Zelai, Huang , Yinying, Jenkinson O. , and Li , Zhiqiang, Joint typical periodic optimization. Preprint, (arXiv:2502.12269), 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025

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Hirsch, M. W. , Differentiable Topology, volume 33 of Grad.\ Stud.\ Math., Springer, New York, 1976

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Work in progress

Huang , Wen, Lian , Zeng, Ma , Xiao, Xu , Leiye, and Zhang , Yiwei, Ergodic optimization theory for a class of typical maps. J.\ Eur.\ Math.\ Soc. doi:10.4171/JEMS/1652

work page doi:10.4171/jems/1652

[13] [13]

, Ergodic optimization

Jenkinson, O. , Ergodic optimization. Discrete Contin.\ Dyn.\ Syst. 15 (2006), 197--224

work page 2006

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, Ergodic optimization in dynamical systems

Jenkinson, O. , Ergodic optimization in dynamical systems. Ergodic Theory Dynam.\ System 39 (2019), 2593--2618

work page 2019

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, and Hasselblatt, B

Katok, A. , and Hasselblatt, B. , Introduction to the Modern Theory of Dynamical Systems, volume 54 of Encyclopedia Math.\ Appl., Cambridge Univ.\ Press, Cambridge, 1995

work page 1995

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, Classical Descriptive Set Theory, Springer, New York, 1995

Kechris, A.S. , Classical Descriptive Set Theory, Springer, New York, 1995

work page 1995

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, Shen , Weixiao, and van Strien, S

Kozlovski, O. , Shen , Weixiao, and van Strien, S. , Density of hyperbolicity in dimension one. Ann.\ Math.\ (2) 166 (2007), 145--182

work page 2007

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Introduction to Riemannian manifolds, 2nd ed., Springer, Cham, 2018

Lee, J.M. Introduction to Riemannian manifolds, 2nd ed., Springer, Cham, 2018

work page 2018

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Adv.\ Math

Li , Zhiqiang and Sun , Yiqing, Tropical thermodynamic formalism. Adv.\ Math. 491 (2026), 110864

work page 2026

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Math.\ Ann

Li , Zhiqiang and Zhang , Yiwei, Ground states and periodic orbits for expanding Thurston maps. Math.\ Ann. 391 (2025), 3913--3985

work page 2025

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, Some typical properties of the dynamics of rational maps

Lyubich, M.Yu. , Some typical properties of the dynamics of rational maps. Russian Math.\ Surveys 38 (1983), 154--155

work page 1983

[22] [22]

, The dynamics of rational transforms: the topological picture

Lyubich, M.Yu. , The dynamics of rational transforms: the topological picture. Russian Math.\ Surveys 41 (1986), 43--117

work page 1986

[23] [23]

, Dynamics of quadratic polynomials, I-II

Lyubich, M.Yu. , Dynamics of quadratic polynomials, I-II. Acta Math. 178 (1997), 185--297

work page 1997

[24] [24]

, Six Lectures on Real and Complex Dynamics

Lyubich, M.Yu. , Six Lectures on Real and Complex Dynamics. Available at https://www.math.stonybrook.edu/ mlyubich/Archive/Selected/lectures.pdf

work page

[25] [25]

, Sad, P

Ma\ n\'e, R. , Sad, P. , and Sullivan, D.P. , On the dynamics of rational maps. Ann.\ Sci.\ \'Ec.\ Norm.\ Sup\'er.\ (4) 16 (1983), 193--217

work page 1983

[26] [26]

, Frontiers in complex dynamics

McMullen, C. , Frontiers in complex dynamics. Bull.\ Amer.\ Math.\ Soc.\ (N.S.) 31 (1994), 155--172

work page 1994

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and Sullivan, D.P

McMullen, C. and Sullivan, D.P. , Quasiconformal homeomorphisms and dynamics III. The Teichm\"uller space of a holomorphic dynamical system Adv.\ Math. 135 (1998), 351--395

work page 1998

[28] [28]

, and van Strien, S

de Melo, M. , and van Strien, S. , One-dimensional Dynamics, volume 25 of Ergeb.\ Math.\ Grenzgeb.\ (3), Springer, New York, 1993

work page 1993

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, Geometry and dynamics of quadratic rational maps

Milnor, J. , Geometry and dynamics of quadratic rational maps. Exp.\ Math. 2 (1993), 37--83

work page 1993

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, Dynamics in One Complex Variable, 3rd ed., Princeton Univ.\ Press, 2006

Milnor, J. , Dynamics in One Complex Variable, 3rd ed., Princeton Univ.\ Press, 2006

work page 2006

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, Maximizing measures of generic H\"older functions have zero entropy

Morris, I.D. , Maximizing measures of generic H\"older functions have zero entropy. Nonlinearity 21 (2008), 993--1000

work page 2008

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, A note on -stability, in Global analysis, edited by Smale, S

Palis, J. , A note on -stability, in Global analysis, edited by Smale, S. and Chern, S.S. , pp. 220--222, volume XIV of Proc.\ Sympos.\ Pure Math., Amer.\ Math.\ Soc., Providence, RI., 1970

work page 1970

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, On the C^1 -stability conjecture

Palis, J. , On the C^1 -stability conjecture. Publ.\ Math.\ Inst.\ Hautes \'Etudes Sci. 66 (1987), 211--215

work page 1987

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and Urba\'nski, M

Przytycki, F. and Urba\'nski, M. , Conformal Fractals: Ergodic Theory Methods, Cambridge Univ.\ Press, Cambridge, 2010

work page 2010

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and Siefken, J

Quas, A. and Siefken, J. , Ergodic optimization of supercontinuous functions on shift spaces. Ergodic Theory Dynam.\ Systems 32 (2012), 2071--2082

work page 2012

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, The topology of spaces of rational functions

Segal, G. , The topology of spaces of rational functions. Acta Math. 143 (1979), 39--72

work page 1979

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, Differentiable dynamical systems

Smale, S. , Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747--817

work page 1967

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, The -stability theorem

Smale, S. , The -stability theorem. Global Analysis (Proc.\ Sympos.\ Pure Math., Vols. XIV, XV, XVI, Berkeley, CA, 1968), pp. 289–297, Amer.\ Math.\ Soc., Providence, RI, 1970

work page 1968

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, and Yu , Wenzhe, Lipschitz sub-actions for locally maximal hyperbolic sets of a C^1 map

Su , Xifeng, Thieullen, P. , and Yu , Wenzhe, Lipschitz sub-actions for locally maximal hyperbolic sets of a C^1 map. Discrete Contin.\ Dyn.\ Syst. 44 (2024), 656--677

work page 2024

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, Roy, M

Urba\' n ski, M. , Roy, M. , and Munday, S. , Non-invertible Dynamical Systems. Volume 1.\ Ergodic Theory -- Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps, volume 69.1 of De Gruyter Exp.\ Math., De Gruyter, Berlin, 2022

work page 2022

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, An Introduction to Ergodic Theory, Springer, New York, 1982

Walters, P. , An Introduction to Ergodic Theory, Springer, New York, 1982

work page 1982

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, Lipschitz Algebras, 2nd ed., World Sci.\ Publ., Hackensack, NJ, 2018

Weaver, N. , Lipschitz Algebras, 2nd ed., World Sci.\ Publ., Hackensack, NJ, 2018

work page 2018

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Wen , Lan, Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolicity, volume 173 of Grad.\ Stud.\ Math., Amer.\ Math.\ Soc., Providence, RI, 2016

work page 2016

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, and Ott, E

Yang , Tsung-Hsun, Hunt, B.R. , and Ott, E. , Optimal periodic orbits of continuous time chaotic systems. Phys.\ Rev.\ E 62 (2000), 1950--1959

work page 2000

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, Optimal orbits of hyperbolic systems

Yuan , Guocheng and Hunt, B.R. , Optimal orbits of hyperbolic systems. Nonlinearity 12 (1999), 1207--1224

work page 1999