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Bowen's equations for invariance pressure of control systems

Rui Yang, Ercai Chen, Jiao Yang, Xiaoyao Zhou

Abstract

We aim to establish Bowen's equations for upper capacity invariance pressure and Pesin-Pitskel invariance pressure of discrete-time control systems. We first introduce a new invariance pressure called induced invariance pressure on partitions that specializes the upper capacity invariance pressure on partitions, and then show that the two types of invariance pressures are related by a Bowen's equation. Besides, to establish Bowen's equation for Pesin-Pitskel invariance pressure on partitions we also introduce a new notion called BS invariance dimension on subsets. Moreover, a variational principle for BS invariance dimension on subsets is established.

Bowen's equations for invariance pressure of control systems

Abstract

We aim to establish Bowen's equations for upper capacity invariance pressure and Pesin-Pitskel invariance pressure of discrete-time control systems. We first introduce a new invariance pressure called induced invariance pressure on partitions that specializes the upper capacity invariance pressure on partitions, and then show that the two types of invariance pressures are related by a Bowen's equation. Besides, to establish Bowen's equation for Pesin-Pitskel invariance pressure on partitions we also introduce a new notion called BS invariance dimension on subsets. Moreover, a variational principle for BS invariance dimension on subsets is established.
Paper Structure (9 sections, 11 theorems, 119 equations)

This paper contains 9 sections, 11 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. The setup of discrete-time control systems
  3. Bowen's equation for upper capacity invariance pressure
  4. Induced invariance pressure on partitions
  5. A dimensional characterization for $P_{inv,\psi }(\varphi,Q,\mathcal{C})$
  6. Proof of Theorem 1.2
  7. Bowen's equation for Pesin-Pitskel invariance pressure
  8. Pesin-Pitskel invariance pressure on subsets
  9. Variational principle for BS invariance dimension

Key Result

Theorem 1.1

Let $\varSigma=(\mathbb{N},X,U,\mathcal{U},\phi)$ be a discrete-time control system. Let $Q$ be a controlled invariant set and $\mathcal{C} =(\mathcal{A}, \tau,\nu)$ be an invariant partition of $Q$, and let $\varphi,\psi \in C(U,\mathbb{R})$ with $\psi>0$. Then where $P_{inv,\psi }(\varphi,Q,\mathcal{C})$ denotes $\psi$-induced invariance pressure of $\varphi$ on $Q$ w.r.t. $\mathcal{C}$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 27 more