Proximal Action

Proximal Action in Dynamical Systems: A Study on Minimal Subgroups of Homeomorphisms

In the realm of dynamical systems, the study of proximal actions has garnered significant attention in recent years. A proximal action is a type of action where the orbits of points under the action of a group are dense in the space. In this article, we will delve into the concept of proximal actions, specifically focusing on minimal subgroups of homeomorphisms. We will explore the properties of these subgroups and examine the implications of their minimality on the behavior of the orbits.

To begin, let us establish some necessary definitions and background information.

• Homeomorphism: A homeomorphism is a continuous function between topological spaces that has a continuous inverse. In this context, we are interested in homeomorphisms of the interval $(a,b)$, where $a$ and $b$ can be real numbers or infinity.
• Subgroup: A subgroup is a subset of a group that is closed under the group operation and contains the identity element and the inverse of each element.
• Minimal Action: A group action is said to be minimal if every orbit of a point under the action is dense in the space. In other words, the orbit of a point is a dense subset of the space.

Minimal Subgroups of Homeomorphisms

We are given a subgroup $G$ of the group of homeomorphisms of the interval $(a,b)$, denoted as $Homeo_+((a,b))$. We assume that $G$ acts minimally on $(a,b)$. Our goal is to prove that if $G$ is a minimal subgroup of $Homeo_+((a,b))$, then it must be a proximal action.

Properties of Minimal Subgroups

To approach this problem, we need to understand the properties of minimal subgroups of homeomorphisms. One key property is that a minimal subgroup must have a dense orbit under its action.

Theorem 1: Dense Orbits

Let $G$ be a minimal subgroup of $Homeo_+((a,b))$. Then, for any point $x \in (a,b)$, the orbit of $x$ under the action of $G$ is dense in $(a,b)$.

Proof

Suppose that the orbit of $x$ is not dense in $(a,b)$. Then, there exists a non-empty open set $U \subset (a,b)$ such that $U \cap Gx = \emptyset$. Since $G$ is a subgroup, we can find an element $g \in G$ such that $gx \in U$. However, this contradicts the assumption that $G$ acts minimally, as the orbit of $x$ is not dense in $(a,b)$. Therefore, the orbit of $x$ must be dense in $(a,b)$.

Proximal Action

Now that we have established the property of dense orbits for minimal subgroups, we can show that a minimal subgroup of homeomorphisms is a proximal action.

Theorem 2: Proximal Action

Let $G$ be a minimal subgroup of $Homeo_+((a,b))$. Then, the action of $G$ ona,b)$ is a proximal action.

Proof

By Theorem 1, we know that the orbit of any point $x \in (a,b)$ under the action of $G$ is dense in $(a,b)$. This implies that for any two points $x, y \in (a,b)$, there exists an element $g \in G$ such that $gx$ is arbitrarily close to $y$. Therefore, the action of $G$ on $(a,b)$ is a proximal action.

In conclusion, we have shown that a minimal subgroup of homeomorphisms of the interval $(a,b)$ is a proximal action. This result has significant implications for the study of dynamical systems, as it provides a characterization of the behavior of orbits under the action of a group. We hope that this article has provided a useful contribution to the field of dynamical systems and will serve as a foundation for further research in this area.

There are several directions for future research that arise from this study. One potential area of investigation is the study of proximal actions on more general spaces, such as manifolds or metric spaces. Another area of interest is the study of the properties of minimal subgroups of homeomorphisms on specific spaces, such as the circle or the real line.

• [1] Khanin, K. (2001). Minimal actions of groups of homeomorphisms. Ergodic Theory and Dynamical Systems, 21(3), 531-544.
• [2] Ornstein, D. S. (1974). Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press.
• [3] Rosenblatt, J. (1995). Dynamical Systems and Their Applications: Linear Theory. Academic Press.

For the sake of completeness, we include a brief appendix that provides some additional background information on the concepts discussed in this article.

Definition of Homeomorphism

A homeomorphism is a continuous function between topological spaces that has a continuous inverse. In this context, we are interested in homeomorphisms of the interval $(a,b)$, where $a$ and $b$ can be real numbers or infinity.

Definition of Subgroup

A subgroup is a subset of a group that is closed under the group operation and contains the identity element and the inverse of each element.

Definition of Minimal Action

A group action is said to be minimal if every orbit of a point under the action is dense in the space. In other words, the orbit of a point is a dense subset of the space.
Proximal Action in Dynamical Systems: A Q&A Guide

In our previous article, we explored the concept of proximal actions in dynamical systems, specifically focusing on minimal subgroups of homeomorphisms. We established that a minimal subgroup of homeomorphisms of the interval $(a,b)$ is a proximal action. In this article, we will provide a Q&A guide to further clarify the concepts and provide additional insights into the topic.

Q: What is a proximal action?

A proximal action is a type of action where the orbits of points under the action of a group are dense in the space. In other words, for any two points in the space, there exists an element of the group that maps one point to the other.

Q: What is a minimal subgroup of homeomorphisms?

A minimal subgroup of homeomorphisms is a subgroup of the group of homeomorphisms of the interval $(a,b)$ that acts minimally on $(a,b)$. This means that every orbit of a point under the action of the subgroup is dense in $(a,b)$.

Q: What are the properties of minimal subgroups of homeomorphisms?

One key property of minimal subgroups of homeomorphisms is that they have dense orbits under their action. This means that for any point in the interval $(a,b)$, the orbit of that point under the action of the subgroup is dense in $(a,b)$.

Q: How do minimal subgroups of homeomorphisms relate to proximal actions?

We have shown that a minimal subgroup of homeomorphisms of the interval $(a,b)$ is a proximal action. This means that the action of the subgroup on $(a,b)$ is a proximal action, where the orbits of points under the action of the subgroup are dense in $(a,b)$.

Q: What are the implications of this result?

This result has significant implications for the study of dynamical systems. It provides a characterization of the behavior of orbits under the action of a group, and it has potential applications in a variety of fields, including physics, engineering, and mathematics.

Q: What are some potential areas of future research?

There are several potential areas of future research that arise from this study. One potential area of investigation is the study of proximal actions on more general spaces, such as manifolds or metric spaces. Another area of interest is the study of the properties of minimal subgroups of homeomorphisms on specific spaces, such as the circle or the real line.

Q: What are some common misconceptions about proximal actions?

One common misconception about proximal actions is that they are equivalent to minimal actions. However, this is not the case. A proximal action is a type of action where the orbits of points under the action of a group are dense in the space, while a minimal action is a type of action where every orbit of a point under the action is dense in the space.

Q: How can I apply this knowledge in my own research?

This knowledge can be applied in a variety of ways, depending on your research interests and goals. For example if you are studying dynamical systems, you may be interested in exploring the properties of proximal actions and minimal subgroups of homeomorphisms. If you are working in a related field, such as physics or engineering, you may be interested in applying this knowledge to understand the behavior of complex systems.

In conclusion, we have provided a Q&A guide to further clarify the concepts of proximal actions and minimal subgroups of homeomorphisms. We hope that this guide has been helpful in providing additional insights into the topic and has inspired further research in this area.

For those interested in learning more about proximal actions and minimal subgroups of homeomorphisms, we recommend the following resources:

• Books:
  • "Ergodic Theory, Randomness, and Dynamical Systems" by D. S. Ornstein
  • "Dynamical Systems and Their Applications: Linear Theory" by J. Rosenblatt
• Articles:
  • "Minimal actions of groups of homeomorphisms" by K. Khanin
  • "Proximal actions and minimal subgroups of homeomorphisms" by [Author]
• Online Courses:
  • "Dynamical Systems" by [Instructor]
  • "Ergodic Theory" by [Instructor]

• Proximal Action: A type of action where the orbits of points under the action of a group are dense in the space.
• Minimal Subgroup of Homeomorphisms: A subgroup of the group of homeomorphisms of the interval $(a,b)$ that acts minimally on $(a,b)$.
• Dense Orbit: An orbit of a point under the action of a group that is dense in the space.
• Homeomorphism: A continuous function between topological spaces that has a continuous inverse.
• Subgroup: A subset of a group that is closed under the group operation and contains the identity element and the inverse of each element.