Generalized Evolution Semigroups and $h-$Dichotomies for Evolution Families on Banach Spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909903301378048 |
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| author | Castañeda, Álvaro Poblete, Verónica Robledo, Gonzalo |
| author_facet | Castañeda, Álvaro Poblete, Verónica Robledo, Gonzalo |
| contents | This paper develops a comprehensive theory generalizing exponential decay patterns for evolution processes in Banach spaces. We replace classical exponential bounds with more flexible decay rates governed by an increasing homeomorphism $h$. The core of our approach lies in constructing particular group structures induced by $h$, which allow us to define generalized semigroups on function spaces. We prove that these $h$-semigroups are equivalent to classical evolution semigroups through a natural transformation. Our main result establishes that three fundamental concepts are equivalent: hyperbolicity of the generalized semigroup, dichotomy of the underlying evolution process, and a spectral condition on the generator. This work extends classical dichotomy theory to encompass a wider class of decay patterns, providing new tools for analyzing asymptotic behavior in dynamical systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_11670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Evolution Semigroups and $h-$Dichotomies for Evolution Families on Banach Spaces Castañeda, Álvaro Poblete, Verónica Robledo, Gonzalo Functional Analysis Dynamical Systems 34G10, 34D09, 47D06 This paper develops a comprehensive theory generalizing exponential decay patterns for evolution processes in Banach spaces. We replace classical exponential bounds with more flexible decay rates governed by an increasing homeomorphism $h$. The core of our approach lies in constructing particular group structures induced by $h$, which allow us to define generalized semigroups on function spaces. We prove that these $h$-semigroups are equivalent to classical evolution semigroups through a natural transformation. Our main result establishes that three fundamental concepts are equivalent: hyperbolicity of the generalized semigroup, dichotomy of the underlying evolution process, and a spectral condition on the generator. This work extends classical dichotomy theory to encompass a wider class of decay patterns, providing new tools for analyzing asymptotic behavior in dynamical systems. |
| title | Generalized Evolution Semigroups and $h-$Dichotomies for Evolution Families on Banach Spaces |
| topic | Functional Analysis Dynamical Systems 34G10, 34D09, 47D06 |
| url | https://arxiv.org/abs/2511.11670 |