# On the cycles of components of disconnected Julia sets

## TL;DR

This paper constructs specific hyperbolic rational maps with multiple cycles of connected Julia set components, advancing understanding of the complex structure of Julia sets in dynamical systems.

## Contribution

It introduces a method to explicitly construct hyperbolic rational maps with a prescribed number of cycles of non-trivial Julia set components.

## Key findings

- Existence of hyperbolic rational maps with multiple cycles of connected Julia components.
- Construction method applicable for any degree $d \\ge 3$ and number of cycles $n \\ge 1$.
- Enhances understanding of the topological complexity of Julia sets.

## Abstract

For any integers $d\ge 3$ and $n\ge 1$, we construct a hyperbolic rational map of degree $d$ such that it has $n$ cycles of the connected components of its Julia set except single points and Jordan curves.