Jörg Neunhäuserer

If we understand philosophy as a rational reference to the world as a whole, then a philosophy of mathematics is a rational reference to mathematics as a whole. We expect such a reference to be understandable and meaningful in every case, ideally, it is also justified. A philosophy of mathematics appears understandable to us if the terms used are clearly explained or at least satisfactorily interpreted and if central statements are clearly and unambiguously formulated or can at least be formulated in this way.

In diesem Buch entwickeln wir die mathematische Theorie zeitdiskreter dynamischer Systeme. Unser Schwerpunkt liegt hierbei auf der Definition sowie der qualitativen und quantitativen Untersuchung chaotischer Systeme. Zu Beginn stellen wir Grundbegriffe der Theorie zeitdiskreter dynamischer Systeme vor und erläutern diese anhand von Beispielen. Im Folgenden gehen wir gängige Definitionen chaotischer Systeme an und diskutieren den Zusammenhang dieser Begriffe. Mit der symbolischen Dynamik und der symbolischen Kodierung besprechen wir daraufhin den Kern der topologischen Theorie chaotischer dynamischer Systeme. Wir führen die symbolische Kodierung einiger paradigmatischer Systeme durch. Die maßtheoretische Untersuchung chaotischer Systeme ist ein Teil der Ergodentheorie. Diese stellen wir kompakt vor und wenden sie auf eine Reihe von dynamischen Systemen an. Des Weiteren stellen wir numerische Methoden dar, die Indizien für eine chaotische Entwicklung eines dynamischen Systems bereitstellen können. Im letzten Kapitel des Buches stellen wir eine Reihe von chaotischen Systemen vor, die aus den empirischen Wissenschaften stammen. Zusätzlich stellen wir in einem abschließenden Kapitel den Zusammenhang zeitdiskreter und zeitkontinuierlicher dynamischer Systeme dar. Ein Anhang des Buches dient dem Nachschlagen der mathematischen Grundbegriffe, die wir verwenden. Das Buch wendet sich an Studierende, die Grundkenntnisse der Analysis und Linearen Algebra haben, wie sie in entsprechenden Vorlesungen und Lehrbüchern vermittelt wird. Das Buch kann als Grundlage einer Vorlesung im letzten Jahr eines Bachelorstudiengangs oder einer Vorlesung in einem Master Studiengang in Mathematik, Physik oder Informatik verwendet werden.

What kind of objects does mathematics investigate, and in what sense do these objects exist? Why are we justified in considering mathematical statements as part of our knowledge, and how can they be validated? A philosophy of mathematics seeks to answer such questions. In this introduction, we present the major positions in the philosophy of mathematics and formulate their core ideas into clear, accessible theses. Readers will learn which philosophers developed each position and the historical context in which they emerged. Drawing on fundamental intuitions and scientific findings, one can argue for or against these theses – such arguments form the second focus of this book. The book aims to encourage readers to reflect on the philosophy of mathematics, to develop their own stance, and to learn how to argue for it. This book is a translation of the original German 2nd edition. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content, so that the book may read stylistically differently from a conventional translation.

Wenn wir Philosophie als vernünftige Bezugnahme auf die Welt als Ganzes begreifen, so ist eine Philosophie der Mathematik eine vernünftige Bezugnahme auf die Mathematik als Ganzes. Wir erwarten von solch einer Bezugnahme in jedem Fall, dass sie verständlich und aussagekräftig ist, im Idealfall ist sie auch noch begründet. Eine Philosophie der Mathematik erscheint uns verständlich, wenn die verwendeten Begriffe einleuchtend erläutert werden oder sich zumindest zufriedenstellend deuten lassen und wenn zentrale Aussagen klar und unmissverständlich formuliert werden oder sich zumindest so formulieren lassen.

Die Grundannahme des Rationalismus ist, dass wir als rationale Wesen apriorisches Wissen besitzen, das unabhängig von jedweder Erfahrung ist und keiner Begründung durch Erfahrung bedarf. Rationalismus Wissen ist unserer Vernunft entweder unmittelbar gegeben oder es wird durch die Tätigkeit der reinen Vernunft, ohne Rückgriff auf Erfahrung, erschlossen. Vernunft Tätigkeit der reinen Vernunft wird gewöhnlich als Deduktion beschrieben, d. h., unsere Vernunft zieht aus gegebenen Voraussetzungen gültige Schlüsse.

In this work we study dimensional theoretical properties of some a±ne dynamical systems. By dimensional theoretical properties we mean Hausdor® dimension and box- counting dimension of invariant sets and ergodic measures on theses sets. Especially we are interested in two problems. First we ask whether the Hausdor® and box- counting dimension of invariant sets coincide. Second we ask whether there exists an ergodic measure of full Hausdor® dimension on these invariant sets. If this is not the case we ask the question, whether at least the variational principle for Haus- dor® dimension holds, which means that there is a sequence of ergodic measures such that their Hausdor® dimension approximates the Hausdor® dimension of the invariant set. It seems to be well accepted by experts that these questions are of great importance in developing a dimension theory of dynamical systems (see the book of Pesin about dimension theory of dynamical systems [PE2]). Dimensional theoretical properties of conformal dynamical systems are fairly well understood today. For example there are general theorems about conformal repellers and hyperbolic sets for conformal di®eomorphisms (see chapter 7 of [PE2]). On the other hand the existence of two di®erent rates of expansion or contraction forces problems that are not captured by a general theory this days. At this stage of de- velopment of the dimension theory of dynamical systems it seems natural to study non conformal examples. This is the ¯rst step to understand the mechanisms that determine dimensional theoretical properties of non conformal dynamical systems. A±ne dynamical systems represent simple examples of non conformal systems. They are easy to de¯ne, but studying their dimensional theoretical properties does never- theless provide challenging mathematical problems and exemplify interesting phe- nomena. We consider here a special class of self-a±ne repellers in dimension two, depending on four parameters (see 2.1.). Furthermore we study a class of attractors of piecewise a±ne maps in dimension three depending on four parameters as well. The last object of our work are projections of these maps that are known as gener- alized Baker's transformations (see 2.2.). The contents of our work is the following: In chapter two we give an overview about some main results in the area of di- mension theory of a±ne dynamical systems and de¯ne the systems we study in this work. We will explain, what is known about the dimensional theoretical properties of these systems and describe what our new results are. In chapter three we then apply symbolic dynamics to our systems. We will introduce explicit shift codings 4 and ¯nd representations of all ergodic measures for our systems using these codings. From chapter four to chapter eight we study dimensional theoretical properties, which our systems generally or generically have. In chapter four we will prove a formula for the box-counting dimension of the repellers and the attractors (see the- orem 4.1.). Then in chapter ¯ve we apply general dimensional theoretical results for ergodic measures found by Ledrappier and Young [LY] and Barreira, Schmeling and Pesin [BPS] to our systems. These results relate the dimension of ergodic measures to metric entropy and Lyapunov exponents. Using this approach we will be able to reduce questions about the dimension of ergodic measures in our context to ques- tions about certain overlapping and especially overlapping self-similar measures on the line. These overlapping self-similar measures are studied in chapter six. Our main theorem extends a result of Peres and Solomyak [PS2] concerning the absolute continuity resp. singularity of symmetric self-similar measures to asymmetric ones (see theorem 6.1.3.). In chapter seven we bring our results together. We prove that we generically (in the sense of Lebesgue measure on a part of the parameter space) have the iden- tity of box-counting and Hausdor® dimension for the repellers and the attractors. (see theorem 7.1.1. and corollary 7.1.2.). This result suggest that one can expect that the identity of box-counting dimension and Hausdor® dimension holds at least generically in some natural classes of non conformal dynamical systems. Furthermore we will see in chapter seven that there generically exists an ergodic measure of full Hausdor® dimension for the repellers. On the other hand the vari- ational principle for Hausdor® dimension is not generic for the attractors. It holds only if we assume a certain symmetry (see theorem 7.1.1.). For generalized Baker's transformations we will ¯nd a part of the parameter space where there generically is an ergodic measure of full dimension and a part where the variational principle for Hausdor® dimension does not hold (see theorem 7.1.3.). Roughly speaking the reason why the variational principle does not hold here is, that if there exists both a stable and an unstable direction one can not generically maximize the dimension in the stable and in the unstable direction at the same time. In an other setting this phenomenon was observed before by Manning and McCluskey [MM]. In chapter eight we extend some results of the last section to invariant sets that correspond to special Markov chains instead of full shifts (see theorem 8.1.1.). In the last two chapters of our work we are interested in number theoretical excep- tions to our generic results. The starting point of our considerations in section nine are results of ErdÄos [ER1] and Alexander and Yorke [AY] that establish singularity and a decrease of dimension for in¯nite convolved Bernoulli measures under special conditions. Using a generalized notion of the Garsia entropy ([GA1/2]) we are able 5 to understand the consequences of number theoretical peculiarities in broader class of overlapping measures (see theorem 9.1.1.). In chapter ten we then analyze number theoretical peculiarities in the context of our dynamical systems. We restrict our attention to a symmetric situation where we generically have the existence of a Bernoulli measure of full dimension and the identity of Hausdor® and box-counting dimension for all of our systems. In the ¯rst section of chapter ten we ¯nd parameter values such that the variational principle for Hausdor® dimension does not hold for the attractors and for the Fat Baker's transformations (see theorem 10.1.1.). These are the ¯rst known examples of dynamical systems for which the variational principle for Hausdor® dimension does not hold because of number theoretical peculiarities of parameter values. For the repellers we have been able to show that under certain number theoretical conditions there is at least no Bernoulli measure of full Hausdor® dimension; the question if the variational principle for Hausdor® dimension holds remains open in this situation. In the second section of chapter ten we will show that the identity for Hausdor® and box-counting dimension can drops because there are number theoretical pecu- liarities. In the context of Weierstrass-like functions this phenomenon was observed by Przytycki and Urbanski [PU]. Our theorem extends this result to a larger class of sets, invariant under dynamical systems (see theorem 10.2.1). At the end of this work the reader will ¯nd two appendices, a list of notations and the list of references. In appendix A we introduce the notions of dimension we use in this work and collect some general facts in dimension theory. In appendix B we state the facts about Pisot-Vijayarghavan number, we need in our analysis of number theoretical peculiarities. The list of notations contains general notations and a table with a summary of notations we use to describe the dynamical systems that we study. Acknowledgments I wish to thank my supervisor JÄorg Schmeling for a lot of valuable discussion and all his help. Also thanks to Luis Barreira for his great hospitality in Lisboa and many interesting comments. This work was done while I was supported by "Promotionstipendium gem. NaFÄoG der Freien UniversitÄat Berlin".

Dieses Buch ist ein wichtiges studienbegleitendes Hilfsmittel für alle die Mathematik-Lehrveranstaltungen besuchen. Die Lektüre dieses Buch ermöglicht Ihnen, begriffliche Sicherheit für Mathematik-Vorlesungen und Prüfungen aufzubauen. Die Lektüre jedes Kapitel dieses Buches erlaubt Ihnen, einen Überblick über die Begriffe eines Teilgebiets der Mathematik zu erhalten und diese Begriffe nachhaltig zu erfassen. Wenn Sie als Student einen mathematischen Begriff nicht richtig verstehen oder sich an seine Definition nicht erinnern, können Sie in diesem Buch nachschlagen und erhalten durch paradigmatische Beispiele und Bilder ein fundiertes Verständnis des Begriffs. Arbeiten Sie ein Kapitel dieses Buches in Vorbereitung einer Prüfung durch, so können Sie sich in begrifflicher Hinsicht in der Prüfung sicher fühlen. Insgesamt finden sich in diesem Buch mehr als tausend Definitionen von Begriffen aus vierzehn Teilgebieten der Mathematik. Die Auswahl der Begriffe orientiert sich in jedem Kapitel an den Vorlesungen zum behandelten Thema, die an deutschen Hochschulen gehalten werden. Alle wesentlichen Begriffe, die in Mathematik-Vorlesungen in Bachelorstudiengängen vorkommen und auch alle grundlegenden Begriffe der Mathematik-Vorlesungen in Masterstudiengängen sind in diesem Buch enthalten. Dieses Buch stellt also einen Kanon mathematischer Begriffe vor, der auch für Lehrende von Interesse ist.