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Author: Jason P. Bell Publisher: American Mathematical Soc. ISBN: 1470424088 Category : Mathematics Languages : en Pages : 297
Book Description
The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. It predicts the behavior of the orbit of a point x under the action of an endomorphism f of a quasiprojective complex variety X. More precisely, it claims that for any point x in X and any subvariety V of X, the set of indices n such that the n-th iterate of x under f lies in V is a finite union of arithmetic progressions. In this book the authors present all known results about the Dynamical Mordell-Lang Conjecture, focusing mainly on a p-adic approach which provides a parametrization of the orbit of a point under an endomorphism of a variety.
Author: Jason P. Bell Publisher: American Mathematical Soc. ISBN: 1470424088 Category : Mathematics Languages : en Pages : 297
Book Description
The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. It predicts the behavior of the orbit of a point x under the action of an endomorphism f of a quasiprojective complex variety X. More precisely, it claims that for any point x in X and any subvariety V of X, the set of indices n such that the n-th iterate of x under f lies in V is a finite union of arithmetic progressions. In this book the authors present all known results about the Dynamical Mordell-Lang Conjecture, focusing mainly on a p-adic approach which provides a parametrization of the orbit of a point under an endomorphism of a variety.
Author: Junyi Xie Publisher: ISBN: 9782856298695 Category : Affine algebraic groups Languages : en Pages : 110
Book Description
In this paper we prove the Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane over the algebraic numbers. More precisely, let f be an endomorphism of the affine plan over the algebraic numbers. Let x be a point in the affine plan and C be a curve. If the intersection of C and the orbits of x is infinite, then C is periodic.
Author: Christian Elsholtz Publisher: Springer ISBN: 3319553577 Category : Mathematics Languages : en Pages : 447
Book Description
This volume is dedicated to Robert F. Tichy on the occasion of his 60th birthday. Presenting 22 research and survey papers written by leading experts in their respective fields, it focuses on areas that align with Tichy’s research interests and which he significantly shaped, including Diophantine problems, asymptotic counting, uniform distribution and discrepancy of sequences (in theory and application), dynamical systems, prime numbers, and actuarial mathematics. Offering valuable insights into recent developments in these areas, the book will be of interest to researchers and graduate students engaged in number theory and its applications.
Author: Wayne Peng Publisher: ISBN: Category : Languages : en Pages : 127
Book Description
"I study three problems in my thesis, X-base Fibonacci-Wieferich primes, dynamical unlikely intersection problems, and dynamical isogeny problem. In Chapter 1, I define X-base Fibonacci-Wieferich primes, which generalize Wieferich primes, where X is a finite set of algebraic numbers. I show that there are infinitely many non-X-base Fibonacci-Wieferich primes, assuming the abc-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of the free part of the abelian group generated by all elements in X and give some heuristics that support the conjecture. In Chapter 2, I explore the unlikely intersection problems in the field of dynamical systems. I give a friendly introduction to perfectoid spaces. I then use the theory of perfectoid spaces to prove the dynamical Manin-Mumford conjecture, dynamical Tate-Voloch conjecture, and dynamical inverse Mordell-Lang conjecture on restricted lift of p-th power. In Chapter 3, I give a new definition of the isogeny to dynamical trees. The definition is an analog of the isogeny of abelian varieties. Then, I prove a theorem toward the dynamical version of the isogeny theorem. I also study the intersection of two dynamical fields. I show that if the intersection of two dynamical fields is nontrivial, then there is no isomorphic between the corresponding trees in the sense of Arboreal representation"--Page vi.
Author: Carlo Gasbarri Publisher: American Mathematical Soc. ISBN: 1470414589 Category : Mathematics Languages : en Pages : 176
Book Description
This volume contains papers from the Short Thematic Program on Rational Points, Rational Curves, and Entire Holomorphic Curves and Algebraic Varieties, held from June 3-28, 2013, at the Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada. The program was dedicated to the study of subtle interconnections between geometric and arithmetic properties of higher-dimensional algebraic varieties. The main areas of the program were, among others, proving density of rational points in Zariski or analytic topology on special varieties, understanding global geometric properties of rationally connected varieties, as well as connections between geometry and algebraic dynamics exploring new geometric techniques in Diophantine approximation. This book is co-published with the Centre de Recherches Mathématiques.
Author: Keping Huang Publisher: ISBN: Category : Languages : en Pages : 70
Book Description
"In this thesis we study two problems on arithmetic dynamics of algebraic varieties. In Chapter 2 we study the generalized greatest common divisor for orbits of rational points, under rational functions with rational coefficients. We prove an upper bound of the greatest common divisor assuming Vojta's Conjecture and under some very mild conditions. In Chapter 3 we study the Dynamical Mordell-Lang problem for varieties with finitely many periodic points. We show a gap principle for the iterates when the initial point returns to a certain subvariety"--Page vi.
Author: Ruy Exel Publisher: American Mathematical Soc. ISBN: 1470437856 Category : Mathematics Languages : en Pages : 330
Book Description
Partial dynamical systems, originally developed as a tool to study algebras of operators in Hilbert spaces, has recently become an important branch of algebra. Its most powerful results allow for understanding structural properties of algebras, both in the purely algebraic and in the C*-contexts, in terms of the dynamical properties of certain systems which are often hiding behind algebraic structures. The first indication that the study of an algebra using partial dynamical systems may be helpful is the presence of a grading. While the usual theory of graded algebras often requires gradings to be saturated, the theory of partial dynamical systems is especially well suited to treat nonsaturated graded algebras which are in fact the source of the notion of “partiality”. One of the main results of the book states that every graded algebra satisfying suitable conditions may be reconstructed from a partial dynamical system via a process called the partial crossed product. Running in parallel with partial dynamical systems, partial representations of groups are also presented and studied in depth. In addition to presenting main theoretical results, several specific examples are analyzed, including Wiener–Hopf algebras and graph C*-algebras.
Author: Tushar Das Publisher: American Mathematical Soc. ISBN: 1470434652 Category : Mathematics Languages : en Pages : 321
Book Description
This book presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit sets and on behavior not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. The book contains both introductory material to help beginners as well as new research results, and closes with a list of attractive unsolved problems.
Author: Jason P. Bell
Author: Jason P. Bell
Author: Junyi Xie
Author: Christian Elsholtz
Author: Wayne Peng
Author: Carlo Gasbarri
Author: Keping Huang
Author: Ruy Exel
Author: Enrico Bombieri
Author: Tushar Das
Author: Joseph H. Silverman