Tuesday Seminar on Topology

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Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.)
Organizer(s) KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya

2020/01/14

18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tsukasa Ishibashi (The University of Tokyo)
Algebraic entropy of sign-stable mutation loops (JAPANESE)
[ Abstract ]
Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics, unifying several combinatorial operations as well as their positivity notions. A mutation loop induces several dynamical systems via cluster transformations, and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.
We introduce a new property of mutation loops called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.