# Relationship among several types of sensitivity in general semi-flows

## TL;DR

This paper explores the relationships among different types of sensitivity in semi-flows, demonstrating the existence of specific monoids and semi-flows with particular sensitivity properties, thereby answering open questions in the field.

## Contribution

It provides counterexamples showing certain sensitivity properties do not imply others, advancing understanding of sensitivity in semi-flows and answering open questions from prior research.

## Key findings

- Existence of a monoid without syndetic or dual syndetic property.
- Existence of a strongly mixing semi-flow lacking various sensitivities.
- A thickly sensitive cascade that is not multi-sensitive.

## Abstract

In this paper, we show that there exists a monoid, on which neither the syndetic property nor the dual syndetic property holds, and there exists a strongly mixing semi-flow with this monoid action which does not have thick sensitivity, syndetic sensitivity, thickly syndetic sensitivity, or thickly periodical sensitivity. Meanwhile, we show that there exists a thickly sensitive cascade which is not multi-sensitive. The first result answers positively Question 2, and the first and the second results answer negatively Question 3 in [A. Miller, A note about various types of sensitivity in general semiflows, Appl. Gen. Topol., 2018].