This thesis is dedicated to studying the theory of entropy and its relation to the Kolmogorov complexity. Originating in physics, the notion of entropy was introduced to mathematics by C. E. Shannon as a way of measuring the rate at which information is coming from a data source. There are, however, a few different ways of telling how much information there is: an alternative approach to quantifying the amount of information is the Kolmogorov complexity, which was proposed by A. N. Kolmogorov. The Shannon entropy is the key ingredient in the definition of the Kolmogorov-Sinai entropy of a measure-preserving systems. Roughly speaking, the Kolmogorov-Sinai entropy is the expected amount of information in `Shannon sense' that one obtains per unit of time by observing the evolution of a measure-preserving system. In topological dynamics, the topological entropy takes place of the Kolmogorov-Sinai entropy. For metrizable systems, the topological entropy measures the exponential growth rate of the number of distinguishable partial orbits of length n as n tends to infinity. Originally defined for Z-actions, the 'classical' theories of entropy were later extended to actions of amenable groups.
We provide a necessary background on amenable groups, topological/measurepreserving dynamics and the entropy theory in Chapters 1, 2, 3 and 5.
The main focus of this thesis is extending the following results. First of all, a common generalization of the topological and the Kolmogorov-Sinai entropy theories for Z-systems was suggested by G. Palm. We provide an abstract generalization of the work of Palm for actions of discrete amenable groups in the language of measurement functors in Chapter 6.
Secondly, we investigate the connection of entropy and Kolmogorov complexity.
Originally, the equality between the topological entropy and a certain quantity measuring maximal asymptotic Kolmogorov complexity of the trajectories was established by A. A. Brudno for subshifts over Z. Later, he proved the equality of the Kolmogorov-Sinai entropy and the asymptotic Kolmogorov complexity of almost every trajectory for ergodic subshifts over Z. We provide a generalization of these results as follows. Firstly, in Chapter 4 we give a background on computability and Kolmogorov complexity and, further, introduce computable Fölner monotilings, which are central in our extensions of Brudno's results. We treat the 'first' and the 'second' theorems of Brudno in Chapter 7.
The first theorem is generalized for subshifts over computable groups admitting computable Fölner monotilings, while the second theorem is proved under the assertion of regularity of the monotiling, which we introduce in Chapter 7 as well.@en

This thesis is dedicated to the study of several aspects of the theory of functional calculus. This theory considers the combination of an operator A and a function f(z) of a variable z, resulting in an operator f(A). One then attempts to study properties of the operator f(A) in terms of properties of the operator A and the function f. A classical example of a functional calculus is the calculus for diagonalizable matrices. This calculus is based on the idea that for a diagonal matrix D and a function f, f(D) should be defined by simply applying f to the entries of D. One then extends this definition to diagonalizable matrices in the obvious way. Despite its relatively straightforward construction, many nontrivial questions arise when studying this calculus. Recently, functional calculus theory has also proven useful when studying partial differential equations from a functional analytic perspective. The functional analytic viewpoint on a large class of partial differential equation leads to the study of differential operators A on infinite-dimensional spaces X. Functional calculus theory allows one to make this formal intuition precise and provides a convenient framework for studying the differential operator A as well as operators related to A. In this thesis both the functional calculus for diagonalizable matrices and the calculus for differential operators are studied. Using transference principles and double operator integrals, we link the theory of functional calculus to the area of harmonic analysis. Then we use theorems from harmonic analysis to deduce new results in functional calculus theory. We also apply functional calculus theory to a problem in the theory of the numerical approximation of the solutions of evolution equations. Since the solution of such an equation is usually hard to deal with analytically, one tries to approximate it by simpler expressions. One would then like to know whether this approximation converges, and functional calculus theory is a useful tool with which one can deal with this question. A more detailed description of the contents of this thesis is as follows. In Part I we treat some of the basic tools which will be used throughout the thesis. These include: the basics of the theory of functional calculus, some function space theory and preliminaries on vector-valued harmonic analysis, as well as the transference principles which link these three topics together. In Part II we study the functional calculus theory associated for operator semigroups. We obtain new links between harmonic analysis and functional calculus theory . In Chapter 3 this allows us to deduce properties of f(A) for a wide class of functions f. This class of functions depends heavily on geometrical aspects of the underlying space, and therefore so do the results which we obtain. By contrast, in Chapter 4 we study f(A) for a class of functions f which does not depend on geometrical properties of the underlying space. However, here the results are only valid when considering equations with initial values in so-called interpolation spaces. In Part III we study the functional calculus for diagonalizable matrices. For diagonalizable matrices A and B we determine how properties of f(B)-f(A) relate to properties of B-A. Again we relate a question in functional calculus theory to harmonic analysis, this time using the technique of double operator integration. In Part IV we apply functional calculus theory to the study of numerical approximation methods for the solutions to evolution equations. In particular, we consider a recently proposed numerical approximation method. We show that this approximation method converges for a large set of initial values and determine the corresponding rates of convergence. Then, using the theory from earlier chapters, we improve these rates of convergence for specific classes of operators. Finally, two appendices contain results which are used in earlier Chapters \ref{functional calculus for semigroup generators}. These results are of a technical nature and have been placed in appendices to improve the readability of the main text.@en