Two-variable polynomials with dynamical Mahler measure zero

Carter, Annie; Laĺın, Matilde; Miller, Alison Beth; Mocz, Lucia

doi:10.1007/s40993-022-00322-z

Cited by 3 publications

(1 citation statement)

References 28 publications

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“…We remark that the growth of the Mahler measure for the iterates exhibited here is essentially due to the intrinsic connection of the Mahler measure to the unit circle. A more suitable version of the Mahler measure for the dynamical setting is known, see the recent papers [5] and [4], where the last one surveys many developments in the area. Another related notion is dynamical (or canonical) height, see [14] for a comprehensive exposition.…”

Section: Resultsmentioning

confidence: 99%

Mahler measure of polynomial iterates

Pritsker

2023

Can. Math. Bull.

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Granville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial 𝑓 (𝑧) = 𝑧 𝑑 + . . . ∈ C[𝑧 ], deg( 𝑓 ) ≥ 2, we show that the Mahler measure of the iterates 𝑓 𝑛 grows geometrically fast with the degree 𝑑 𝑛 , and find the exact base of that exponential growth. This base is expressed via an integral of log + | 𝑧 | with respect to the invariant measure of the Julia set for the polynomial 𝑓 . Moreover, we give sharp estimates for such an integral when the Julia set is connected.

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Section: Resultsmentioning

confidence: 99%

Mahler measure of polynomial iterates

Pritsker

2023

Can. Math. Bull.

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Dynamical Mahler Measure: A Survey and Some Recent Results

Carter,

Lalín,

Manes

et al. 2024

Association for Women in Mathematics Series

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Limits of Mahler measures in multiple variables

Brunault,

Guilloux,

Mehrabdollahei

et al. 2024

Annales De l'Institut Fourier

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We prove that certain sequences of Laurent polynomials, obtained from a fixed multivariate Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P . This generalises previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, extending work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.Résumé. -Nous prouvons que certaines suites de polynômes de Laurent, obtenues à partir d'un polynôme de Laurent P fixé par substitutions monomiales, donnent des suites de mesures de Mahler qui convergent vers la mesure de Mahler de P . Ce résultat généralisent des travaux antérieurs de Boyd et Lawton, qui considéraient des substitutions univariées. Nous obtenons aussi une borne supérieure explicite pour le terme d'erreur dans cette convergence, généralisant des travaux de Dimitrov et Habegger. Nous donnons enfin un développement asymptotique complet pour une famille particulière de polynômes bivariés, dont les mesures de Mahler avaient été étudiées indépendamment par la troisième autrice. d j=1 (z − α j ) for its complex factorisation, the product ∆ k (P ) = d j=1 (α k j − 1) is an integer for every k ∈ N. The case of the polynomial

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Two-variable polynomials with dynamical Mahler measure zero

Cited by 3 publications

References 28 publications

Mahler measure of polynomial iterates

Mahler measure of polynomial iterates

Dynamical Mahler Measure: A Survey and Some Recent Results

Limits of Mahler measures in multiple variables

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