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arxiv: 2603.17091 · v3 · pith:5DONGLU7new · submitted 2026-03-17 · 🧮 math.DS

On quantization and the classical variational principle for the metric mean dimension

Maria Carvalho , Gustavo Pessil This is my paper

Pith reviewed 2026-05-21 10:12 UTC · model grok-4.3

classification 🧮 math.DS
keywords metric mean dimensionmean quantization dimensionvariational principleinvariant measurestopological dynamical systemsinfinite entropyergodic theory
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The pith

Metric mean dimension of a dynamical system equals the supremum of mean quantization dimensions over its invariant measures.

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

The paper establishes that the metric mean dimension equals the largest mean quantization dimension achieved by any invariant probability measure on the system. This supplies a variational principle that works even when entropy is infinite by letting one measure encode the scaling of information at every length scale. The equality also permits interchanging limits and suprema inside related variational principles for metric mean dimension, a step most entropy-like quantities cannot take because they lack suitable convexity properties. The authors further verify that Katok entropy and Shapira entropy obey the required conditions, guaranteeing that maximizing measures exist.

Core claim

For a topological dynamical system that admits a well-defined metric mean dimension, this quantity coincides exactly with the supremum of the mean quantization dimensions taken over all invariant probability measures. The mean quantization dimension of a measure is obtained by averaging, across scales, the minimal number of bits needed to quantize typical orbits according to that measure. This identification yields a classical variational principle and shows that limits and suprema may be exchanged in the Lindenstrauss-Tsukamoto formulations.

What carries the argument

Mean quantization dimension of an invariant measure, defined by averaging the minimal quantization error across successive scales and used to recover the global metric mean dimension via supremum.

If this is right

  • Systems with infinite entropy can be analyzed by locating a single invariant measure whose mean quantization dimension equals the overall metric mean dimension.
  • Limits and suprema may be interchanged inside the Lindenstrauss-Tsukamoto variational principles for metric mean dimension.
  • Katok entropy and Shapira entropy each produce a classical variational principle for metric mean dimension with the existence of maximizing measures.
  • The approach supplies a measure-theoretic witness for the scaling complexity of the system at every length scale.

Where Pith is reading between the lines

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

  • Numerical optimization over invariant measures could furnish practical approximations to metric mean dimension in concrete examples.
  • The same quantization construction might extend to other dimension-type invariants that currently lack variational principles.
  • The result may link quantization techniques from information theory more directly to questions of orbit complexity in topological dynamics.

Load-bearing premise

The system possesses a well-defined metric mean dimension together with a sufficiently rich collection of invariant measures for which the mean quantization dimension is defined and the supremum is approachable.

What would settle it

A concrete topological dynamical system in which the metric mean dimension is strictly larger than the mean quantization dimension of every invariant probability measure, or in which the supremum is never attained.

read the original abstract

This paper investigates the relationship between quantization of measures and metric mean dimension of topological dynamical systems. We introduce the concept of mean quantization dimension for invariant probability measures and establish a classical variational principle: the metric mean dimension of a dynamical system is equal to the maximum mean quantization dimension among all invariant measures. This approach effectively characterizes the complexity of systems with infinite entropy by identifying a measure that captures information across all scales; and yields a fundamental property that allows for the exchange of limits and suprema in the Lindenstrauss-Tsukamoto variational principles, a feat that most known entropy-like maps fail to achieve due to convexity. Nevertheless, we show that the Katok and Shapira entropies do satisfy this property and, therefore, a classical variational principle for the metric mean dimension, for which maximizing measures always exist.

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.

Referee Report

0 major / 3 minor

Summary. The paper introduces the mean quantization dimension for invariant probability measures on compact metric dynamical systems and proves a variational principle asserting that the metric mean dimension equals the supremum of these mean quantization dimensions over all invariant measures. It further shows that this characterization permits the interchange of limits and suprema in the Lindenstrauss-Tsukamoto variational principles, a property that fails for most entropy-like maps owing to convexity, and verifies that the Katok and Shapira entropies satisfy the requisite non-convexity, thereby guaranteeing the existence of maximizing measures.

Significance. If the central equality is established without circularity, the result supplies a new, measure-theoretic description of metric mean dimension that is particularly useful for systems of infinite entropy. The explicit use of non-convexity to justify limit-supremum exchange constitutes a technical advance that may extend to other dimension-like invariants. The confirmation that Katok and Shapira entropies admit maximizing measures strengthens the applicability of classical variational principles in this setting.

minor comments (3)
  1. The abstract refers to 'the classical variational principle' without a one-sentence reminder of its standard statement; adding this would improve accessibility for readers outside the immediate subfield.
  2. Notation for the mean quantization dimension (introduced in paragraph 2) should be fixed early and used consistently; the current text occasionally reverts to the full descriptive phrase after the definition.
  3. The discussion of the non-convexity property of the quantization map would benefit from a short explicit verification or reference to the precise lemma establishing that the map is not convex, rather than relying solely on the general statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. We are pleased that the report recommends minor revision and accurately summarizes the main results on the mean quantization dimension and its relation to metric mean dimension via a variational principle.

Circularity Check

0 steps flagged

No significant circularity; variational principle derived from independent definitions

full rationale

The paper defines metric mean dimension via standard covering-based scaling of orbit segments and introduces mean quantization dimension independently via distortion-rate scaling for invariant measures. The central claim equates the former to the supremum of the latter over all invariant measures, with both directions of the inequality proved separately from the respective definitions; exchange of limits and suprema follows from non-convexity of the quantization map rather than any definitional reduction. No load-bearing step collapses to a self-citation, fitted parameter renamed as prediction, or ansatz smuggled via prior work by the same authors. The result is a genuine relation between two independently defined quantities on compact metric dynamical systems, consistent with the absence of circularity indicators in the abstract and derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard background facts from topological dynamics plus the new definition of mean quantization dimension; no free parameters or invented physical entities appear.

axioms (1)
  • domain assumption Topological dynamical systems possess invariant probability measures and a well-defined metric mean dimension
    Invoked implicitly when the variational equality is stated for general systems.
invented entities (1)
  • mean quantization dimension no independent evidence
    purpose: to quantify scale-dependent approximation quality of invariant measures
    New quantity introduced to obtain the variational principle; no independent empirical test supplied.

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discussion (0)

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