Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the algorithmic descriptive complexity of attractors in topological dynamics

Cristobal Rojas, Mathieu Sablik

TL;DR

This work develops a computable-analysis framework to quantify the algorithmic descriptive complexity of attractors in topological dynamics, focusing on the three limiting objects: topological, metric, and statistical attractors. It derives general upper bounds: $\Pi_2$-computability for topological/metric attractors and $\Sigma_2$-computability for statistical attractors, with $\Pi_1$-computability under strong attraction; and shows these bounds are tight through explicit symbolic constructions. The authors build computable symbolic systems realizing $\Pi_1$-, $\Pi_2$-, and $\Sigma_2$-complete attractors, including wild attractors with differing complexities across attractor notions. They further demonstrate how to transfer these dynamics to the unit interval via a computable Cantor-set embedding, producing interval maps with corresponding complex attractors. Overall, the paper clarifies how dynamical constraints shape the computability of asymptotic descriptions and provides concrete, computable examples of highly noncomputable attractors within both symbolic and interval settings.

Abstract

We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constrains and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non computable.

On the algorithmic descriptive complexity of attractors in topological dynamics

TL;DR

This work develops a computable-analysis framework to quantify the algorithmic descriptive complexity of attractors in topological dynamics, focusing on the three limiting objects: topological, metric, and statistical attractors. It derives general upper bounds: -computability for topological/metric attractors and -computability for statistical attractors, with -computability under strong attraction; and shows these bounds are tight through explicit symbolic constructions. The authors build computable symbolic systems realizing -, -, and -complete attractors, including wild attractors with differing complexities across attractor notions. They further demonstrate how to transfer these dynamics to the unit interval via a computable Cantor-set embedding, producing interval maps with corresponding complex attractors. Overall, the paper clarifies how dynamical constraints shape the computability of asymptotic descriptions and provides concrete, computable examples of highly noncomputable attractors within both symbolic and interval settings.

Abstract

We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constrains and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non computable.
Paper Structure (13 sections, 11 theorems, 73 equations, 3 figures)

This paper contains 13 sections, 11 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary definitions
  3. On the concept of Attractor
  4. Rudiments of Computable Analysis
  5. Computable metric spaces.
  6. Computational structure of closed sets
  7. Computability obstructions in attractors realization: complexity upper bounds
  8. Realizing complex attractors within symbolic systems
  9. Symbolic Systems
  10. A $\Pi_1$-complete Strong Attractor
  11. $\Pi_{2}$-complete wild attractors
  12. A $\Sigma_{2}$-complete statistical attractor
  13. Interval maps with computationally complex attractors

Key Result

Proposition 1

Let $\mu$ be a probability measure over a computable metric space $X$. Then, $\mu$ is computable if and only if $\mu(B_{i_1}\cup B_{i_2}\cup \dots \cup B_{i_n})$ is lower computable uniformly in $i_1,\dots, i_n$ in the sense that there exists an algorithm which, upon input $i_1,\dots, i_n$ and $k$,

Figures (3)

  • Figure 1: An illustration of the action of $T$ on a configuration $x$. It is assumed that machines $M_1$ and $M_3$ halt in 8 and 4 steps respectively, while machine $M_2$ doesn't halt at all.
  • Figure 2: An illustration of the action of $T'$ on a configuration $x$ following that it belongs to $U_{s,s+2t}$ or $U_{s,s+2t+1}$. The red rectangle corresponds to the word $a^{s+2t}$ and the blue rectangle to the word $a^{s+2t+1}$ following that $x$ is $U_{s,s+2t}$ or $U_{s,s+2t+1}$.
  • Figure 3: The function $f$ on the gap $[a,b]$ between intervals $I_{w0}$ and $I_{w1}$ of $C$.

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 1
  • Proposition 1
  • Example 2
  • Example 3
  • Proposition 2
  • proof
  • ...and 19 more