Subshifts on Infinite Alphabets and Their Entropy
We analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-11-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/22/11/1293 |
| _version_ | 1827702270605656064 |
|---|---|
| author | Sharwin Rezagholi |
| author_facet | Sharwin Rezagholi |
| author_sort | Sharwin Rezagholi |
| collection | DOAJ |
| description | We analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by countably infinite graphs, our approach yields the same entropy as the approach of Gurevich We give formulae for the entropy of countable topological Markov chains in terms of the spectral radius in <inline-formula><math display="inline"><semantics><msup><mi>l</mi><mn>2</mn></msup></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T14:51:56Z |
| format | Article |
| id | doaj.art-895215f1311d49e1befc2d098a409abf |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T14:51:56Z |
| publishDate | 2020-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-895215f1311d49e1befc2d098a409abf2023-11-20T20:55:57ZengMDPI AGEntropy1099-43002020-11-012211129310.3390/e22111293Subshifts on Infinite Alphabets and Their EntropySharwin Rezagholi0Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, GermanyWe analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by countably infinite graphs, our approach yields the same entropy as the approach of Gurevich We give formulae for the entropy of countable topological Markov chains in terms of the spectral radius in <inline-formula><math display="inline"><semantics><msup><mi>l</mi><mn>2</mn></msup></semantics></math></inline-formula>.https://www.mdpi.com/1099-4300/22/11/1293infinite graphssymbolic dynamicstopological entropyword complexity |
| spellingShingle | Sharwin Rezagholi Subshifts on Infinite Alphabets and Their Entropy Entropy infinite graphs symbolic dynamics topological entropy word complexity |
| title | Subshifts on Infinite Alphabets and Their Entropy |
| title_full | Subshifts on Infinite Alphabets and Their Entropy |
| title_fullStr | Subshifts on Infinite Alphabets and Their Entropy |
| title_full_unstemmed | Subshifts on Infinite Alphabets and Their Entropy |
| title_short | Subshifts on Infinite Alphabets and Their Entropy |
| title_sort | subshifts on infinite alphabets and their entropy |
| topic | infinite graphs symbolic dynamics topological entropy word complexity |
| url | https://www.mdpi.com/1099-4300/22/11/1293 |
| work_keys_str_mv | AT sharwinrezagholi subshiftsoninfinitealphabetsandtheirentropy |