---
title: Conditional Local Topological Intricacy
url: https://www.emergentmind.com/topics/conditional-local-topological-intricacy
type: topic
---

# Conditional Local Topological Intricacy

Searching arXiv for the focal paper and closely related work on intricacy, average sample complexity, and local weighted pressure.
Conditional local topological intricacy is a fiberwise dynamical invariant for amenable group actions that localizes the notions of intricacy and average sample complexity relative to a factor system. For two $G$-systems $(X,G)$ and $(Y,G)$, where $G$ is an infinite countable discrete amenable group and $X$, $Y$ are compact metric spaces, the theory introduced in "Local intricacy and average sample complexity for amenable group actions" formalizes how combinatorial and entropic complexity should be measured not on $X$ globally, but along the fibers of a factor map onto $Y$ [2509.20738]. In this framework, conditional local topological intricacy and conditional local average sample complexity are defined from finite covers, their measure-theoretic analogs are defined from conditional entropies, and the resulting quantities satisfy identities, existence theorems, an ergodic decomposition formula, equality results for the cover-based $\mathrm{Asc}_\mu^\pm$ variants, and a local variational principle [2509.20738]. When $(Y,G)$ is trivial, these conditional definitions coincide with the unconditional notions of dynamical intricacy and average sample complexity introduced earlier [1512.01143].

## 1. Setting, notation, and conceptual position

The ambient objects are $G$-systems $(X,G)$ and $(Y,G)$, with $G$ countable, discrete, infinite, and amenable, together with a factor map $\varphi:(X,G)\to (Y,G)$ or $\pi:(X,G)\to (Y,G)$, depending on the formulation used in the definition [2509.20738]. The theory uses finite covers and finite partitions:
- $\mathcal{C}_X$: finite covers of $X$
- $\mathcal{C}_X^o$: finite open covers
- $\mathcal{P}_X$: finite partitions

A Følner sequence is written $\{F_n\}_{n\in\mathbb N}$, and the paper fixes a uniform system of coefficients
\[
c^{F_n}_S = 2^{-|F_n|}, \qquad S\subseteq F_n.
\]
For finite $S\subset G$, joins of translates are denoted by
\[
\mathcal{U}_S = \bigvee_{g\in S} g^{-1}\mathcal U,
\qquad
\alpha_S = \bigvee_{g\in S} g^{-1}\alpha.
\]
This places the theory in direct continuity with the earlier dynamical definitions of intricacy and average sample complexity, where analogous expressions were formed from $\log N(\mathscr U_S)$ or $H_\mu(\alpha_S)$ and then averaged over subsets $S$ with symmetric coefficients [1512.01143].

The conditional aspect is explicit: the relevant combinatorial quantity is not $N(\mathcal U_S)$ on $X$ itself, but $N(\mathcal U_S\mid Y)$, defined through the fibers of the factor map. The paper states that these formulas localize intricacy and average sample complexity by measuring them “along the fibers over $Y$,” thereby producing a conditional, local version sensitive to the factor $Y$ [2509.20738]. This suggests that the invariant is best viewed as a relative local complexity measure situated between local entropy theory and the earlier unconditional intricacy formalism.

## 2. Conditional local topological definitions

For a finite cover $\mathcal U\in \mathcal C_X$ and $y\in Y$, the fiberwise cover number is
\[
N(\mathcal U\mid y):=N(\mathcal U\mid \varphi^{-1}(y)),
\]
where $N(\mathcal U\mid K)$ is the minimal cardinality of a subcover of $\mathcal U$ covering $K$ [2509.20738]. The uniformized conditional cover number is then
\[
N(\mathcal U_S\mid Y):=\sup_{y\in Y} N(\mathcal U_S\mid y).
\]

With this notation, the conditional local topological intricacy is
\[
\mathrm{Int}_{\mathrm{top}}(G,\mathcal U\mid Y)
=
\lim_{n\to\infty}\frac{1}{|F_n|}
\sum_{S\subset F_n} c_S^{F_n}
\log\!\left(
\frac{N(\mathcal U_S\mid Y)\,N(\mathcal U_{F_n\setminus S}\mid Y)}
{N(\mathcal U_{F_n}\mid Y)}
\right),
\]
and the corresponding conditional local topological average sample complexity is
\[
\mathrm{Asc}_{\mathrm{top}}(G,\mathcal U\mid Y)
=
\lim_{n\to\infty}\frac{1}{|F_n|}
\sum_{S\subset F_n} c_S^{F_n}\log N(\mathcal U_S\mid Y)
\]
[2509.20738].

The system-level quantities are obtained by taking suprema over finite open covers:
\[
\mathrm{Int}_{\mathrm{top}}(G,X\mid Y)
=
\sup_{\mathcal U\in \mathcal C_X^o}
\mathrm{Int}_{\mathrm{top}}(G,\mathcal U\mid Y),
\]
\[
\mathrm{Asc}_{\mathrm{top}}(G,X\mid Y)
=
\sup_{\mathcal U\in \mathcal C_X^o}
\mathrm{Asc}_{\mathrm{top}}(G,\mathcal U\mid Y).
\]

A compact summary of the topological part is given below.

| Quantity | Input | Definition pattern |
|---|---|---|
| $\mathrm{Int}_{\mathrm{top}}(G,\mathcal U\mid Y)$ | finite cover $\mathcal U$ | averaged log-ratio built from $N(\mathcal U_S\mid Y)$ |
| $\mathrm{Asc}_{\mathrm{top}}(G,\mathcal U\mid Y)$ | finite cover $\mathcal U$ | averaged $\log N(\mathcal U_S\mid Y)$ |
| $\mathrm{Int}_{\mathrm{top}}(G,X\mid Y)$ | system $(X,G)\to(Y,G)$ | supremum over $\mathcal C_X^o$ |
| $\mathrm{Asc}_{\mathrm{top}}(G,X\mid Y)$ | system $(X,G)\to(Y,G)$ | supremum over $\mathcal C_X^o$ |

The formal similarity with the original topological intricacy and average sample complexity is exact at the level of structure: the conditional theory replaces absolute cover growth by fiberwise cover growth relative to the factor [1512.01143; 2509.20738].

## 3. Conditional local measure-theoretic formulations

Let $\mu\in\mathcal M(X,G)$ be a $G$-invariant Borel probability measure and let $\pi:(X,G)\to(Y,G)$ be a factor map. For a finite partition $\alpha\in\mathcal P_X$, the conditional entropy is defined by
\[
H_\mu(\alpha\mid Y)
:=
\sum_{A\in\alpha}
\int_X
-\mathbb E(1_A\mid \pi^{-1}(\mathcal B(Y)))
\log \mathbb E(1_A\mid \pi^{-1}(\mathcal B(Y)))
\,d\mu
\]
[2509.20738].

Using $\alpha_S=\bigvee_{g\in S} g^{-1}\alpha$, the conditional local measure-theoretical intricacy is
\[
\mathrm{Int}_\mu(G,\alpha\mid Y)
=
\lim_{n\to\infty}\frac{1}{|F_n|}
\sum_{S\subset F_n} c_S^{F_n}
\Big[
H_\mu(\alpha_S\mid Y)
+
H_\mu(\alpha_{F_n\setminus S}\mid Y)
-
H_\mu(\alpha_{F_n}\mid Y)
\Big],
\]
and the conditional local measure-theoretical average sample complexity is
\[
\mathrm{Asc}_\mu(G,\alpha\mid Y)
=
\lim_{n\to\infty}\frac{1}{|F_n|}
\sum_{S\subset F_n} c_S^{F_n} H_\mu(\alpha_S\mid Y)
\]
[2509.20738].

The supremum versions are
\[
\mathrm{Int}_\mu(G,X\mid Y)
=
\sup_{\alpha\in\mathcal P_X}\mathrm{Int}_\mu(G,\alpha\mid Y),
\qquad
\mathrm{Asc}_\mu(G,X\mid Y)
=
\sup_{\alpha\in\mathcal P_X}\mathrm{Asc}_\mu(G,\alpha\mid Y).
\]

The paper also introduces cover-based measure-theoretic quantities. For a finite cover $\mathcal U\in\mathcal C_X$,
\[
H_\mu(\mathcal U\mid Y)
:=
\inf_{\alpha\in\mathcal P_X,\ \alpha\succeq\mathcal U}
H_\mu(\alpha\mid Y),
\]
where $\alpha\succeq\mathcal U$ means that $\alpha$ is finer than $\mathcal U$ [2509.20738]. From this one defines
\[
\mathrm{Asc}_\mu^{-}(G,\mathcal U\mid Y)
=
\lim_{n\to\infty}\frac{1}{|F_n|}
\sum_{S\subset F_n} c_S^{F_n} H_\mu(\mathcal U_S\mid Y),
\]
and
\[
\mathrm{Asc}_\mu^{+}(G,\mathcal U\mid Y)
=
\inf_{\alpha\in\mathcal P_X,\ \alpha\succeq\mathcal U}
\mathrm{Asc}_\mu(G,\alpha\mid Y).
\]
Their system-level versions are obtained by taking suprema over finite covers [2509.20738].

The distinction between $\mathrm{Asc}_\mu^{-}$ and $\mathrm{Asc}_\mu^{+}$ is structural. The paper states that $\mathrm{Asc}_\mu^{-}$ uses the cover directly, while $\mathrm{Asc}_\mu^{+}$ first replaces the cover by all finer partitions and then computes average sample complexity [2509.20738]. This is the measure-theoretic analog of the standard cover-versus-partition tension in local entropy theory.

## 4. Identities, existence, and regularity properties

A central identity connects conditional local topological intricacy to conditional local topological average sample complexity and conditional topological entropy:
\[
\mathrm{Int}_{\mathrm{top}}(G,\mathcal U\mid Y)
=
2\mathrm{Asc}_{\mathrm{top}}(G,\mathcal U\mid Y)
-
h_{\mathrm{top}}(G,\mathcal U\mid Y),
\]
where
\[
h_{\mathrm{top}}(G,\mathcal U\mid Y)
=
\lim_{n\to\infty}\frac{1}{|F_n|}\log N(\mathcal U_{F_n}\mid Y)
\]
is the conditional topological entropy [2509.20738]. This reproduces, in conditional local form, the same algebraic relation that ties intricacy, average sample complexity, and entropy in the earlier unconditional setting [1512.01143].

The paper further states that the relevant limits exist and are independent of the choice of Følner sequence, due to sub-additivity and the Ornstein-Weiss theorem [2509.20738]. This point is technically important: it means the invariants belong to the intrinsic asymptotic structure of the amenable action, rather than to a particular averaging scheme.

For open covers, both $\mathrm{Asc}_\mu^+_\,$ and $\mathrm{Asc}_\mu^-_\,$ are upper semi-continuous on the space of invariant measures $\mathcal M(X,G)$ [2509.20738]. The paper also records that the function $H_\mu(\mathcal U_S\mid Y)$ is increasing as a function of the coarseness of the cover [2509.20738]. In the broader architecture of local invariants, these regularity properties place average sample complexity close to local entropy and local pressure constructions, where upper semi-continuity and open-cover approximation play a foundational role. A plausible implication is that the theory is designed to interact naturally with localized variational principles of the kind developed for local weighted topological pressure [2307.08021].

## 5. Ergodic decomposition and the equality of $\mathrm{Asc}_\mu^{-}$ and $\mathrm{Asc}_\mu^{+}$

If $\mu=\int \mu_\omega\,dm(\omega)$ is the ergodic decomposition of a $G$-invariant measure, then the paper proves
\[
\mathrm{Asc}_\mu^+(G,\mathcal U\mid Y)
=
\int \mathrm{Asc}_{\mu_\omega}^+(G,\mathcal U\mid Y)\,dm(\omega),
\]
and similarly for $\mathrm{Asc}_\mu^{-}$ [2509.20738]. The accompanying interpretation in the paper is that average sample complexity is affine in measure. This is one of the principal structural facts of the theory.

The equality problem for the two cover-based measure-theoretic versions is resolved in two stages. For $G=\mathbb Z$, Theorem 4.3 states that for every cover $\mathcal U$ and invariant measure $\mu$,
\[
\mathrm{Asc}_\mu^{-}(T,\mathcal U\mid Y)
=
\mathrm{Asc}_\mu^{+}(T,\mathcal U\mid Y)
\]
[2509.20738]. The proof is described as combining the Rohlin tower technique, the local variational principle, and the uniquely ergodic model [2509.20738].

The general amenable-group case is then obtained by using orbital entropy and the extension of local entropy theory in the amenable-group setting, yielding
\[
\mathrm{Asc}_\mu^{-}(G,\mathcal U\mid Y)
=
\mathrm{Asc}_\mu^{+}(G,\mathcal U\mid Y)
\]
for general amenable groups; this is Theorem 4.6 and is identified in the summary as the key technical development of the paper [2509.20738]. The abstract also emphasizes a related conclusion: the paper verifies that $\mathrm{Asc}_\mu^{-}(G,\mathcal U)$ is equal to $\mathrm{Asc}_\mu^{+}(G,\mathcal U)$ in the general case, and establishes the equivalence of the two conditional variants when $G=\mathbb Z$ [2509.20738].

This equality has conceptual significance. It shows that the two natural ways of passing from covers to measure-theoretic average sample complexity—directly via conditional cover entropy, or indirectly via refinements by partitions—ultimately encode the same invariant.

## 6. Variational principle, special cases, and related directions

The paper culminates in a local variational principle. For $\mathcal U\in\mathcal C_X^o$ and uniform coefficients,
\[
\mathrm{Asc}_{\mathrm{top}}(G,\mathcal U)
=
\max_{\mu\in\mathcal M(X,G)} \mathrm{Asc}_\mu(G,\mathcal U)
=
\max_{\mu\in\mathcal M^e(X,G)} \mathrm{Asc}_\mu(G,\mathcal U)
\]
[2509.20738]. The summary interprets this as showing that topological average sample complexity localizes via a maximum over invariant measures of the corresponding measure-theoretic quantity, in parallel with the classical variational principle for entropy.

Several contextual remarks delimit the scope of the invariant. When $(Y,G)$ is trivial, the conditional definitions reduce to the unconditional intricacy and average sample complexity of Petersen et al. [2509.20738; 1512.01143]. The paper also states that its approach and results mirror those of local entropy theory, but in the intricacy and sample-complexity setting [2509.20738]. This places conditional local topological intricacy within a larger localization program in topological dynamics, alongside local entropy and local weighted topological pressure [2307.08021].

The relationship to the 2015 theory is especially direct. In the unconditional case, dynamical intricacy and average sample complexity were introduced for topological and measure-preserving systems, with formulas based on weighted averages over subsets and with the relation
\[
\Int = 2\Asc - h
\]
in both topological and measure-theoretic settings [1512.01143]. The 2025 amenable-group paper extends that framework in three directions at once: from $\mathbb Z$-actions to amenable group actions, from global to conditional fiberwise formulations, and from absolute to local variational statements [2509.20738].

A concise comparison of the core objects is useful.

| Level | Intricacy-type quantity | Average-sample-complexity quantity |
|---|---|---|
| Topological, conditional local | $\mathrm{Int}_{\mathrm{top}}(G,\mathcal U\mid Y)$ | $\mathrm{Asc}_{\mathrm{top}}(G,\mathcal U\mid Y)$ |
| Measure-theoretic, partition form | $\mathrm{Int}_\mu(G,\alpha\mid Y)$ | $\mathrm{Asc}_\mu(G,\alpha\mid Y)$ |
| Measure-theoretic, cover form | — | $\mathrm{Asc}_\mu^{-}(G,\mathcal U\mid Y)$ and $\mathrm{Asc}_\mu^{+}(G,\mathcal U\mid Y)$ |

A common misconception is to read “local” here as referring to pointwise or small-ball locality in the metric sense. In the present theory, locality is instead encoded by finite covers or partitions and by conditioning along the fibers of a factor map [2509.20738]. Another possible misunderstanding is to regard intricacy as independent of entropy. The paper does not support that reading: at the cover level, conditional local topological intricacy is explicitly linked to conditional topological entropy through the identity above [2509.20738]. What is new is not an entropy replacement, but a refined conditional statistic of dynamical organization.

In that sense, conditional local topological intricacy may be understood as a localized relative invariant for amenable actions, built to measure how combinatorial freedom and coherence distribute across fibers of a factor. The results of [2509.20738] establish that this invariant admits a coherent topological theory, a matching conditional measure-theoretic theory, robust limit behavior, affine ergodic decomposition, equality of the two cover-based measurable constructions, and a local variational principle.

Source: https://www.emergentmind.com/topics/conditional-local-topological-intricacy