The Ph.D. Thesis “Strong mixing measures and invariant sets in linear dynamics”
has three differenced parts. Chapter 0 introduces the notation,
definitions and the basic results that will be needed troughout the thesis.
There is a first part consisting of Chapters 1 and 2, where we study the
relation between the Frequent Hypercyclicity Criterion and the existence of
strongly-mixing Borel probability measures. A third chapter, where we focus
our attention on frequent hypercyclicity for translation C0-semigroups,
and the last part corresponding to Chapters 4 and 5, where we study dynamical
properties satisfied by autonomous and non-autonomous linear dynamical
systems on certain invariant sets. In what follows, we give a brief
description of each chapter:
In Chapter 1, we construct strongly mixing Borel probability T-invariant
measures with full support for operators on F-spaces which satisfy the
Frequent Hypercyclicity Criterion. Moreover, we provide examples of operators
that verify this criterion and we also show that this result can be
improved in the case of chaotic unilateral backward shifts. The contents of
this chapter have been published in [88] and [12].
In Chapter 2, we show that the Frequent Hypercyclicity Criterion for C0-
semigroups, which was given by Mangino and Peris in [82], ensures the
existence of invariant strongly mixing measures with full support. We will
provide several examples, that range from birth-and-death models to the
Black-Scholes equation, which illustrate these results. All the results of this
chapter have been published in [86].
In Chapter 3, we focus our attention on one of the most important tests
C0-semigroups, the translation semigroup. Inspired in the work of Bayart
and Ruzsa in [22], where they characterize frequent hypercyclicity of
weighted backward shifts we characterize frequently hypercyclic translation
C0-semigroups on C
ρ
0
(R) and L
ρ
p(R). Moreover, we first review some
known results on the dynamics of the translation C0-semigroups. Later we
state and prove a characterization of frequent hypercyclicity for weighted
pseudo shifts in terms of the weights that will be used later to obtain a
characterization of frequent hypercyclicity for translation C0-semigroups
on C
ρ
0
(R). Finally we study the case of L
ρ
p(R). We will also establish an
analogy between the study of frequent hypercyclicity for the translation
C0-semigroup in L
ρ
p(R) and the corresponding one for backward shifts on
weighted sequence spaces. The contents of this chapter have been included
in [81].
Chapter 4 is devoted to study hypercyclicity, Devaney chaos, topological
mixing properties and strong mixing in the measure-theoretic sense for operators
on topological vector spaces with invariant sets. More precisely, we
establish links between the fact of satisfying any of our dynamical properties
on certain invariant sets, and the corresponding property on the closed
linear span of the invariant set, or on the union of the invariant sets. Viceversa,
we give conditions on the operator (or C0-semigroup) to ensure that,
when restricted to the invariant set, it satisfies certain dynamical property.
Particular attention is given to the case of positive operators and semigroups
on lattices, and the (invariant) positive cone. The contents of this
chapter have been published in [85].
In the last chapter, motivated by the work of Balibrea and Oprocha [4],
where they obtained several results about weak mixing and chaos for nonautonomous
discrete systems on compact sets, we study mixing properties for
nonautonomous linear dynamical systems that are induced by the corresponding
dynamics on certain invariant sets. All the results of this chapter
have been published in [87].
Murillo Arcila, M. (2015). Strong mixing measures and invariant sets in linear dynamics [Tesis doctoral no publicada]. Universitat Politècnica de València. doi:10.4995/Thesis/10251/48519.
TESIS