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SYNCHRONIZATION POINTS AND ASSOCIATED DYNAMICAL INVARIANTS
"... Abstract. This paper introduces new invariants for multiparameter dynamical systems. This is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms. We call these points synchronization points. The resulting sequences of cou ..."
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Abstract. This paper introduces new invariants for multiparameter dynamical systems. This is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms. We call these points synchronization points. The resulting sequences of counts together with generating functions and growth rates are subsequently investigated for homeomorphisms of compact metric spaces, toral automorphisms and compact abelian group epimorphisms. Synchronization points are also used to generate invariant measures and the distribution properties of these are analysed for the algebraic systems considered. Furthermore, these systems reveal strong connections between the new invariants and problems of active interest in number theory, relating to heights and greatest common divisors. 1.
ON THE GREATEST PRIME FACTOR OF SOME DIVISIBILITY SEQUENCES
"... ABSTRACT. Let P (m) denote the greatest prime factor of m. For integer a> 1, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, P (an − 1),a n2− for any > 0. We study analogues results for the corresponding divisibility sequence over the function field Fq(t) and ..."
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ABSTRACT. Let P (m) denote the greatest prime factor of m. For integer a> 1, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, P (an − 1),a n2− for any > 0. We study analogues results for the corresponding divisibility sequence over the function field Fq(t) and for some divisibility sequences associated to elliptic curves over the rational field Q. In honor of M. Ram Murty on his sixtieth birthday 1.