Discrete dynamical systems from real valued mutation. (English) Zbl 07873973
Summary: We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated with integrability, namely preservation of a symplectic form, and in the tropical case, the existence of a conserved quantity. We show in certain cases that the orbits are unbounded. The tropical dynamics are related to matrix mutation, from the theory of cluster algebras. We are able to show that in certain special cases, the tropical map is periodic. We also explain how our dynamics imply the asymptotic sign-coherence observed by Gekhtman and Nakanishi in the two-dimensional situation.
MSC:
| 39A36 | Integrable difference and lattice equations; integrability tests |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 37J37 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures |
| 13F60 | Cluster algebras |
References:
| [1] | Beineke, A., Brüstle, T., Hille, L. (2011). Cluster-cyclic quivers with three vertices and the Markov equation. Algebras Represent. Theory14(1): 97-112. · Zbl 1239.16017 |
| [2] | Di Francesco, P. (2011). Discrete integrable systems, positivity, and continued fraction rearrangements. Lett. Math. Phys. 96(1-3): 299-324. doi: · Zbl 1277.13019 |
| [3] | Felikson, A., Lampe, P. (2019). Exchange graphs for mutation-finite non-integer quivers of rank 3. arXiv:1904.03928. |
| [4] | Felikson, A., Tumarkin, P. (2019). Mutation-finite quivers with real weights. arXiv:1902.01997. |
| [5] | Felikson, A., Tumarkin, P. (2019). Geometry of Mutation Classes of rank 3 Quivers. Arnold Math. J.5(1): 37-55. doi: · Zbl 1442.13066 |
| [6] | Fomin, S., Zelevinsky, A. (2002). Cluster algebras. I. Foundations. J. Amer. Math. Soc.15(2): 497-529. doi: · Zbl 1021.16017 |
| [7] | Fomin, S., Zelevinsky, A. (2003). Cluster algebras. II. Finite type classification. Invent. Math.154(1): 63-121. doi: · Zbl 1054.17024 |
| [8] | Fomin, S., Zelevinsky, A. (2007). Cluster algebras. IV. Coefficients. Compos. Math. 143(1): 112-164. doi: · Zbl 1127.16023 |
| [9] | Gekhtman, M., Nakanishi, T.Asymptotic sign coherence conjecture. Exp. Math.1-9. Accepted, available online. doi: · Zbl 1492.13031 |
| [10] | Gekhtman, M., Nakanishi, T., Rupel, D. (2017). Hamiltonian and Lagrangian formalisms of mutations in cluster algebras and application to dilogarithm identities. J. Integrable Syst., 2(1): xyx005. doi.org/ doi: · Zbl 1431.37057 |
| [11] | Glick, M., Rupel, D. (2017). Introduction to cluster algebras. In: Levi, D., Rebelo, R., Winternitz, P., eds. Symmetries and Integrability of Difference Equations, CRM Series in Mathematical Physics. Cham: Springer, pp. 325-357. · Zbl 1373.15040 |
| [12] | Hoffmann, T., Kellendonk, J., Kutz, N., Reshetikhin, N. (2000). Factorization dynamics and Coxeter-Toda lattices. Commun. Math. Phys. 212(2): 297-321. doi: · Zbl 0989.37074 |
| [13] | Hone, A. N. W., Lampe, P., Kouloukas, T. E. (2019). Cluster algebras and discrete integrability. In: Euler, N., Nicci, M. C. eds. Nonlinear Systems and Their Remarkable Mathematical Structures. Vol. 2. Boca Raton, FL: Chapman and Hall/CRC. doi.org/ doi: |
| [14] | Kuniba, A., Nakanishi, T., Suzuki, J. (2011). T-systems and Y-systems in integrable systems. J. Phys. A44(10): 103001. doi: · Zbl 1222.82041 |
| [15] | Lampe, P. (2018). On the approximate periodicity of sequences attached to non-crystallographic root systems. Exp. Math. 27(3): 265-271. doi: · Zbl 1423.13126 |
| [16] | Nakanishi, T. (2021). Synchronicity phenomenon in cluster patterns. J. London Math. Soc. 103(3): 1120-1152, 2021. doi.org/ doi:. · Zbl 1473.13022 |
| [17] | Reading, N. (2014). Universal geometric cluster algebras. Math. Z.277(1-2): 499-547. doi: · Zbl 1328.13034 |
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