On <i>α</i>-Limit Sets in Lorenz Maps
The aim of this paper is to show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-limit sets in Lorenz maps do not have to be completely...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-09-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/23/9/1153 |
| _version_ | 1797519449073385472 |
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| author | Łukasz Cholewa Piotr Oprocha |
| author_facet | Łukasz Cholewa Piotr Oprocha |
| author_sort | Łukasz Cholewa |
| collection | DOAJ |
| description | The aim of this paper is to show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. On the basis of provided examples, we also present how the performed study on the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-limit sets is closely connected with the calculation of the topological entropy. |
| first_indexed | 2024-03-10T07:42:56Z |
| format | Article |
| id | doaj.art-d2c4803387dd4933bc9b5d341ac781cf |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T07:42:56Z |
| publishDate | 2021-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-d2c4803387dd4933bc9b5d341ac781cf2023-11-22T12:57:27ZengMDPI AGEntropy1099-43002021-09-01239115310.3390/e23091153On <i>α</i>-Limit Sets in Lorenz MapsŁukasz Cholewa0Piotr Oprocha1Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, PolandFaculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, PolandThe aim of this paper is to show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. On the basis of provided examples, we also present how the performed study on the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-limit sets is closely connected with the calculation of the topological entropy.https://www.mdpi.com/1099-4300/23/9/1153Lorenz maprenormalizable maplimit setcompletely invariant setentropy |
| spellingShingle | Łukasz Cholewa Piotr Oprocha On <i>α</i>-Limit Sets in Lorenz Maps Entropy Lorenz map renormalizable map limit set completely invariant set entropy |
| title | On <i>α</i>-Limit Sets in Lorenz Maps |
| title_full | On <i>α</i>-Limit Sets in Lorenz Maps |
| title_fullStr | On <i>α</i>-Limit Sets in Lorenz Maps |
| title_full_unstemmed | On <i>α</i>-Limit Sets in Lorenz Maps |
| title_short | On <i>α</i>-Limit Sets in Lorenz Maps |
| title_sort | on i α i limit sets in lorenz maps |
| topic | Lorenz map renormalizable map limit set completely invariant set entropy |
| url | https://www.mdpi.com/1099-4300/23/9/1153 |
| work_keys_str_mv | AT łukaszcholewa oniailimitsetsinlorenzmaps AT piotroprocha oniailimitsetsinlorenzmaps |