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dc.contributor.authorArnold, Vladimir
dc.date.accessioned2011-05-06T10:04:11Z
dc.date.available2011-05-06T10:04:11Z
dc.date.issued2005
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/6202
dc.language.isoenen
dc.subjectGeometrical progressionen
dc.subjectarithmetical dynamicsen
dc.subjectergodic propertiesen
dc.subject.ddc516en
dc.titleErgodic and arithmetical properties of geometrical progression's dynamics and of its orbitsen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenThe multiplication by a constant (say, by 2) acts on the set Z/nZ of residues (mod n) as a dynamical system, whose cycles relatively prime to n all have a common period T(n) and whose orbits consist each of elements, forming a geometrical progression or residues. The paper provides many new facts on the arithmetical properties of these periods and orbits (generalizing the Fermat's small theorem, extended by Euler to the case where n is not a prime number). The chaoticity of the orbit is measured by some randomness parameter, comparing the distances distribution of neighbouring points of the orbit with a similar distribution for T randomly chosen residues (which is binominal). The calculations show some kind of repulsion of neighbours, avoiding to be close to other members of the same orbit. A similar repulsion is also observed for the prime numbers, providing their distributions nonrandomness, and for the arithmetical progressions of the residues, whose nonrandomness degree is similar to that of the primes. The paper contains also many conjectures, including that of the infinity of the pairs of prime numbers of the form (q, 2q+1), like (3,7),(11,23) ,(23,47)on one side and that on the structure of some ideals in the multiplicative semigroup of odd integers – on the other.en
dc.relation.isversionofjnlnameMoscow Mathematical Journal
dc.relation.isversionofjnlvol5en
dc.relation.isversionofjnlissue1en
dc.relation.isversionofjnldate2005
dc.relation.isversionofjnlpages5-22en
dc.description.sponsorshipprivateouien
dc.relation.isversionofjnlpublisherSteklov Mathematical Institute RASen
dc.subject.ddclabelGéométrieen


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