Cluster algebras and discrete integrability

Hone, Andrew N. W.; Lampe, Philipp; Kouloukas, Theodoros E.

doi:10.1201/9780429263743-10

Cited by 6 publications

(7 citation statements)

References 57 publications

(159 reference statements)

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“…In general, period 2 B matrices lead to a coupling between two different relations, depending on the parity of n, but in this case µ 1 (B) = −B, so the exponents on the right-hand side are the same in both mutations, and there is just a single recurrence. It is shown in [36] that E = log 1 2 (1 + √ 5) ≈ 0.4812118246 for (1.13), and an analogous calculation with the sequence (1.15) shows that E H (1) takes the same value [30]. As is well known [8], certain combinations of involutions of the Markoff surface generate solutions to Pell equations, corresponding to linear dynamics and subexponential degree growth, and it turns out that (up to symmetry) the composition of a mutation together with a permutation in ϕ corresponds to the maximum possible growth rate that can be produced by mutations alone, so E = E(A) also.…”

Section: Recurrence For Markoff Numbersmentioning

confidence: 99%

Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property

Hone¹

2021

Preprint

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We consider the growth rate of the Mahler measure in discrete dynamical systems with the Laurent property, and in cluster algebras, and compare this with other measures of growth. In particular, we formulate the conjecture that the growth rate of the logarithmic Mahler measure coincides with the algebraic entropy, which is defined in terms of degree growth. Evidence for this conjecture is provided by exact and numerical calculations of the Mahler measure for a family of Laurent polynomials generated by rank 2 cluster algebras, for a recurrence of third order related to the Markoff numbers, and for the Somos-4 recurrence. Also, for the sequence of Laurent polynomials associated with the Kronecker quiver (the cluster algebra of affine type Ã1 ) we prove a precise formula for the leading order asymptotics of the logarithmic Mahler measure, which grows linearly.

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Section: Recurrence For Markoff Numbersmentioning

confidence: 99%

Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property

Hone¹

2021

Preprint

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“…Indeed, in [1], Bellon indicated that the vanishing of entropy should be a necessary condition for the integrability in the Liouville-Arnold sense. For more details, see [22] and references loc. cit.…”

Section: Cluster Transformations and Their Algebraic Entropymentioning

confidence: 99%

“…The algebraic entropy of the cluster Aand X -transformations induced by a mutation loop has been studied by several authors [10,21,22]. In [10,22], the authors computed the algebraic entropies of mutation loops of length one, which have been classified by Fordy-Marsh [13]. They determined the mutation loops with vanishing entropy among the Fordy-Marsh construction.…”

Section: Cluster Transformations and Their Algebraic Entropymentioning

confidence: 99%

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Algebraic entropy of sign-stable mutation loops

Ishibashi

Kano

2021

Geom Dedicata

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In the theory of cluster algebras, a mutation loop induces discrete dynamical systems via its actions on the cluster Aand X -varieties. In this paper, we introduce a property of mutation loops, called the sign stability, with a focus on the asymptotic behavior of the iteration of the tropical X -transformation. The sign stability can be thought of as a cluster algebraic analogue of the pseudo-Anosov property of a mapping class on a surface. 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 Aand X -transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This gives a cluster algebraic analogue of the classical theorem which relates the topological entropy of a pseudo-Anosov mapping class with its stretch factor.

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“…Our definition also works for the ordinary difference equations which correspond to finite quivers. Their Poisson structure were studied by Hone and collaborators [9,12,13].…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian Structures for Integrable Nonabelian Difference Equations

Casati

Wang²

2022

Commun. Math. Phys.

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In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson algebras and vertex algebras) and geometric (in terms of nonabelian Poisson bivectors) definitions. We introduce multiplicative double Poisson vertex algebras (PVAs) as the suitable noncommutative counterpart to multiplicative PVAs, used to describe Hamiltonian differential-difference equations in the commutative setting, and prove that these algebras are in one-to-one correspondence with the Poisson structures defined by difference operators, providing a sufficient condition for the fulfilment of the Jacobi identity. Moreover, we define nonabelian polyvector fields and their Schouten brackets, for both finitely generated noncommutative algebras and infinitely generated difference ones: this allows us to provide a unified characterisation of Poisson bivectors and double quasi-Poisson algebra structures. Finally, as an application we obtain some results towards the classification of local scalar Hamiltonian difference structures and construct the Hamiltonian structures for the nonabelian Kaup, Ablowitz-Ladik and Chen-Lee-Liu integrable lattices.

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Cluster algebras and discrete integrability

Cited by 6 publications

References 57 publications

Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property

Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property

Algebraic entropy of sign-stable mutation loops

Hamiltonian Structures for Integrable Nonabelian Difference Equations

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