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arxiv: 2507.18736 · v1 · submitted 2025-07-24 · 🧮 math.DS

Maximizing measures for countable alphabet shifts via blur shift spaces

Eduardo Garibaldi , Jo\~ao T A Gomes , Marcelo Sobottka This is my paper

Pith reviewed 2026-05-19 02:15 UTC · model grok-4.3

classification 🧮 math.DS
keywords maximizing measurescountable alphabet shiftsblur shift spacesupper semi-continuous potentialsinvariant probabilitiescompactificationsymbolic dynamics
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The pith

Upper semi-continuous potentials on countable alphabet shifts have maximizing measures under blur shift compactification.

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

The paper shows that for upper semi-continuous potentials defined on shifts over countable alphabets, sufficient conditions can be given for the existence of a maximizing measure. It does this by using blur shift spaces, a method of compactification that adds new symbols corresponding to blurred subsets of the alphabet. This allows the results to apply even when the shift map is discontinuous on the original space. A sympathetic reader would care because countable alphabets arise in many models of symbolic dynamics and ergodic theory where standard compactness assumptions fail, so this extends the applicability of maximizing measure theory to more general settings.

Core claim

For upper semi-continuous potentials defined on shifts over countable alphabets, sufficient conditions are ensured for the existence of a maximizing measure using blur shift spaces. The approach extends beyond the Markovian case to encompass more general countable alphabet shifts. In particular, it guarantees a convex characterization and compactness for the set of blur invariant probabilities with respect to the discontinuous shift map.

What carries the argument

The blur shift space, a compactification obtained by adding new symbols given by blurred subsets of the alphabet, which preserves topological and dynamical properties to enable existence proofs for maximizing measures.

If this is right

  • The set of blur invariant probabilities is compact.
  • The set of maximizing measures admits a convex characterization.
  • The existence result holds for general countable alphabet shifts beyond just the Markovian ones.

Where Pith is reading between the lines

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

  • This compactification technique could be applied to study equilibrium states or other thermodynamic quantities in non-compact symbolic spaces.
  • Blur shifts might provide a way to approximate measures numerically by considering finite blur resolutions.
  • Similar compactification ideas could be useful for other discontinuous maps in dynamical systems.

Load-bearing premise

The blur shift construction provides a compactification that preserves the necessary topological and dynamical properties to apply existence results for maximizing measures even when the shift map is discontinuous on the original space.

What would settle it

Finding a specific upper semi-continuous potential on a countable alphabet shift for which no maximizing measure exists even after applying the blur shift compactification would falsify the claim.

Figures

Figures reproduced from arXiv: 2507.18736 by Eduardo Garibaldi, Jo\~ao T A Gomes, Marcelo Sobottka.

Figure 1
Figure 1. Figure 1: Graphical representations of convex decompositions of the set of 𝜎ˆ -invariant measures for blur shifts with one-symbol and three-symbol resolutions. Proof. By the convexity of the set of blur invariant probabilities, it follows Conv M (Σ,𝜎) ∪  𝛿(𝐵𝑟 ,𝐵𝑟 ,...) : 1 ≤ 𝑟 ≤ 𝑠  ⊂ M Σˆ,𝜎ˆ  . On the other hand, assume that 𝜇ˆ ∈ M Σˆ,𝜎ˆ  and remember that Σ = Σ ˆ ⊔𝜕Σ. We will consider three cases [PITH_FULL_IM… view at source ↗
Figure 2
Figure 2. Figure 2: Graph representation of conditions (C2) and (C3) in Definition 7.1. In the following results, we will show that any shift (Λ,𝜎) verifying the conditions in the above Definition 7.1 possesses the properties: • it is non-Markovian (Lemma 7.1); • it obeys the finite cyclic predecessor assumption (Lemma 7.2); • it is non-locally compact and the set 𝜕Λ\ Lˆ 0 is non-empty (Lemma 7.3); • it is topologically trans… view at source ↗
read the original abstract

For upper semi-continuous potentials defined on shifts over countable alphabets, this paper ensures sufficient conditions for the existence of a maximizing measure. We resort to the concept of blur shift, introduced by T. Almeida and M. Sobottka as a compactification method for countable alphabet shifts consisting of adding new symbols given by blurred subsets of the alphabet. Our approach extends beyond the Markovian case to encompass more general countable alphabet shifts. In particular, we guarantee a convex characterization and compactness for the set of blur invariant probabilities with respect to the discontinuous shift map.

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

1 major / 2 minor

Summary. The paper claims that for upper semi-continuous potentials on shifts over countable alphabets, the blur shift compactification (adding blurred subsets as new symbols) yields sufficient conditions for the existence of a maximizing measure. It extends prior Markovian results to general countable-alphabet shifts, providing a convex characterization of the relevant measures and proving compactness of the set of blur-invariant probabilities with respect to the discontinuous shift map.

Significance. If the compactness and existence arguments hold, the work would supply a concrete compactification tool for ergodic optimization on countable alphabets, where standard compactness fails. The convex characterization and extension beyond Markovian shifts could enable further variational analysis of maximizing measures in symbolic dynamics.

major comments (1)
  1. [Compactness of blur invariant probabilities] The compactness proof for the set of blur-invariant probabilities (the key step before applying the existence theorem for upper semi-continuous functions) must explicitly verify that weak*-limits of invariant measures remain invariant. Because the shift map is discontinuous, the push-forward map on measures is not weak*-continuous, so invariance (μ ∘ σ^{-1} = μ) is not automatically closed; without a separate argument using the blur symbols or approximation, the set need not be compact and the central existence claim does not follow from the standard variational argument.
minor comments (2)
  1. [Introduction] Clarify in the introduction how the new convex characterization and non-Markovian extension differ from the original blur-shift construction of Almeida and Sobottka.
  2. [Main results] Add a brief remark on whether the upper semi-continuity of the potential is preserved under the embedding into the blur shift space.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying a key technical point regarding the compactness argument. We address the major comment below and will incorporate revisions to strengthen the exposition.

read point-by-point responses

  1. Referee: The compactness proof for the set of blur-invariant probabilities (the key step before applying the existence theorem for upper semi-continuous functions) must explicitly verify that weak*-limits of invariant measures remain invariant. Because the shift map is discontinuous, the push-forward map on measures is not weak*-continuous, so invariance (μ ∘ σ^{-1} = μ) is not automatically closed; without a separate argument using the blur symbols or approximation, the set need not be compact and the central existence claim does not follow from the standard variational argument.

    Authors: We agree that the discontinuity of the shift map means invariance is not automatically preserved under weak* limits, and an explicit verification is required for the compactness claim. In the manuscript, the proof of compactness for blur-invariant probabilities relies on the blur shift compactification: the blurred symbols are used to approximate the action of the shift and to control the discrepancy in the invariance condition for sequences of measures. Specifically, the convex characterization of blur-invariant probabilities and the topology on the compactification allow us to pass to the limit by testing against continuous functions that are constant on blurred sets. To make this step fully explicit and address the referee's concern, we will add a dedicated lemma in the revised version that directly shows weak* limits of blur-invariant measures remain blur-invariant, using approximation by the blur symbols. This revision will clarify the argument without altering the main results. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent topological properties of blur compactification

full rationale

The paper extends the blur shift compactification (introduced in prior work by Almeida and Sobottka) to non-Markovian countable-alphabet shifts and derives a convex characterization plus compactness for the set of blur-invariant probabilities. These properties are obtained from the explicit construction of the compactified space and the definition of invariance under the (discontinuous) shift, without any reduction of the target existence result to a fitted parameter, self-referential definition, or unverified self-citation chain. The central claim follows from applying standard upper-semicontinuous maximization arguments on the compact space, which remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions in topological dynamics and the previously introduced blur shift construction; no new free parameters or invented entities are apparent from the abstract.

axioms (2)
  • domain assumption Upper semi-continuity of the potential function ensures the existence of maximizers in compact settings.
    Invoked to apply standard variational principles or existence theorems in the compactified space.
  • domain assumption The blur shift space is a valid compactification preserving relevant dynamical properties.
    Central to the method, introduced in prior work.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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