Singular dynamics for discrete weak K.A.M. solutions of exact twist maps
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913609778462720 |
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| author | Du, Jianxing Su, Xifeng |
| author_facet | Du, Jianxing Su, Xifeng |
| contents | For any exact twist map $f$ and any cohomology class $c\in\mathbb{R}$, let $u_c$ be any associated discrete weak K.A.M. solution, and we introduce an inherent Lipschitz dynamics $Σ_+$ given by the discrete forward Lax-Oleinik semigroup. We investigate several properties of $Σ_+$ and show that the non-differentiable points of $u_c$ are globally propagated and forward invariant by $Σ_+$. In particular, such propagating dynamics possesses the same rotation number $α'(c)$ as the associated Aubry-Mather set at cohomology class $c$. As applications, we provide via $Σ_+$ {a discrete analogue of Bernard's regularization theorem \cite{Ber07} and} a detailed exposition of Arnaud's observation \cite{Arnaud_2011}. Furthermore, we construct and analyze the corresponding dynamics on the full pseudo-graphs of discrete weak K.A.M. solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01141 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular dynamics for discrete weak K.A.M. solutions of exact twist maps Du, Jianxing Su, Xifeng Dynamical Systems 37E10, 37E40, 37J10, 37J30, 37J35 For any exact twist map $f$ and any cohomology class $c\in\mathbb{R}$, let $u_c$ be any associated discrete weak K.A.M. solution, and we introduce an inherent Lipschitz dynamics $Σ_+$ given by the discrete forward Lax-Oleinik semigroup. We investigate several properties of $Σ_+$ and show that the non-differentiable points of $u_c$ are globally propagated and forward invariant by $Σ_+$. In particular, such propagating dynamics possesses the same rotation number $α'(c)$ as the associated Aubry-Mather set at cohomology class $c$. As applications, we provide via $Σ_+$ {a discrete analogue of Bernard's regularization theorem \cite{Ber07} and} a detailed exposition of Arnaud's observation \cite{Arnaud_2011}. Furthermore, we construct and analyze the corresponding dynamics on the full pseudo-graphs of discrete weak K.A.M. solutions. |
| title | Singular dynamics for discrete weak K.A.M. solutions of exact twist maps |
| topic | Dynamical Systems 37E10, 37E40, 37J10, 37J30, 37J35 |
| url | https://arxiv.org/abs/2403.01141 |