Explicit formulae and topological descriptions of action-minimizing sets of a full shift with an uncountable alphabet $[0,1]$

Mao Shinoda; Shoya Motonaga; Yuika Kajihara

arxiv: 2510.02678 · v3 · submitted 2025-10-03 · 🧮 math.DS

Explicit formulae and topological descriptions of action-minimizing sets of a full shift with an uncountable alphabet [0,1]

Yuika Kajihara , Shoya Motonaga , Mao Shinoda This is my paper

Pith reviewed 2026-05-18 11:02 UTC · model grok-4.3

classification 🧮 math.DS

keywords ergodic optimizationfull shiftuncountable alphabetMather setAubry setMañé settwist conditionsymbolic dynamics

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The pith

For full shifts over the interval [0,1] and potentials on consecutive pairs that satisfy the twist condition, the Mather and Aubry sets receive explicit formulas while the Mañé set contains cubes of every finite dimension.

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

This paper solves the ergodic optimization problem for the full shift with uncountable alphabet [0,1], a system known for positive mean dimension and infinite topological entropy. The solution holds for potentials that depend only on the first two coordinates and obey the twist condition. Under these restrictions the authors supply closed-form expressions for the associated Mather set and Aubry set. They further establish that the quotient Aubry set is totally disconnected when the potential is differentiable, yet the larger Mañé set contains embedded cubes of arbitrary finite dimension. The proofs rely on new estimates for connecting orbits obtained by merging weak KAM subaction methods, Mañé’s Lagrangian formulation, and Bangert’s variational approach.

Core claim

We completely solve ergodic optimization of a full shift with an uncountable alphabet [0,1] for potentials depending only on the first two coordinates with the twist condition as well as giving explicit formulae of the associated Mather set and the Aubry set. Moreover, we investigate the total disconnectedness of the (quotient) Aubry set, in which case the differentiability of the potential function makes a crucial difference. Although these results imply that the (quotient) Aubry set is small enough, we give a complete characterization of an analogical object of the Aubry set, called the Mañé set, and show that it is much larger than the Aubry set so that it contains cubes of any finite.

What carries the argument

The twist condition imposed on potentials that depend only on two consecutive coordinates, together with explicit estimates for connecting orbits constructed via weak KAM subactions, Mañé’s Lagrangian formulation, and Bangert’s variational methods.

If this is right

The Mather set and Aubry set admit explicit coordinate-wise descriptions under the stated hypotheses.
Differentiability of the potential forces the quotient Aubry set to be totally disconnected.
The Mañé set always contains embedded copies of the unit cube in every finite dimension.
Connecting-orbit estimates suffice to determine all action-minimizing invariant measures.

Where Pith is reading between the lines

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

The same two-coordinate twist framework may yield explicit solutions for other shifts on compact metric spaces of positive dimension.
The size gap between the Aubry set and the Mañé set could serve as a diagnostic for positive mean dimension in broader classes of infinite-alphabet systems.
The combined variational techniques may extend to optimization problems on sequence spaces equipped with non-product topologies.

Load-bearing premise

The potentials are assumed to depend only on the first two coordinates and to satisfy the twist condition.

What would settle it

An explicit counter-example potential that depends on three or more coordinates or violates the twist condition for which no closed-form description of the Mather set exists or the Mañé set fails to contain a three-dimensional cube.

read the original abstract

We completely solve ergodic optimization of a full shift with an uncountable alphabet $[0,1]$, which is one of the most well-known examples of infinite dimensional dynamical systems with positive mean dimension (and thus with infinite topological entropy), for potentials depending only on the first two coordinates with the twist condition as well as giving explicit formulae of the associated Mather set and the Aubry set. Moreover, we investigate the total disconnectedness of the (quotient) Aubry set, in which case the differentiability of the potential function makes a crucial difference. Although these results imply that the (quotient) Aubry set is small enough, we give a complete characterization of an analogical object of the Aubry set, called the Ma\~{n}\'e set, and show that it is much larger than the Aubry set so that it contains cubes of any finite dimension. In our proofs, estimates for ``connecting orbits" play key roles, and we establish them by combining three different perspectives in our symbolic setting; weak KAM subaction approach for symbolic dynamics, Ma\~{n}\'e's formulation in Lagrangian systems, and Bangert's variational approach for twist maps.

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 supplies explicit formulas for the Mather and Aubry sets on the uncountable full shift under two-coordinate twist potentials, using connecting-orbit estimates from three classical perspectives.

read the letter

The main result is a complete solution to ergodic optimization for potentials on the full shift over [0,1] that depend only on consecutive symbols and satisfy the twist condition. The authors give closed-form descriptions of the Mather and Aubry sets and show that the Mañé set is substantially larger, containing cubes of arbitrary finite dimension while the Aubry set remains totally disconnected when the potential is differentiable. They obtain the formulas by combining weak-KAM subactions, Mañé’s Lagrangian formulation, and Bangert’s variational methods to control connecting orbits in the symbolic setting. That combination is the concrete advance; it turns the usual abstract existence statements into explicit objects for this infinite-alphabet example. The topological distinction between the Aubry and Mañé sets is also cleanly drawn and worth noting. The estimates for connecting orbits are the load-bearing step. With positive mean dimension the usual compactness arguments are unavailable, so the proofs must supply a uniform bound on orbit length or cost that works for every pair of symbols in [0,1]. If that modulus is established without hidden dependence on the particular potential, the explicit formulas follow; if not, the global minimizers may not be captured as stated. The paper is written for researchers already working on variational methods in systems with infinite entropy or mean dimension. Readers who want templates for extending finite-alphabet results to continuous alphabets will find the constructions useful. It deserves a serious referee because the claims are specific and the methods are standard tools applied in a new regime; a careful check of the uniformity of the connecting estimates would settle whether the explicit descriptions hold.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to completely solve the ergodic optimization problem for the full shift on the uncountable alphabet [0,1] (a system with positive mean dimension and infinite topological entropy) when the potential depends only on the first two coordinates and satisfies the twist condition. It derives explicit formulae for the associated Mather set and Aubry set, investigates the total disconnectedness of the quotient Aubry set (with differentiability of the potential playing a key role), and gives a complete characterization of the Mañé set, showing that it contains cubes of arbitrary finite dimension. The proofs rely on estimates for connecting orbits obtained by combining the weak KAM subaction approach, Mañé’s Lagrangian formulation, and Bangert’s variational approach for twist maps.

Significance. If the connecting-orbit estimates and resulting explicit formulae hold, the work would be a substantial advance: it extends ergodic optimization and weak KAM theory from finite-alphabet or finite-dimensional twist-map settings to an infinite-dimensional symbolic system with positive mean dimension, replacing abstract existence statements with concrete descriptions of the minimizing sets. The explicit distinction between the (small) Aubry set and the (much larger) Mañé set, together with the observation that the latter contains finite-dimensional cubes, provides new structural insight. The methodological synthesis of three classical perspectives in the symbolic setting is a clear strength.

major comments (1)

[Connecting-orbit estimates (proofs section)] The explicit formulae for the Mather and Aubry sets rest on uniform estimates for connecting orbits that are asserted to follow from the combination of weak KAM, Mañé, and Bangert methods. In the uncountable-alphabet case the underlying space has positive mean dimension, which removes the compactness or finite-dimensional reduction that makes these perspectives interchangeable in the classical settings. The manuscript must therefore supply, in the section deriving the connecting-orbit estimates, an explicit uniform modulus (independent of the choice of symbols in [0,1]) that controls both length and action cost for every pair; without such a bound the passage from subaction to global minimizer is not guaranteed.

minor comments (1)

[Abstract / Introduction] The abstract refers to the “(quotient) Aubry set” without defining the equivalence relation used for the quotient; a brief clarification in the introduction or notation section would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its potential contribution to ergodic optimization in systems with positive mean dimension. We address the single major comment below.

read point-by-point responses

Referee: [Connecting-orbit estimates (proofs section)] The explicit formulae for the Mather and Aubry sets rest on uniform estimates for connecting orbits that are asserted to follow from the combination of weak KAM, Mañé, and Bangert methods. In the uncountable-alphabet case the underlying space has positive mean dimension, which removes the compactness or finite-dimensional reduction that makes these perspectives interchangeable in the classical settings. The manuscript must therefore supply, in the section deriving the connecting-orbit estimates, an explicit uniform modulus (independent of the choice of symbols in [0,1]) that controls both length and action cost for every pair; without such a bound the passage from subaction to global minimizer is not guaranteed.

Authors: We agree that an explicit uniform modulus, independent of the symbols chosen from [0,1], is required to rigorously justify the transition from subactions to global minimizers in this setting with positive mean dimension. While the manuscript derives the connecting-orbit estimates by synthesizing the weak KAM subaction approach, Mañé’s Lagrangian formulation, and Bangert’s variational method for twist maps, the uniformity of the resulting bounds with respect to the uncountable alphabet is currently implicit. In the revised version we will add a dedicated paragraph (or short subsection) in the proofs section that extracts and states an explicit uniform modulus. This modulus will bound both the length of connecting orbits and their action cost by a constant depending only on the twist condition and the Lipschitz constant of the potential, with no dependence on the specific symbols in [0,1]. The derivation will track the constants appearing in the weak KAM estimates and verify their independence from alphabet elements via the twist hypothesis, thereby making the passage to global minimizers fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation combines standard external tools

full rationale

The paper derives explicit formulae for Mather and Aubry sets in the uncountable-alphabet shift by establishing connecting-orbit estimates via the combination of weak KAM subaction methods, Mañé’s Lagrangian formulation, and Bangert’s variational approach for twist maps. These three perspectives are cited as independent, established frameworks from the literature rather than prior self-work by the authors. The central assumptions (potentials depending only on the first two coordinates and satisfying the twist condition) are stated upfront and do not presuppose the explicit formulae or the size characterizations of the Aubry/Mañé sets. No step reduces a derived quantity to a fitted parameter, self-defined object, or load-bearing self-citation chain; the results remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background facts of symbolic dynamics and the explicit twist assumption for the chosen class of potentials; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Full shift on the uncountable alphabet [0,1] is a well-defined dynamical system
Invoked throughout as the underlying space.
domain assumption Potentials depend only on the first two coordinates and satisfy the twist condition
Stated as the setting in which the explicit formulae hold.

pith-pipeline@v0.9.0 · 5758 in / 1355 out tokens · 38430 ms · 2026-05-18T11:02:22.402391+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

We completely solve ergodic optimization of a full shift with an uncountable alphabet [0,1] … for potentials depending only on the first two coordinates with the twist condition … explicit formulae of the associated Mather set and the Aubry set.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

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

estimates for connecting orbits … weak KAM subaction approach … Mañé’s formulation … Bangert’s variational approach for twist maps

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