In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We substantiate that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:14-qucosa-129909 |
| Date | 09 December 2013 |
| Creators | Behrisch, Mike, Kerkhoff, Sebastian, Pöschel, Reinhard, Schneider, Friedrich Martin, Siegmund, Stefan |
| Contributors | Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften, Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften, Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften |
| Publisher | Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:preprint |
| Format | application/pdf, text/plain, text/plain, text/plain, application/pdf, application/zip |
| Relation | dcterms:isPartOf:Preprint / Technische Universität Dresden ; MATH-AL 2013,05 |
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