The shadowing property for piecewise monotone interval maps
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909483112857600 |
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| author | Bura, Adarsh Good, Chris Samuel, Tony |
| author_facet | Bura, Adarsh Good, Chris Samuel, Tony |
| contents | The property of shadowing has been shown to be fundamental in both the theory of symbolic dynamics as well as continuous dynamical systems. A quintessential class of discontinuous dynamical systems are those driven by transitive piecewise monotone interval maps and in particular $β$-transformations, namely transformations of the form $T_{β, α} : x \mapsto βx + α\; (\operatorname{mod} \, 1)$ acting on $[0,1]$. We provide a short elegant proof showing that this class of dynamical systems does not possess the property of shadowing, complementing and extending the work of Chen and Portela. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The shadowing property for piecewise monotone interval maps Bura, Adarsh Good, Chris Samuel, Tony Dynamical Systems 37E05, 37B65 The property of shadowing has been shown to be fundamental in both the theory of symbolic dynamics as well as continuous dynamical systems. A quintessential class of discontinuous dynamical systems are those driven by transitive piecewise monotone interval maps and in particular $β$-transformations, namely transformations of the form $T_{β, α} : x \mapsto βx + α\; (\operatorname{mod} \, 1)$ acting on $[0,1]$. We provide a short elegant proof showing that this class of dynamical systems does not possess the property of shadowing, complementing and extending the work of Chen and Portela. |
| title | The shadowing property for piecewise monotone interval maps |
| topic | Dynamical Systems 37E05, 37B65 |
| url | https://arxiv.org/abs/2502.05058 |