MathematischeGesellschaft Descriptive theory of chaos |

Speaker:Olexandr Mikolajowytsch Sharkovsky, Institute of Mathematics, NAS of Ukraine

Organizer:Fakultät für Mathematik und Informatik

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Descriptive theory of sets is a classical section of mathematics, which arose at the beginning of the last century. The talk proposes the basis of the descriptive theory of

chaos. We consider dynamical systems on a compact X, generated by a continuous map f : X →X, mainly in the case of when X is an interval I ⊂R. The asymptotic behavior of every trajectory is usually determined through the so- called ω-limit set, or, more simply, the attractor of this trajectory. The set of all trajectories attracted by the same attractor is called the basin of this attractor.

Dynamical system if its topological entropy is positive :

1) has a lot of diﬀerent attractors of trajectories, namely, the continuum of attractors;

2) basins of most attractors have a very complex structure, namely, they are sets of the 3rd class in the terminology of the descriptive theory of sets;

3) basins of diﬀerent attractors are very intertwined and they can not be separated from each other by open or closed sets, but only by sets of the 2nd class of complexity, and

4) in the space of all closed subsets of the state space (with the Hausdorﬀ metric), the set of all attractors is an attractor net (network) whose cells are formed by Cantor sets whose points are themselves attractors.

chaos. We consider dynamical systems on a compact X, generated by a continuous map f : X →X, mainly in the case of when X is an interval I ⊂R. The asymptotic behavior of every trajectory is usually determined through the so- called ω-limit set, or, more simply, the attractor of this trajectory. The set of all trajectories attracted by the same attractor is called the basin of this attractor.

Dynamical system if its topological entropy is positive :

1) has a lot of diﬀerent attractors of trajectories, namely, the continuum of attractors;

2) basins of most attractors have a very complex structure, namely, they are sets of the 3rd class in the terminology of the descriptive theory of sets;

3) basins of diﬀerent attractors are very intertwined and they can not be separated from each other by open or closed sets, but only by sets of the 2nd class of complexity, and

4) in the space of all closed subsets of the state space (with the Hausdorﬀ metric), the set of all attractors is an attractor net (network) whose cells are formed by Cantor sets whose points are themselves attractors.

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Type:Colloquium

Language:English

Category:Research

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