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The Specification Property for \(C_0\)-Semigroups

  • S. Bartoll
  • F. Martínez-Giménez
  • A. PerisEmail author
  • F. Rodenas
Article
  • 113 Downloads

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.

Mathematics Subject Classification

47A16 47B37 

Notes

Acknowledgements

The authors were supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P. The second and third authors were also supported by Generalitat Valenciana, Projects PROMETEOII/2013/013 and PROMETEO/2017/102. We are indebted to the referee whose valuable comments produced an improvement in the presentation of the paper.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut Universitari de Matemàtica Pura i AplicadaUniversitat Politècnica de ValènciaValènciaSpain

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