Correlation integral and determinism for a family of $2^\infty$ maps

TL;DR

This paper investigates the correlation integral and determinism in a family of non-chaotic maps, revealing that while determinism is perfect in finite horizons, its behavior over infinite horizons is complex and counter-intuitive.

Contribution

It provides sharp bounds on the determinism for $2^inite$ non-chaotic maps, extending understanding of predictability in these systems.

Findings

Determinism equals 1 in finite horizons for these maps.

Behavior of determinism over infinite horizons is counter-intuitive.

Sharp bounds on determinism are established.

Abstract

The correlation integral and determinism are quantitative characteristics of a dynamical system based on the recurrence of orbits. For strongly non-chaotic interval maps, the determinism equals 1 for every small enough threshold. This means that trajectories of such systems are perfectly predictable in the infinite horizon. In this paper we study the correlation integral and determinism for the family of $2^\infty$ non-chaotic maps, first considered by Delahaye in 1980. The determinism in a finite horizon equals 1. However, the behaviour of the determinism in the infinite horizon is counter-intuitive. Sharp bounds on the determinism are provided.