Die vorliegende Arbeit befasst sich mit einer Vermutung von Sarnak aus dem Jahre 2010 über die Orthogonalität von durch deterministische dynamische Systeme induzierte Folgen zur Möbiusschen μ-Funktion. Ihre Hauptresultate sind zum einen der Ergodensatz mit Möbiusgewichten, welcher eine maßtheoretische (schwächere) Version von Sarnaks Vermutung darstellt, und zum anderen die bereits gesicherte Gültigkeit der genannten Vermutung in Spezialfällen, wobei hier exemplarisch unter anderem der Thue–Morse Shift und Schiefprodukterweiterungen von rationalen Rotationen auf dem Kreis gewählt worden sind. Zum Zwecke der Motivation zeigen wir, dass eine gewisse Wachstumsabschätzung für die Mertensfunktion äquivalent ist zum Primzahlsatz und skizzieren ein Resultat, welches die Äquivalenz einer weiteren solchen Abschätzung zur Riemannschen Vermutung liefert, um auf diese Weise die Bedeutung der Möbiusfunktion für die Zahlentheorie herauszustellen. Da sie für das Verständnis von Sarnaks Vermutung unerlässlich ist, geben wir eine Einführung in die Theorie der Entropie dynamischer Systeme auf Grundlage der Definitionen von Adler–Konheim–McAndrew, Bowen–Dinaburg und Kolmogorov–Sinai. Ferner berechnen wir die topologische Entropie des Thue–Morse Shifts und von Schiefprodukterweiterungen von Rotatione auf dem Kreis. Wir studieren die ergodische Zerlegung T-invarianter Maße auf kompakten metrischen Räumen mit stetiger Transformation T, welche wir für den Beweis des Ergodensatzes mit Möbiusgewichten benötigen. Sodann beweisen wir den genannten gewichteten Ergodensatz. Wir geben eine hinreichende Bedingung an für das Erfülltsein von Sarnaks Vermutung in einem gegebenen dynamischen System, welche im anschließenden Kapitel Anwendung findet. So wird nachgewiesen, dass Sarnaks Vermutung im Falle des Thue–Morse Shifts und von Schiefprodukterweiterungen von rationalen Rotationen auf dem Kreis erfüllt ist. Abschließend wird gezeigt, dass Sarnaks Vermutung sich als Konsequenz aus einer Vermutung von Chowla ergibt. / The thesis in hand deals with a conjecture of Sarnak from 2010 about the orthogonality of sequences induced by deterministic dynamical systems to the Möbius μ-function. Its main results are the ergodic theorem with Möbius weights, which is a measure theoretic (weaker) version of Sarnak’s conjecture, and the already assured validity of Sarnak’s conjecture in special cases, where we have exemplarily chosen the Thue–Morse shift and skew product extensions of rational rotations on the significance of the Möbius function for number theory. Since it is essential for the understanding of Sarnak’s conjecture we give an introduction to the theory of entropy of dynamical systems based on the definitions of Adler–Konheim–McAndrew, Bowen–Dinaburg and Kolmogorov–Sinai. Furthermore, we calculate the topological entropy of the Thue–Morse shift and of skew product extensions of rotations on the circle. We study the ergodic decomposition for T-invariant measures on compact metric spaces with continuous transformations T, which we will need for the proof of the ergodic theorem with Möbius weights. Thereafter, we prove the namely weighted ergodic theorem. We give a sufficient condition for Sarnak’s conjecture to hold for a given dynamical system, which we make use of in the following chapter. Thereupon, it is varified that Sarnak’s conjecture holds for the Thue–Morse shift and for skew product extensions of rational rotations on the circle. Lastly, it is shown that Sarnak’s conjecture from one of Chowla.

Many complexity measures are defined as the size of a minimal representation in a specific model class. One such complexity measure, which is important because it is widely applied, is statistical complexity. It is defined for discrete-time, stationary stochastic processes within a theory called computational mechanics. Here, a mathematically rigorous, more general version of this theory is presented, and abstract properties of statistical complexity as a function on the space of processes are investigated. In particular, weak-* lower semi-continuity and concavity are shown, and it is argued that these properties should be shared by all sensible complexity measures. Furthermore, a formula for the ergodic decomposition is obtained. The same results are also proven for two other complexity measures that are defined by different model classes, namely process dimension and generative complexity. These two quantities, and also the information theoretic complexity measure called excess entropy, are related to statistical complexity, and this relation is discussed here. It is also shown that computational mechanics can be reformulated in terms of Frank Knight's prediction process, which is of both conceptual and technical interest. In particular, it allows for a unified treatment of different processes and facilitates topological considerations. Continuity of the Markov transition kernel of a discrete version of the prediction process is obtained as a new result.

Many complexity measures are defined as the size of a minimal representation in a specific model class. One such complexity measure, which is important because it is widely applied, is statistical complexity. It is defined for discrete-time, stationary stochastic processes within a theory called computational mechanics. Here, a mathematically rigorous, more general version of this theory is presented, and abstract properties of statistical complexity as a function on the space of processes are investigated. In particular, weak-* lower semi-continuity and concavity are shown, and it is argued that these properties should be shared by all sensible complexity measures. Furthermore, a formula for the ergodic decomposition is obtained. The same results are also proven for two other complexity measures that are defined by different model classes, namely process dimension and generative complexity. These two quantities, and also the information theoretic complexity measure called excess entropy, are related to statistical complexity, and this relation is discussed here. It is also shown that computational mechanics can be reformulated in terms of Frank Knight''s prediction process, which is of both conceptual and technical interest. In particular, it allows for a unified treatment of different processes and facilitates topological considerations. Continuity of the Markov transition kernel of a discrete version of the prediction process is obtained as a new result.