Sharkovsky's Ordering in the Mandelbrot Set

TL;DR

This paper explores Sharkovsky's ordering, originally for interval maps, and extends it to trees and the Mandelbrot set, revealing new insights into orbit forcing in complex dynamics.

Contribution

It introduces analogous Sharkovsky orderings for trees and demonstrates their occurrence within the Mandelbrot set, extending classical results to complex dynamics.

Findings

Sharkovsky's ordering is applicable to trees.

Analogous orderings are identified in the Mandelbrot set.

The work connects orbit forcing in interval maps to complex dynamics.

Abstract

Sharkovsky's ordering describes orbit forcing of interval maps, and generalizations of Sharkovsky's ordering exist for maps of trees. In this paper I will describe Sharkovsky's ordering and analogous orderings for trees, and their occurrence on the Mandelbrot set.