dc.contributor.author Breuillard, Emmanuel dc.contributor.author Varjú, PP dc.date.accessioned 2019-01-11T00:31:21Z dc.date.available 2019-01-11T00:31:21Z dc.date.issued 2020-03 dc.identifier.issn 0021-7670 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/287816 dc.description.abstract The exponential growth rate of non polynomially growing subgroups of $GL_d$ is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture from number theory. In this note, we show that it is equivalent to the Lehmer conjecture. This is done by establishing a lower bound for the entropy of the random walk on the semigroup generated by the maps $x\mapsto \lambda\cdot x\pm 1$, where $\lambda$ is an algebraic number. We give a bound in terms of the Mahler measure of $\lambda$. We also derive a bound on the dimension of Bernoulli convolutions. dc.description.sponsorship Simons Foundation Royal Society ERC dc.publisher Springer Science and Business Media LLC dc.title Entropy of Bernoulli convolutions and uniform exponential growth for linear groups dc.type Article prism.endingPage 481 prism.issueIdentifier 2 prism.publicationDate 2020 prism.publicationName Journal d'Analyse Mathematique prism.startingPage 443 prism.volume 140 dc.identifier.doi 10.17863/CAM.35131 dcterms.dateAccepted 2018-06-10 rioxxterms.versionofrecord 10.1007/s11854-020-0100-0 rioxxterms.version AM rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved rioxxterms.licenseref.startdate 2020-03-01 dc.identifier.eissn 1565-8538 rioxxterms.type Journal Article/Review pubs.funder-project-id Royal Society (UF140146) pubs.funder-project-id Simons Foundation (LETTER DATED 10-NOV-09) cam.issuedOnline 2020-04-20 cam.orpheus.success Tue May 26 08:17:26 BST 2020 - Embargo updated cam.orpheus.counter 9 rioxxterms.freetoread.startdate 2021-03-01
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