Bowen’s Formula for a Dynamical Solenoid
More than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen’s formula, that relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function defined by...
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2024-11-01
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| author | Andrzej Biś Wojciech Kozłowski Agnieszka Marczuk |
| author_facet | Andrzej Biś Wojciech Kozłowski Agnieszka Marczuk |
| author_sort | Andrzej Biś |
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| description | More than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen’s formula, that relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function defined by a single conformal map. In this paper, we extend the result of Bowen to a sequence of conformal maps. We present a dynamical solenoid, i.e., a generalized dynamical system obtained by backward compositions of a sequence of continuous surjections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>f</mi><mi>n</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></msub></semantics></math></inline-formula> defined on a compact metric space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> Under mild assumptions, we provide a self-contained proof that Bowen’s formula holds for dynamical conformal solenoids. As a corollary, we obtain that the Bowen’s formula holds for a conformal surjection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></semantics></math></inline-formula> of a compact |
| format | Article |
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| institution | OA Journals |
| issn | 1099-4300 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj-art-20790cb33fed4b249c2caedecc9827d62025-08-20T02:08:00ZengMDPI AGEntropy1099-43002024-11-01261197910.3390/e26110979Bowen’s Formula for a Dynamical SolenoidAndrzej Biś0Wojciech Kozłowski1Agnieszka Marczuk2Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Łódź, PolandFaculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Łódź, PolandFaculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Łódź, PolandMore than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen’s formula, that relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function defined by a single conformal map. In this paper, we extend the result of Bowen to a sequence of conformal maps. We present a dynamical solenoid, i.e., a generalized dynamical system obtained by backward compositions of a sequence of continuous surjections <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>f</mi><mi>n</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></msub></semantics></math></inline-formula> defined on a compact metric space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> Under mild assumptions, we provide a self-contained proof that Bowen’s formula holds for dynamical conformal solenoids. As a corollary, we obtain that the Bowen’s formula holds for a conformal surjection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></semantics></math></inline-formula> of a compacthttps://www.mdpi.com/1099-4300/26/11/979topological pressureconformal mapstopological entropy |
| spellingShingle | Andrzej Biś Wojciech Kozłowski Agnieszka Marczuk Bowen’s Formula for a Dynamical Solenoid Entropy topological pressure conformal maps topological entropy |
| title | Bowen’s Formula for a Dynamical Solenoid |
| title_full | Bowen’s Formula for a Dynamical Solenoid |
| title_fullStr | Bowen’s Formula for a Dynamical Solenoid |
| title_full_unstemmed | Bowen’s Formula for a Dynamical Solenoid |
| title_short | Bowen’s Formula for a Dynamical Solenoid |
| title_sort | bowen s formula for a dynamical solenoid |
| topic | topological pressure conformal maps topological entropy |
| url | https://www.mdpi.com/1099-4300/26/11/979 |
| work_keys_str_mv | AT andrzejbis bowensformulaforadynamicalsolenoid AT wojciechkozłowski bowensformulaforadynamicalsolenoid AT agnieszkamarczuk bowensformulaforadynamicalsolenoid |