Convergence and combinatorics of the Reverse algorithm
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908835373907968 |
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| author | Ito, Hiroaki Langeveld, Niels Thuswaldner, Jörg |
| author_facet | Ito, Hiroaki Langeveld, Niels Thuswaldner, Jörg |
| contents | We study the Reverse algorithm, a multidimensional continued fraction algorithm, which is not unimodular. We show that the Reverse algorithm is ergodic and, by proving that its second Lyapunov exponent is negative, that it is a.e. exponentially convergent. In addition to that, we attach substitutions to this algorithm and study the $S$-adic languages generated by sequences of these substitutions. The negativity of the second Lyapunov exponent implies that almost all of these languages are balanced. By a thorough study of the combinatorics of the substitutions, we are even able to obtain a concrete generic family of balanced languages that is characterized in terms of a simple condition on the underlying sequence of substitutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14142 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence and combinatorics of the Reverse algorithm Ito, Hiroaki Langeveld, Niels Thuswaldner, Jörg Dynamical Systems 37B10, 37A25, 68R15, 11J70 We study the Reverse algorithm, a multidimensional continued fraction algorithm, which is not unimodular. We show that the Reverse algorithm is ergodic and, by proving that its second Lyapunov exponent is negative, that it is a.e. exponentially convergent. In addition to that, we attach substitutions to this algorithm and study the $S$-adic languages generated by sequences of these substitutions. The negativity of the second Lyapunov exponent implies that almost all of these languages are balanced. By a thorough study of the combinatorics of the substitutions, we are even able to obtain a concrete generic family of balanced languages that is characterized in terms of a simple condition on the underlying sequence of substitutions. |
| title | Convergence and combinatorics of the Reverse algorithm |
| topic | Dynamical Systems 37B10, 37A25, 68R15, 11J70 |
| url | https://arxiv.org/abs/2602.14142 |