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arxiv: 2606.30998 · v1 · pith:W2RXFQRHnew · submitted 2026-06-30 · 🧮 math.DS · math.PR

Distributional results for the shortest distance between trajectories of different dynamics

Romain Aimino , Th\'eophile Caby , Jorge Milhazes Freitas , Duarte S\'a Pinho This is my paper

Pith reviewed 2026-07-01 01:25 UTC · model grok-4.3

classification 🧮 math.DS math.PR
keywords extreme value distributionsdynamical systemsstrongly mixing mapsextremal indexinvariant measuresco-dimensiontrajectory encounters
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The pith

Extreme value distributions describe the shortest distance between trajectories of different strongly mixing maps.

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

The paper shows that the minimal distance between points on trajectories from two different maps on the same space obeys an extreme value law in the limit of long trajectories. This law depends on the lengths of the two trajectories, the co-dimension of their invariant measures, and an extremal index that measures how quickly nearby points separate under the two dynamics. A formula is provided for the extremal index in the case of chaotic interval maps, and for the co-dimension when the measures have densities featuring isolated zeros and singularities. Examples of such systems are given with explicit computations of the modulating parameters. This framework quantifies the compatibility of distinct dynamical systems through their closest encounters.

Core claim

We establish Extreme Value Distributions for the closest encounter between trajectories generated by different maps defined in the same reference phase space. For a class of strongly mixing maps, we show that the limit distribution depends on the length of the different trajectories and the co-dimension of the associated invariant measures. It is also modulated by an Extremal Index, that informs on the tendency of nearby points to diverge along with the evolution of their respective dynamics, serving as an indicator of their compatibility. We give a formula for this quantity for a class of chaotic maps of the interval and for the co-dimension in the case when the respective measures admit de

What carries the argument

The extremal index modulating the extreme value distribution for minimal inter-trajectory distances, reflecting divergence tendency of nearby points under different dynamics.

Load-bearing premise

The maps are strongly mixing and their invariant measures have densities with isolated zeros and singularities for the co-dimension formula.

What would settle it

For two concrete strongly mixing maps on the interval with computed co-dimensions, the histogram of minimal distances over many pairs of long trajectories fails to converge to the predicted extreme value distribution.

read the original abstract

We establish Extreme Value Distributions for the closest encounter between trajectories generated by different maps defined in the same reference phase space. For a class of strongly mixing maps, we show that the limit distribution depends on the length of the different trajectories and the co-dimension of the associated invariant measures. It is also modulated by an Extremal Index, that informs on the tendency of nearby points to diverge along with the evolution of their respective dynamics, serving as an indicator of their compatibility. We give a formula for this quantity for a class of chaotic maps of the interval and for the co-dimension in the case when the respective measures admit densities with isolated zeros and singularities. We present diverse examples of systems satisfying these assumptions and compute the different parameters modulating the limit distribution.

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 manuscript establishes extreme value distributions for the shortest distance between trajectories of two different maps acting on the same phase space. Under the assumption that the maps are strongly mixing, the limiting distribution is shown to depend on the lengths of the two trajectories, the codimension of the associated invariant measures, and an extremal index that quantifies the tendency of nearby points to diverge under the respective dynamics. Explicit formulas are derived for the extremal index in the case of a class of interval maps and for the codimension when the invariant measures admit densities possessing isolated zeros and singularities. Several concrete examples satisfying the hypotheses are worked out and the modulating parameters are computed explicitly.

Significance. If the derivations hold, the results supply a new class of limit theorems that compare orbits generated by distinct dynamical systems rather than within a single system. The explicit dependence on trajectory length, codimension, and the extremal index, together with closed-form expressions under stated regularity conditions on the densities, offers a concrete tool for quantifying dynamical compatibility. The provision of worked examples strengthens the applicability of the theory.

minor comments (3)
  1. [Introduction] The abstract states that formulas are derived, yet the manuscript would benefit from a brief outline of the main steps in the proof of the extreme-value limit (e.g., the role of the mixing assumption in controlling the dependence between the two orbits) already in the introduction.
  2. Notation for the codimension of the invariant measures should be introduced once and used consistently; the current alternation between “co-dimension” and “codimension” is minor but distracting.
  3. [Examples] In the examples section, the numerical verification of the extremal index could be accompanied by a short table comparing the theoretical value with the empirical estimate obtained from finite trajectories.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report, so we have no points to address individually. We will incorporate any minor editorial or presentational suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives extreme value limit laws for shortest distances between trajectories of distinct strongly mixing maps under explicit hypotheses on mixing rates and on the densities of the invariant measures (isolated zeros/singularities). These assumptions are stated up front and the limit formulas (including the extremal index for interval maps and the codimension term) are obtained from the mixing and density conditions rather than by fitting parameters to the target quantities or by reducing to a self-citation chain. No equation is shown to be definitionally equivalent to its input, and no load-bearing uniqueness theorem is imported from the authors' prior work. The derivation therefore remains self-contained against the stated external assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger is necessarily incomplete; only the abstract was available. No free parameters or invented entities are mentioned. The two domain assumptions below are the explicit prerequisites stated for the main theorems.

axioms (2)
  • domain assumption The maps are strongly mixing
    Required to obtain the extreme value limit distributions for the shortest distance.
  • domain assumption Invariant measures admit densities with isolated zeros and singularities
    Needed to obtain an explicit formula for the co-dimension modulating the limit law.

pith-pipeline@v0.9.1-grok · 5662 in / 1205 out tokens · 45265 ms · 2026-07-01T01:25:27.411997+00:00 · methodology

discussion (0)

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