The Specification Property for $C_0$-Semigroups

TL;DR

This paper extends the concept of the specification property, a strong form of chaos, to $C_0$-semigroups and explores its relationship with other dynamical properties, providing examples from linear PDEs.

Contribution

It introduces the specification property for $C_0$-semigroups and investigates its connections with mixing, chaos, and hypercyclicity, with applications to linear PDE solution semigroups.

Findings

$C_0$-semigroups can exhibit the specification property.

The specification property relates to mixing, chaos, and hypercyclicity.

Examples include semigroups from heat and Black-Scholes equations.

Abstract

We study one of the strongest versions of chaos for continuous dynamical systems, namely the specification property. We extend the definition of specification property for operators on a Banach space to strongly continuous one-parameter semigroups of operators, that is, $C_0$-semigroups. In addition, we study the relationships of the specification property for $C_0$-semigroups (SgSP) with other dynamical properties: mixing, Devaney's chaos, distributional chaos and frequent hypercyclicity. Concerning the applications, we provide several examples of semigroups which exhibit the SgSP with particular interest on solution semigroups to certain linear PDEs, which range from the hyperbolic heat equation to the Black-Scholes equation.