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Description
Dynamical systems are models of all kind of phenomena evolving in time.
Thereby, the best possible situation we could imagine is to completely classify all possible outcomes
of processes with arbitrary accuracy. Indeed, in a sense, this is the quintessentially goal of the theory
of dynamical systems, namely, to classify all possible dynamical systems via their long-term behavior
in a meaningful and absolute fashion. A very common scheme and successful approach to obtain useful classification of dynamical systems is the following: first, we have to define when we want to consider two dynamical systems to be the same. Here, two very prominent concepts are topological conjugacy and measure-theoretic isomorphism. Second, we have to come up with convenient dynamical notions which do not change for systems which are isomorphic. Such notions are called dynamical invariants and they usually reflect diffrent kinds of dynamical behaviour and complexity of a system. In the best case possible, these invariants can be expressed as a single number.
Now, one of the most prominent dynamical invariants is the notion of entropy. It measures how
much disorder is present in a system, by quantifying the exponential growth rate for the number of
initial states that can be separated within a certain accuracy while time passes and accuracy increases.
As it turns out, entropy is especially useful for classifying systems that exhibited a lot of complexity
(sometimes referred to as chaotic systems). However, this project is devoted to explore and study
dynamical invariants for systems showing only low-complexity behavior (zero entropy). For these
systems one needs to develop new invariants. One aim of our project will be to demonstrate that these dynamical invariants induced by pseudometrics are suitable to study low-complexity systems and to take a step towards providing fundamental classiffications-schemes for these systems.