pith. machine review for the scientific record. sign in

arxiv: 2605.13342 · v1 · submitted 2026-05-13 · 🧮 math.DS · math-ph· math.MP· math.PR

Recognition: unknown

Ergodic Optimization and Ground States: a brief Introduction

Artur O. Lopes
Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:21 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.MPmath.PR
keywords somedescribesubjectwillconceptsergodicoptimizationproperties
0
0 comments X

The pith

The paper delivers a brief, non-technical overview of ergodic optimization and its relationship to statistical mechanics without new results or proofs.

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

Ergodic optimization looks for the best possible long-run average behavior in systems that change over time according to fixed rules. It seeks special measures or configurations that maximize a given quantity, much like finding the lowest-energy arrangements in a physical system. The note explains these ideas in simple terms, gives a few examples, and shows how they relate to concepts such as temperature and energy minimization from statistical mechanics.

Core claim

Our goal in this short note is to briefly and succinctly describe some basic concepts and properties of Ergodic Optimization for readers unfamiliar with the subject.

Load-bearing premise

That a schematic, non-technical description with selected examples can convey the essential relationship between ergodic optimization and statistical mechanics without technical details or proofs.

Figures

Figures reproduced from arXiv: 2605.13342 by Artur O. Lopes.

Figure 1
Figure 1. Figure 1: The graph of x → A(x) “defined” for x ∈ [0, 1] corresponding to the potential A = I01. 0.2 0.4 0.6 0.8 1.0 -1.0 -0.5 0.5 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The graph of the subaction x → u(x) “defined” for x ∈ [0, 1] obtained from the 1/2-iteration method when A = I01. From the computer￾generated picture we guess that the subaction of A is u = −I0 + 1/2 I1 . One can verify rigorously (by hand) that such u is indeed a solution to equation (7). In this case 1 2 δ(01)∞ + 1 2 δ(10)∞ is the maximizing probability and α(A) = 1/2. Not many iterations were required t… view at source ↗
Figure 3
Figure 3. Figure 3: The graph of x → u(x) on [0, 1] obtained from the 1/2-iteration method when A = I01111. From the above picture one can guess the expres￾sion for a subaction u in the case of this potential A. One can verify that such u is a solution to the subaction equation (7), where α(A) = 0.2. 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The graph on [0, 1] corresponding to the function x → R(x) for the potential I01111 (using the subaction u obtained from [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: From left to right: the graph of the potential [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Case A(x) = sin2 (2πx) - From the 1/2 iterative procedure: on the left side, we show the approximated subaction u which is given by the supremum of the two functions in red and in blue. The graph of R using the approximation of the subaction u is shown in the right-hand picture. The orbit of period 2 given by 1 2 δ1/3 + 1 2 δ2/3 is inside the set R = 0, and therefore is a maximizing probability µ A for A. … view at source ↗
read the original abstract

Our goal in this short note is to briefly and succinctly describe some basic concepts and properties of Ergodic Optimization for readers unfamiliar with the subject. We avoid technical issues in order to provide a global overview of this topic. We will not attempt to cover all of the many contributions of various authors, who have greatly enriched the theory with invaluable results. The author has made a personal selection of the topics to be addressed, keeping in mind two main objectives: to motivate the reasons for studying the subject, and to describe schematically and pictorially its relationship with relevant concepts and properties of Statistical Mechanics, which is one of the sources of inspiration for the theory. We will not present new results or detailed proofs. Some examples will be provided. We describe some procedures that may help in obtaining explicit solutions. We present some references that by no means aim to exhaust the bibliography on the subject, where possible, minimizing the number of references.

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.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript is framed explicitly as a non-technical overview whose goal is to provide a schematic, motivational description of ergodic optimization concepts and their relation to statistical mechanics. It states it will avoid technical issues, proofs, new results, and detailed derivations. No equations, predictions, fitted parameters, or load-bearing derivations are advanced that could reduce to self-referential inputs or self-citations. The central claim stands as an independent selection of topics for readers unfamiliar with the subject.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an expository overview with no derivations, so it introduces no free parameters, axioms, or invented entities beyond standard background concepts in ergodic theory and statistical mechanics.

pith-pipeline@v0.9.0 · 5455 in / 863 out tokens · 99701 ms · 2026-05-14T18:21:03.034553+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific (2000)

    V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific (2000)

    work page 2000

  2. [2]

    Baraviera, A

    A. Baraviera, A. O. Lopes and Ph. Thieullen, A Large Deviation Prin- ciple for Gibbs states of H¨ older potentials: the zero temperature case. Stoch. and Dyn. (6), 77-96, (2006)

    work page 2006

  3. [3]

    Baraviera, R

    A. Baraviera, R. Leplaideur and A. O. Lopes, Selection of ground states in the zero temperature limit for a one-parameter family of potentials, SIAM Journal on Appl. Dyn. Systems, Vol. 11, no. 1, 243-260 (2012)

    work page 2012

  4. [4]

    Baraviera, R

    A. Baraviera, R. Leplaideur and A. O. Lopes, Ergodic Optimization, Zero Temperature Limits and the Max-Plus Algebra, XXIX Col´ oquio Brasileiro de Matem´ atica (2013) 15

    work page 2013

  5. [5]

    Bochi, Ergodic optimization of Birkhoff averages and Lyapunov ex- ponents

    J. Bochi, Ergodic optimization of Birkhoff averages and Lyapunov ex- ponents. In Proc. Internat. Congr. Math. (Rio de Janeiro 2018), Volume III, World Sci. Publ., Singapore, pp. 1843-1866 (2018)

    work page 2018

  6. [6]

    Bochi and Y

    J. Bochi and Y. Zhang, Ergodic optimization of prevalent super- continuous functions. Int. Math. Res. Not. IMRN, 19, 5988-6017 (2016)

    work page 2016

  7. [7]

    Bissacot, E

    R. Bissacot, E. Garibaldi and Ph. Thieullen, Zero-temperature phase diagram for double-well type potentials in the summable variation class, ETDS 38, no. 3, 863-885 (2018)

    work page 2018

  8. [8]

    Chazottes, J.-M

    J.-R. Chazottes, J.-M. Gambaudo and E. Ugalde, Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials, Erg. Theo. and Dyn. Sys. 1109-1161(2010)

    work page 2010

  9. [9]

    Contreras, A

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

    work page 2001

  10. [10]

    J. P. Conze and Y. Guivarc’h. Croissance des sommes ergodiques et principe variationnel. manuscript circa (1993)

    work page 1993

  11. [11]

    Coronel and J

    D. Coronel and J. Rivera-Letelier, Sensitive dependence of Gibbs mea- sures at low temperatures, Jou. St. Phys, 160, Issue 6, 1658-1683 (2015)

    work page 2015

  12. [12]

    Cioletti, M

    L. Cioletti, M. Denker, A. O. Lopes and M. Stadlbauer, Spectral Proper- ties of the Ruelle Operator for Product Type Potentials on Shift Spaces, Jour. of the London Math. Soc., Volume 95, Issue 2, 684-704 (2017)

    work page 2017

  13. [13]

    Contreras, Ground states are generically a periodic orbit, Invent

    G. Contreras, Ground states are generically a periodic orbit, Invent. Math, no. 2, 383-412 (2016)

    work page 2016

  14. [14]

    Contreras and R

    G. Contreras and R. Iturriaga. Global minimizers of autonomous La- grangians, 22◦ Col´ oquio Brasileiro de Matem´ atica, IMPA (1999)

    work page 1999

  15. [15]

    Fathi, Weak KAM Theorem in Lagrangian Dynamics, Lecture Notes, Pisa (2005)

    A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Lecture Notes, Pisa (2005)

    work page 2005

  16. [16]

    Modeling, Dynamics, Optimization IV

    H. H. Ferreira, A. O. Lopes and E. R. Oliveira, An iterative process for approximating subactions, “Modeling, Dynamics, Optimization IV”, Editors: Alberto Pinto and David Zilberman, 187-212 (2021)

    work page 2021

  17. [17]

    H. H. Ferreira, A. O. Lopes and E. R. Oliveira, Explicit examples in Ergodic Optimization, S. P. Jou. of Math. Scien, Vol. 14, 443-489 (2020) 16

    work page 2020

  18. [18]

    H. H. Ferreira, Python Code used to generate the figures for the explicit examples, written in Jupyter notebooks. https://github.com/hermes-hf/Explicitexamples ergodic

    work page

  19. [19]

    R. Gao, W. Shen and R. Zhang, Typicality of periodic optimization over an expanding circle map, arXiv (2025)

    work page 2025

  20. [20]

    Giulietti, B

    P. Giulietti, B. Kloeckner, A. O. Lopes and D. Marcon, The calculus of thermodynamical formalism, Journ. of the European Math Society, Vol 20, Issue 10, pages 2357-2412 (2018)

    work page 2018

  21. [21]

    Garibaldi, A

    E. Garibaldi, A. O. Lopes and P. Thieullen, On calibrated and separating sub-actions, Bull. of the Bras. Math. Soc. Vol. 40, 577-602 (2009)

    work page 2009

  22. [22]

    Garibaldi, Ergodic Optimization in the Expanding Case (2017)

    E. Garibaldi, Ergodic Optimization in the Expanding Case (2017)

    work page 2017

  23. [23]

    Garibaldi and A

    E. Garibaldi and A. O. Lopes, On Aubry–Mather theory for symbolic dynamics, ETDS, Vol. 28, Issue 3, 791-815 (2008)

    work page 2008

  24. [24]

    Garibaldi and A

    E. Garibaldi and A. O. Lopes, Functions for relative maximization, Dy- namical Systems, v. 22, 511-528 (2007)

    work page 2007

  25. [25]

    Huang, L

    W. Huang, L. Xu and D. Yang, Lyapunov Optimizing Measures and Periodic Measures forC 2 Expanding Maps, Acta Mathematica Sinica, Volume 41, pages 2259-2274 (2025)

    work page 2025

  26. [26]

    Jenkinson, Ergodic optimization, Discrete and Continuous Dynami- cal Systems, Series A, V

    O. Jenkinson, Ergodic optimization, Discrete and Continuous Dynami- cal Systems, Series A, V. 15, 197-224 (2006)

    work page 2006

  27. [27]

    Jenkinson, Ergodic optimization in dynamical systems, Ergodic The- ory and Dynamical Systems, pp

    O. Jenkinson, Ergodic optimization in dynamical systems, Ergodic The- ory and Dynamical Systems, pp. 2593–2618 (2019)

    work page 2019

  28. [28]

    Jenkinson, X

    O. Jenkinson, X. Li and Y. Zhang, Typical Uniqueness in Ergodic Op- timization, arXiv (2025)

    work page 2025

  29. [29]

    Kucherenko and C

    T. Kucherenko and C. Wolf, Geometry and entropy of generalized rota- tion sets, Israel Journal of Mathematics 199, 791-829 (2014)

    work page 2014

  30. [30]

    Lalley, Distribution of Periodic Orbits of Symbolic and Axiom A Flows, Adv

    S. Lalley, Distribution of Periodic Orbits of Symbolic and Axiom A Flows, Adv. in Appl. Math, 8, 154-193 (1987)

    work page 1987

  31. [31]

    Leplaideur, A dynamical proof for the convergence of Gibbs measures at temperature zero, Nonlinearity, 18(6):2847–2880 (2005) 17

    R. Leplaideur, A dynamical proof for the convergence of Gibbs measures at temperature zero, Nonlinearity, 18(6):2847–2880 (2005) 17

    work page 2005

  32. [32]

    Leplaideur and J

    R. Leplaideur and J. K. Mengue, On the selection of subaction and measure for perturbed potentials, ETDS, Vol. 46, 3, 660-694 (2026)

    work page 2026

  33. [33]

    A. O. Lopes, J. K. Mengue, J. Mohr and R. R. Souza, Entropy and Variational Principle for one-dimensional Lattice Systems with a general a priori probability: positive and zero temperature, Erg. Theory and Dyn. Systems, 35 (6), 1925-1961 (2015)

    work page 1925

  34. [34]

    A. O. Lopes, E. R. Oliveira and Ph. Thieullen, The dual potential, the involution kernel and transport in ergodic optimization, Dynamics, Games and Science, Springer Verlag, 331-398 (2015)

    work page 2015

  35. [35]

    A. O. Lopes, E. R. Oliveira, W. de S. Pedra and V. Vargas, Grand- canonical Thermodynamic Formalism via IFS: volume, temperature, gas pressure and grand-canonical topological pressure, Sao Paulo Journal of Math. Scien., Volume 19, article number 30 - pp 1-34 (2025)

    work page 2025

  36. [36]

    A. O. Lopes, Thermodynamic Formalism, Maximizing Probabilities and Large Deviations — Notes UFRGS — online homepage

    work page

  37. [37]

    J. K. Mengue, Large Deviations for Equilibrium Measures and Selection of Subaction. Bull. of the Braz. Math. Soc., v. 49, p. 17-42 (2018)

    work page 2018

  38. [38]

    Morris, Prevalent uniqueness in ergodic optimisation, Proc

    I. Morris, Prevalent uniqueness in ergodic optimisation, Proc. Amer. Math. Soc. 149, no. 4, 1631-1639 (2021)

    work page 2021

  39. [39]

    Parry and M

    W. Parry and M. Pollicott. Zeta functions and the periodic orbit struc- ture of hyperbolic dynamics, Ast´ erisque Vol. 187-188 (1990)

    work page 1990

  40. [40]

    Quas and J

    A. Quas and J. Siefken, Ergodic optimization of super-continuous func- tions on shift spaces. ETDS, 32, 6, 2071-2082 (2012)

    work page 2071

  41. [41]

    Sigmund, On dynamical systems with the specification property

    K. Sigmund, On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190, 285-299 (1974)

    work page 1974

  42. [42]

    Souza, Sub-actions for weakly hyperbolic one-dimensional sys- tems, Dynamical Systems, Volume 18, Issue 2, 165-179 (2003)

    Rafael R. Souza, Sub-actions for weakly hyperbolic one-dimensional sys- tems, Dynamical Systems, Volume 18, Issue 2, 165-179 (2003)

    work page 2003

  43. [43]

    Viana and K

    M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge University Press (2016)

    work page 2016

  44. [44]

    Walters, Introduction to Ergodic Theory, Springer Verlag (1982) 18

    P. Walters, Introduction to Ergodic Theory, Springer Verlag (1982) 18

    work page 1982