Recent Topics on Linear Dynamics

TL;DR

This monograph surveys foundational and cutting-edge topics in the dynamics of linear operators on Banach spaces, connecting the Banach contraction principle with asymptotic orbital behavior, notably that a contraction with constant $0<\lambda<1$ yields a unique fixed point and $f^n(x)\to p$ for all $x$. It develops core dynamical properties—hyperbolicity, expansivity, and the shadowing property—and analyzes their interrelations, including that hyperbolicity implies both expansivity and shadowing, while in finite dimensions the three notions are equivalent. The text extends to generalized hyperbolicity, homoclinic structure, and structural stability (Bernardes–Messaoudi Grobman–Hartman-type results), and interrogates spectral-theoretic aspects and the invariant-subspace problem. It also discusses hypercyclic phenomena, weak forms of hyperbolicity, and the role of perturbations, with constructive examples and a unified framework for linear dynamics.

Abstract

Notes from a course on linear dynamics given by the author at the University of Da Nang in January 2024.

Theorems & Definitions (4)

• definition thmcounterdefinition
• Theorem 1.1: Banach Contracting Principle
• proof
• definition thmcounterdefinition