The algebra of semi-flows: a tale of two topologies
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912479675678720 |
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| author | Spendlove, Kelly Vandervorst, Robert |
| author_facet | Spendlove, Kelly Vandervorst, Robert |
| contents | To capture the global structure of a dynamical system we reformulate dynamics in terms of appropriately constructed topologies, which we call flow topologies; we call this process topologization. This yields a description of a semi-flow in terms of a bi-topological space, with the first topology corresponding to the (phase) space and the second to the flow topology. A study of topology is facilitated through discretization, i.e. defining and examining appropriate finite sub-structures. Topologizing the dynamics provides an elegant solution to their discretization by discretizing the associated flow topologies. We introduce Morse pre-orders, an instance of a more general bi-topological discretization, which synthesize the space and flow topologies, and encode the directionality of dynamics. We describe how Morse pre-orders can be augmented with appropriate (co)homological information in order to describe invariance of the dynamics; this ensemble provides an algebraization of the semi-flow. An illustration of the main ingredients of the paper is provided by an application to the theory of discrete parabolic flows. Algebraization yields a new invariant for positive braids in terms of a bi-graded differential module which contains Morse theoretic information of parabolic flows. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_06971 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The algebra of semi-flows: a tale of two topologies Spendlove, Kelly Vandervorst, Robert Algebraic Topology Dynamical Systems 06-04, 06-08, 06D05, 06E05, 37B35, 37M22, 37C70 To capture the global structure of a dynamical system we reformulate dynamics in terms of appropriately constructed topologies, which we call flow topologies; we call this process topologization. This yields a description of a semi-flow in terms of a bi-topological space, with the first topology corresponding to the (phase) space and the second to the flow topology. A study of topology is facilitated through discretization, i.e. defining and examining appropriate finite sub-structures. Topologizing the dynamics provides an elegant solution to their discretization by discretizing the associated flow topologies. We introduce Morse pre-orders, an instance of a more general bi-topological discretization, which synthesize the space and flow topologies, and encode the directionality of dynamics. We describe how Morse pre-orders can be augmented with appropriate (co)homological information in order to describe invariance of the dynamics; this ensemble provides an algebraization of the semi-flow. An illustration of the main ingredients of the paper is provided by an application to the theory of discrete parabolic flows. Algebraization yields a new invariant for positive braids in terms of a bi-graded differential module which contains Morse theoretic information of parabolic flows. |
| title | The algebra of semi-flows: a tale of two topologies |
| topic | Algebraic Topology Dynamical Systems 06-04, 06-08, 06D05, 06E05, 37B35, 37M22, 37C70 |
| url | https://arxiv.org/abs/2207.06971 |