Finite Invariant Sets with Bridging Points in Logistic IFS

TL;DR

This paper studies finite invariant sets with bridging points in logistic iterated function systems, deriving exact conditions for toss-and-catch dynamics and identifying cases with bridging points outside individual invariant sets.

Contribution

It provides exact parameter conditions for toss-and-catch structures in logistic IFS and logistic-tent IFS, revealing novel bridging points.

Findings

Derived exact conditions for toss-and-catch dynamics.

Identified cases with bridging points outside individual invariant sets.

Analyzed logistic and logistic-tent IFS structures.

Abstract

We investigate iterated function systems (IFS) that randomly alternate between two non-identical one-dimensional maps. Our primary focus is on finite invariant sets exhibiting ``toss-and-catch'' dynamics, in which trajectories alternate between fixed points and periodic orbits of the constituent maps. We derive exact parameter conditions for several toss-and-catch structures in a pair of logistic maps (logistic IFS) and a combination of logistic and tent maps (logistic-tent IFS). Notably, we identify cases in which the invariant set contains bridging points that belong to neither of the invariant sets of the individual maps.