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semi-conjugacy skill

This skill helps you analyze and apply semi-conjugacy concepts to understand and simplify long-term behavior of dynamical systems.

npx playbooks add skill plurigrid/asi --skill semi-conjugacy

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---
name: semi-conjugacy
description: Surjective map intertwining two dynamical systems
trit: 1
geodesic: true
moebius: "μ(n) ≠ 0"
---

# Semi-conjugacy

**Trit**: 1 (PLUS)
**Domain**: Dynamical Systems Theory
**Principle**: Surjective map intertwining two dynamical systems

## Overview

Semi-conjugacy is a fundamental concept in dynamical systems theory, providing tools for understanding the qualitative behavior of differential equations and flows on manifolds.

## Mathematical Definition

```
SEMI-CONJUGACY: Phase space × Time → Phase space
```

## Key Properties

1. **Local behavior**: Analysis near equilibria and invariant sets
2. **Global structure**: Long-term dynamics and limit sets  
3. **Bifurcations**: Parameter-dependent qualitative changes
4. **Stability**: Robustness under perturbation

## Integration with GF(3)

This skill participates in triadic composition:
- **Trit 1** (PLUS): Sources/generators
- **Conservation**: Σ trits ≡ 0 (mod 3) across skill triplets

## AlgebraicDynamics.jl Connection

```julia
using AlgebraicDynamics

# Semi-conjugacy as compositional dynamical system
# Implements oapply for resource-sharing machines
```

## Related Skills

- equilibrium (trit 0)
- stability (trit +1)  
- bifurcation (trit +1)
- attractor (trit +1)
- lyapunov-function (trit -1)

---

**Skill Name**: semi-conjugacy
**Type**: Dynamical Systems / Semi-conjugacy
**Trit**: 1 (PLUS)
**GF(3)**: Conserved in triplet composition

## Non-Backtracking Geodesic Qualification

**Condition**: μ(n) ≠ 0 (Möbius squarefree)

This skill is qualified for non-backtracking geodesic traversal:

1. **Prime Path**: No state revisited in skill invocation chain
2. **Möbius Filter**: Composite paths (backtracking) cancel via μ-inversion
3. **GF(3) Conservation**: Trit sum ≡ 0 (mod 3) across skill triplets
4. **Spectral Gap**: Ramanujan bound λ₂ ≤ 2√(k-1) for k-regular expansion

```
Geodesic Invariant:
  ∀ path P: backtrack(P) = ∅ ⟹ μ(|P|) ≠ 0
  
Möbius Inversion:
  f(n) = Σ_{d|n} g(d) ⟹ g(n) = Σ_{d|n} μ(n/d) f(d)
```

Overview

This skill describes semi-conjugacy: a surjective map that intertwines two dynamical systems, relating their time evolution while possibly collapsing some structure. It emphasizes how semi-conjugacies transfer qualitative behavior (limit sets, recurrence, attractors) from one system to another. The presentation highlights local and global consequences for stability, bifurcations, and invariant sets.

How this skill works

The skill inspects a surjective continuous map h between phase spaces that satisfies h ∘ f = g ∘ h, where f and g are the two dynamics. It checks preserved structures: images of invariant sets, correspondences between equilibria and attractors, and how long-term behavior pushes forward under h. It also records algebraic and combinatorial qualifications used for compositional reasoning in families of skills.

When to use it

Relating a complicated system to a simpler model to infer long-term behavior.
Proving existence or persistence of attractors and recurrent sets via factor maps.
Comparing bifurcation scenarios between systems that share a semi-conjugacy.
Analyzing reductions where multiple states collapse onto a single observed variable.
Composing dynamical building blocks with conservation constraints in a toolkit.

Best practices

Verify surjectivity and the intertwining identity h ∘ f = g ∘ h explicitly.
Track images of invariant and recurrent sets rather than expecting pointwise conjugacy.
Use semi-conjugacy to transfer stability and attractor existence, but avoid inferring finer spectral properties.
When composing skills, check algebraic conservation rules (e.g., trit sums) to ensure compatibility.
Document any collapse of degrees of freedom introduced by h to avoid misinterpreting dynamics.

Example use cases

Projecting a high-dimensional flow onto a one-dimensional map to study global attractors.
Using a semi-conjugacy to show that chaotic behavior in a factor implies complexity in the original system.
Relating parametric bifurcations of a simplified model to the full system via the factor map.
Composing skills in a pipeline that enforces GK(3)-style conservation when assembling modular analyses.
Filtering non-backtracking geodesic traversals where Möbius-based cancellation and spectral gap constraints apply.

FAQ

Does semi-conjugacy imply full equivalence of dynamics?

No. Semi-conjugacy preserves forward behavior under the map but can collapse distinct states; it is weaker than conjugacy and may omit inverse-time structure.

What can be transferred through a semi-conjugacy?

Invariant sets, attractors, and recurrence properties typically push forward under the factor map. Spectral details and finer stability margins do not always transfer.

When is semi-conjugacy most useful?

When you need a rigorous reduction or factor model to deduce qualitative long-term behavior of a complex system from a simpler one.