# Reduced dynamical systems

## TL;DR

This paper investigates how the ergodic properties of complex rational maps on the Riemann sphere are maintained when their orbits are simplified to a finite set of distances, providing insights into their dynamical behavior.

## Contribution

It introduces a reduction method for orbits of rational maps that preserves ergodic properties, offering a new approach to studying complex dynamical systems.

## Key findings

- Ergodic properties are preserved under the reduction process.
- Reduction simplifies the analysis of complex rational maps.
- The method applies to finitely many initial orbit elements.

## Abstract

We consider the dynamics of complex rational maps on the Riemann sphere. We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini-Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.