# Dendrites and chaos

## TL;DR

This paper explores the relationships between different types of chaos in dendrite maps, providing constructions that separate these chaos notions and answering open questions from prior research.

## Contribution

It constructs specific dendrite maps to demonstrate the independence of haos, distributional chaos, and LY-scrambled sets, addressing open problems in the field.

## Key findings

- Constructed a dendrite map with haos but no extit{DC3} pairs.
- Built a dendrite map with extit{DC1} pairs but no infinite LY-scrambled set.
- Clarified the relationships between various chaos notions in dendrite dynamics.

## Abstract

We answer the two questions left open in [Z.~Ko\v{c}an, Internat. J. Bifur. Chaos Appl. Sci. Engrg. \textbf{22}, article id: 125025 (2012)] i.e. whether there is a relation between $\omega$-chaos and distributional chaos and whether there is a relation between an infinite LY-scrambled set and distributional chaos for dendrite maps. We construct a continuous self-map of dendrite without any \textit{DC3} pairs but containing an uncountable $\omega$-scrambled set. To answer for the second question we construct dendrite $\mathcal{D}$ and continuous dendrite map without an infinite LY-scrambled set but with \textit{DC1} pairs.