Semiflows on finite topological spaces

TL;DR

The paper analyzes semiflows on finite topological $T_0$ spaces, showing that non-trivial semiflows exist exactly when the space has down beat points, and such semiflows are essentially strong deformation retracts moving high points downward. It uses this characterization to prove that finite flows are always trivial and to connect dynamical behavior to the homotopy type of the space. A counting framework is developed: the number of semiflows $S_F(X)$ satisfies $S_F(X)\ge 2^{|D(X)|}$ and equals 1 iff $D(X)=\emptyset$, with explicit constructions (e.g., spaces $X_n$) realizing a range of values for $S_F(X)$. The work links combinatorial topology with dynamical systems on finite spaces and raises questions about realizability and the role of potential down beat points in determining semiflow richness.

Abstract

In this paper, we study flows and semiflows defined on any given finite topological $T_0$-space $X$. We show that there exist non-trivial semiflows on $X$, unless $X$ is a minimal finite space. Specifically, non-trivial semiflows exist if and only if $X$ contains down beat points, and a non-trivial semiflow is essentially a strong deformation retraction. As a consequence of this result, we provide a new and concise proof that the only flow that can be defined on $X$ is the trivial flow. Finally, we discuss the number of different semiflows that can be defined on $X$ in terms of down beat points and other special points.

Figures (3)

• Figure 1: Hasse diagram of $X$ and schematic representation of the semiflow $\varphi$ from Example \ref{['Ex:notrivial']}.
• Figure 2: Description of the non-trivial semiflows defined on the space $X$ from Example \ref{['Ex:notrivial']}.
• Figure 3: Hasse diagram of $X_3$ from proof of Theorem \ref{['thm_realizacion_todos_los_enteros_positivos']}.

Theorems & Definitions (36)

• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Example 2.5
• Theorem 2.6
• Definition 2.7
• Example 3.1
• Theorem 3.2
• proof