Dynamical systems with infinite mean return times.

Research project P 14734 Dynamical systems with infinite mean return times Maximilian THALER 27.11.2000 During the last decades, scientists have become convinced that many systems following some deterministic evolution in time do by no means allow serious detailed predictions about their behaviour in the distant future. In view of this, the observation that many of these "chaotic" systems seem to precisely follow certain statistical laws - like those we know from actually random processes - is of particular importance. The investigation of the mathematical mechanisms responsible for the emergence of seemingly random behaviour in deterministic systems therefore is fundamental for a thorough understanding of a wide variety of mathematical models in the sciences. lt forms the basis for serious predictions on a statistical level appropriate for these systems. The strategy mathematics successfully pursues here is to first consider specific problems within the framework of suitable prototypical models, where it is possible to investigate new phenomena by isolating essential features and separating them from other technical difficulties. Quite frequently the insights obtained this way are the key to an investigation of more complex models, as they reveal which structures one should look for. The present project is devoted to a detailed study of systems in which phases of chaotic behaviour are separated by long laminar phases. This phenomenon, known as intermittency, can be observed in a variety of different situations. Suitable mathematical models are given by transformations with infinite invariant measures. The probabilistic laws governing them are different from those occuring in finite measure preserving situations and still much less understood. Our aim is to contribute to their theory using tools from ergodic theory and probabilistic renewal theory. Much in the spirit of the strategy outlined above, we focus on the specific class of models given by interval maps with indifferent fixed points.

During the last decades, scientists have become convinced that many systems following some deterministic evolution in time do by no means allow serious detailed predictions about their behaviour in the distant future. In view of this, the observation that many of these `chaotic` systems seem to precisely follow certain statistical laws - like those we know from actually random processes - is of particular importance. The investigation of the mathematical mechanisms responsible for the emergence of seemingly random behaviour in deterministic systems therefore is of fundamental importance for a thorough understanding of a wide variety of mathematical models in the sciences. It forms the basis for serious predictions on a statistical level appropriate for these systems. The strategy mathematics successfully pursues here is to first consider specific problems within the framework of suitable prototypical models, where it is possible to investigate new phenomena by isolating essential features and separating them from other technical difficulties. Quite frequently the insights obtained this way are the key to an investigation of more complex models, as they reveal which structures one should look for. The present project was devoted to a detailed study of systems in which phases of chaotic behaviour are separated by long laminar phases. This phenomenon, known as intermittency, can be observed in a variety of different situations, ranging from particle physics to questions about the capacity of information networks. Suitable mathematical models are given by transformations with infinite invariant measures. Some of the probabilistic laws governing them are surprisingly different from statistical rules of thumb taken for granted in everyday life, and a good understanding of these unusual features can contribute to technical improvements in various applications. We succeeded in answering a number of questions about the long term behaviour, which due to the chaotic nature of such systems can hardly be clarified by simply simulating them on a computer. Besides providing an actual verification which can never be accomplished by empirical investigations, the quest for mathematical proofs also serves another purpose: As it requires us to struggle for as good an understanding as possible, we learn much more about these systems than we possibly could by merely inspecting and recording loads of data. These results enable us to make valid predictions where other approaches can only provide guesswork (no matter how subtle the empirical method it is based on may be). The insights obtained in the process of rigorous mathematical analysis, captured in the formal proofs, enhance our ability to also understand how other, more complicated systems will behave in the long run and why they do so.