Partial classification of spectrum maximizing products for pairs of $2\times2$ matrices
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912354043691008 |
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| author | Laskawiec, Piotr |
| author_facet | Laskawiec, Piotr |
| contents | Experiments suggest that typical finite sets of square matrices admit spectrum maximizing products (SMPs): that is, products that attain the joint spectral radius (JSR). Furthermore, those SMPs are often combinatorially "simple." In this paper, we consider pairs of real $2 \times 2$ matrices. We identify regions in the space of such pairs where SMPs are guaranteed to exist and to have a simple structure. We also identify another region where SMPs may fail to exist (in fact, this region includes all known counterexamples to the finiteness conjecture), but nevertheless a Sturmian maximizing measure exists. Though our results apply to a large chunk of the space of pairs of $2 \times 2$ matrices, including for instance all pairs of non-negative matrices, they leave out certain "wild" regions where more complicated behavior is possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16680 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partial classification of spectrum maximizing products for pairs of $2\times2$ matrices Laskawiec, Piotr Optimization and Control 15A18 (primary), 15A60, 20G05, 37H15 (secondary) Experiments suggest that typical finite sets of square matrices admit spectrum maximizing products (SMPs): that is, products that attain the joint spectral radius (JSR). Furthermore, those SMPs are often combinatorially "simple." In this paper, we consider pairs of real $2 \times 2$ matrices. We identify regions in the space of such pairs where SMPs are guaranteed to exist and to have a simple structure. We also identify another region where SMPs may fail to exist (in fact, this region includes all known counterexamples to the finiteness conjecture), but nevertheless a Sturmian maximizing measure exists. Though our results apply to a large chunk of the space of pairs of $2 \times 2$ matrices, including for instance all pairs of non-negative matrices, they leave out certain "wild" regions where more complicated behavior is possible. |
| title | Partial classification of spectrum maximizing products for pairs of $2\times2$ matrices |
| topic | Optimization and Control 15A18 (primary), 15A60, 20G05, 37H15 (secondary) |
| url | https://arxiv.org/abs/2406.16680 |