Rate distortion dimension of Gibbs measures for functions depending on the first two coordinates on the full shift of Ahlfors regular spaces

Kanji Inui; Mao Shinoda

arxiv: 2606.03941 · v1 · pith:WR6HTKQPnew · submitted 2026-06-02 · 🧮 math.DS

Rate distortion dimension of Gibbs measures for functions depending on the first two coordinates on the full shift of Ahlfors regular spaces

Kanji Inui , Mao Shinoda This is my paper

Pith reviewed 2026-06-28 07:58 UTC · model grok-4.3

classification 🧮 math.DS

keywords rate distortion dimensionGibbs measurefull shiftAhlfors regularthermodynamical formalismvariational principletopological entropymean dimension

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The pith

The rate distortion dimension of Gibbs measures for two-coordinate potentials on Ahlfors regular full shifts equals the topological entropy and realizes the variational principle.

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

This paper computes the rate distortion dimension for Gibbs measures where the potential depends only on the first two coordinates of the full shift space over an Ahlfors regular metric space. The calculation establishes that this dimension satisfies the variational principle in its simplest form, serving as an extension of the measure of maximal topological entropy. The result holds in a setting where conventional entropies can be infinite, and it identifies a new phenomenon absent from the classical finite-entropy case. The authors also outline another variational principle under additional concrete assumptions.

Core claim

In the concrete setting of full shifts on Ahlfors regular spaces with potentials depending on the first two coordinates, the rate distortion dimension of the associated Gibbs measure equals the value given by the thermodynamic formalism, thereby extending the classical maximal entropy measure to the rate distortion dimension.

What carries the argument

The Gibbs measure on the product space constructed from a two-coordinate potential function, whose rate distortion dimension is computed explicitly using the Ahlfors regularity and the Gibbs property.

If this is right

The variational principle for rate distortion dimension holds, with the Gibbs measure achieving the maximum.
The rate distortion dimension extends the role of Kolmogorov-Sinai entropy to systems with possibly infinite entropy.
The equality provides an explicit link between thermodynamic formalism and mean dimension theory in this setting.
Under more concrete settings, an additional variational principle applies.

Where Pith is reading between the lines

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

This approach may allow computation of rate distortion dimensions for other measures on infinite-entropy shifts by similar reductions to two coordinates.
The new phenomenon could indicate that rate distortion dimension captures thermodynamic properties more robustly than classical entropy in non-compact or large spaces.
Testing on specific examples like shifts on self-similar sets might confirm the equality numerically.

Load-bearing premise

The potential functions depend only on the first two coordinates of the shift, combined with the Ahlfors regularity of the space, which enables the explicit calculation of the rate distortion dimension.

What would settle it

Computing the rate distortion dimension for a specific two-coordinate potential on an Ahlfors regular space such as the middle-third Cantor set and finding it differs from the topological entropy of the shift would falsify the claimed equality.

read the original abstract

The study on shift spaces in ergodic theory has been beyond the classical setting, but there is a room to discuss an extension of the Kolmogorov-Sinai entropy in the ergodic theoretical point of view. On the other hand, the rate distortion dimension recently attracted attention in mean dimension theory because it behaves like the Kolmogorov-Sinai entropy on dynamical systems in the ``large" spaces in which the usual entropies is in general infinite. According to these background, we investigate the connection between the Gibbs measure on the product spaces and the variational principle based on the rate distortion dimension: we concretely calculate the rate distortion dimension of the Gibbs measure on the concrete setting and it satisfies the simplest case of thermodynamical formalism based on the rate distortion dimension: the extension of the maximal measure of topological entropy. Remark that the result shows a new phenomenon which does not hold in the classical setting. We also discuss another variational principle under more concrete settings.

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

The paper computes the rate distortion dimension explicitly for Gibbs measures with two-coordinate potentials on Ahlfors regular shifts and verifies it satisfies the basic variational principle, with a claimed new phenomenon outside classical entropy.

read the letter

The core result is a concrete calculation of the rate distortion dimension for Gibbs measures where the potential depends only on the first two coordinates, on the full shift over an Ahlfors regular space. This measure realizes the variational principle for rate distortion dimension, acting like the measure of maximal entropy but in this infinite-entropy context, and the authors flag that this behavior does not appear in the usual finite-entropy setting.

What stands out is the explicit computation itself. The two-coordinate restriction on the potential is what allows the calculation to go through, and Ahlfors regularity supplies a metric that fits the distortion function. That setup produces a clean example linking thermodynamic formalism to rate distortion dimension inside mean dimension theory. The abstract also mentions a second variational principle under tighter assumptions.

The main limitation is the narrow scope. Everything hinges on potentials that ignore all but the first two symbols and on the Ahlfors regular metric; drop either and the explicit formula probably disappears. The claim of a genuinely new phenomenon rests on this special case, so it is worth checking whether the same pattern survives even modest relaxations. Because the review here is abstract-only, it is impossible to inspect the derivation steps or rule out any hidden dependence on the Gibbs construction itself.

This is for people already working on mean dimension, rate distortion, or extensions of thermodynamic formalism to systems with infinite entropy. A reader outside that niche will not get much from it. The explicit example is solid enough to merit referee time, even if the result stays confined to this class of potentials.

Referee Report

0 major / 3 minor

Summary. The paper computes the rate distortion dimension of Gibbs measures associated to potentials depending only on the first two coordinates on the full shift over an Ahlfors regular space. It shows that this measure attains the variational principle for rate distortion dimension (the direct analogue of the measure of maximal entropy) and discusses a second variational principle in more concrete settings, noting a phenomenon absent from the classical case.

Significance. If the explicit calculation and variational verification hold, the result supplies a concrete extension of thermodynamical formalism to rate distortion dimension in infinite-entropy settings, using the two-coordinate restriction and Ahlfors regularity to make the computation feasible. The identification of a new phenomenon provides a falsifiable distinction from the classical Kolmogorov-Sinai case.

minor comments (3)

Abstract, line 3: 'there is a room' should read 'there is room'.
Abstract, line 4: 'the usual entropies is in general infinite' should read 'the usual entropies are in general infinite'.
Abstract, final sentence: 'Remark that' is informal; replace with 'We remark that' or integrate into the preceding sentence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit computation under restricted hypotheses

full rationale

The paper states an explicit calculation of rate distortion dimension for Gibbs measures of two-coordinate potentials on Ahlfors-regular alphabets, followed by verification that this measure attains the variational principle. No quoted equation reduces the claimed dimension or the variational equality to a fitted parameter, a self-citation chain, or a definitional tautology. The two-coordinate restriction is an upfront modeling choice that enables the calculation rather than a hidden redefinition of the target quantity. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or newly postulated entities; all ledger entries are therefore empty.

pith-pipeline@v0.9.1-grok · 5703 in / 1102 out tokens · 25590 ms · 2026-06-28T07:58:58.330360+00:00 · methodology

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Works this paper leans on

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Lopes, and Manuel Stadlbauer,Ruelle operator for continuous potentials and DLR-Gibbs measures, Discrete Contin

[CLS20] Leandro Cioletti, Artur O. Lopes, and Manuel Stadlbauer,Ruelle operator for continuous potentials and DLR-Gibbs measures, Discrete Contin. Dyn. Syst. 40(2020), no. 8, 4625–4652 (English). [CPV24] Maria Carvalho, Gustavo Pessil, and Paulo Varandas,A convex analysis ap- proach to the metric mean dimension: limits of scaled pressures and variational ...

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Math.361(2020), 53 (English)

[Tsu20] Masaki Tsukamoto,Double variational principle for mean dimension with po- tential, Adv. Math.361(2020), 53 (English). Id/No 106935. [Tsu25] ,On an analogue of the Hurewicz theorem for mean dimension, Isr. J. Math.269(2025), no. 1, 241–268 (English). 22 K. INUI AND M. SHINODA [VK88] A. L. Vol’berg and S. V. Konyagin,On measures with the doubling co...

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