Report
Topological entropy of Turing complete dynamics
| Title: | Topological entropy of Turing complete dynamics |
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| Authors: | Bruera Méndez, Renzo, Cardona Aguilar, Robert, Miranda Galcerán, Eva, Peralta-Salas, Daniel |
| Contributors: | Universitat Politècnica de Catalunya. Doctorat en Matemàtica Aplicada, Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. TF-EDP - Grup de Teoria de Funcions i Equacions en Derivades Parcials, Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions |
| Source: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| Publisher Information: | 2024. |
| Publication Year: | 2024 |
| Subject Terms: | Classificació AMS::37 Dynamical systems and ergodic theory::37B Topological dynamics, Dinàmica topològica, Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics, Topological dynamics |
| Description: | We explore the relationship between Turing completeness and topological entropy of dynamical systems. We first prove that a natural class of Turing machines that we call “regular Turing machines” (which includes most of the examples of universal Turing machines) has positive topological entropy. We deduce that any Turing complete dynamics with a continuous encoding that simulates a universal machine in this class is chaotic. This applies to our previous constructions of Turing complete area-preserving diffeomorphisms of the disk and 3D stationary Euler flows. The article concludes with an appendix written by Ville Salo that introduces a method to construct universal Turing machines that are not regular and have zero topological entropy. |
| Document Type: | Report |
| File Description: | application/pdf |
| Language: | English |
| DOI: | 10.48550/arxiv.2404.07288 |
| Access URL: | https://arxiv.org/pdf/2404.07288 https://doi.org/10.48550/arxiv.2404.07288 https://hdl.handle.net/2117/419715 |
| Accession Number: | edsair.dedup.wf.002..e3d37352f76f83d45eb2141aebd53ec2 |
| Database: | OpenAIRE |
| DOI: | 10.48550/arxiv.2404.07288 |
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