Geodesic
Mahler measure of polynomial iterates
Canadian mathematical bulletin, Tome 66 (2023) no. 3, pp. 881-885

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DOI

Granville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial $f(z)=z^d+\cdots \in {\mathbb C}[z],\ \deg (f)\ge 2,$ 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.
DOI : 10.4153/S0008439523000048
Mots-clés : Mahler measure, polynomial dynamics, Julia set, invariant measure 
Pritsker, Igor. Mahler measure of polynomial iterates. Canadian mathematical bulletin, Tome 66 (2023) no. 3, pp. 881-885. doi: 10.4153/S0008439523000048
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     title = {Mahler measure of polynomial iterates},
     journal = {Canadian mathematical bulletin},
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     year = {2023},
     volume = {66},
     number = {3},
     doi = {10.4153/S0008439523000048},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439523000048/}
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