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Documents 11-XX 5 résultats

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Complexity theory in arithmetic dynamical systems - Lecture 3 - Xie, Junyi (Auteur de la Conférence) | CIRM H

Virtualconference

It is a fundamental problem to measure the complexity of a dynamical system. In this lecture, we discuss this problem for arithmetic dynamics in terms of topology, algebra and arithmetic. In particular, the notion of dynamical degrees, which can be viewed as an algebraic analogy of “entropy”, plays a key role. We will see how it applies to study the orbits, periodic points and action of cohomologies.

14-XX ; 37-XX ; 11-XX

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The preceding Etats de la recherche on this topic happened in Rennes, 2006, and the organizers of the present edition asked me to make the bridge between these two sessions. My 2006 talks were devoted to the theory of heights and equidistribution theorems for algebraic dynamical systems. I will start from there by presenting the framework allowed by Arakelov geometry, and explaining the recent manuscript of X. Yuan and S.-W. Zhang who provide a birational perspective to these concepts. The theory is a bit complex and technical but I will try to emphasize the parallel between those ideas and the ones that lie at the ground of pluripotential theory in complex analysis, or in the theory of b-divisors in algebraic geometry.[-]
The preceding Etats de la recherche on this topic happened in Rennes, 2006, and the organizers of the present edition asked me to make the bridge between these two sessions. My 2006 talks were devoted to the theory of heights and equidistribution theorems for algebraic dynamical systems. I will start from there by presenting the framework allowed by Arakelov geometry, and explaining the recent manuscript of X. Yuan and S.-W. Zhang who provide a ...[+]

37-XX ; 14-XX ; 11-XX

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Complexity theory in arithmetic dynamical systems - Lecture 1 - Xie, Junyi (Auteur de la Conférence) | CIRM H

Virtualconference

It is a fundamental problem to measure the complexity of a dynamical system. In this lecture, we discuss this problem for arithmetic dynamics in terms of topology, algebra and arithmetic. In particular, the notion of dynamical degrees, which can be viewed as an algebraic analogy of “entropy”, plays a key role. We will see how it applies to study the orbits, periodic points and action of cohomologies.

14-XX ; 37-XX ; 11-XX

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Complexity theory in arithmetic dynamical systems - Lecture 2 - Xie, Junyi (Auteur de la Conférence) | CIRM H

Virtualconference

It is a fundamental problem to measure the complexity of a dynamical system. In this lecture, we discuss this problem for arithmetic dynamics in terms of topology, algebra and arithmetic. In particular, the notion of dynamical degrees, which can be viewed as an algebraic analogy of “entropy”, plays a key role. We will see how it applies to study the orbits, periodic points and action of cohomologies.

14-XX ; 37-XX ; 11-XX

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Hilbert cubes in arithmetic sets - Elsholtz, Christian (Auteur de la Conférence) | CIRM H

Single angle

Let $S$ be a multiplicatively defined set. Ostmann conjectured, that the set of primes cannot be (nontrivially) written as a sumset $P\sim A+B$ (even in an asymptotic sense, when finitely many deviations are allowed). The author had previously proved that there is no such ternary sumset $P\sim A+B+C$ (with $ \left |A \right |,\left |B \right |,\left |C \right |\geq 2$). More generally, in recent work we showed (with A. Harper) for certain multiplicatively defined sets $S$, namely those which can be treated by sieves, or those with some equidistribution condition of Bombieri-Vinogradov type, that again there is no (nontrivial) ternary decomposition $P\sim A+B+C$. As this covers the case of smooth numbers, this settles a conjecture of A.Sárközy.
Joint work with Adam J. Harper.[-]
Let $S$ be a multiplicatively defined set. Ostmann conjectured, that the set of primes cannot be (nontrivially) written as a sumset $P\sim A+B$ (even in an asymptotic sense, when finitely many deviations are allowed). The author had previously proved that there is no such ternary sumset $P\sim A+B+C$ (with $ \left |A \right |,\left |B \right |,\left |C \right |\geq 2$). More generally, in recent work we showed (with A. Harper) for certain ...[+]

11-XX ; 05-XX

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