Diophantine Approximation And The Geometry Of Limit Sets In Gromov Hyperbolic Metric Spaces
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Diophantine Approximation And The Geometry Of Limit Sets In Gromov Hyperbolic Metric Spaces
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Author : Lior Fishman
language : en
Publisher:
Release Date : 2018
Diophantine Approximation And The Geometry Of Limit Sets In Gromov Hyperbolic Metric Spaces written by Lior Fishman and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018 with Diophantine approximation categories.
In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic '76 paper to more recent results of Hersonsky and Paulin ('02, '04, '07). Concrete examples of situations we consider which have not been considered before include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which we are aware, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones ('97) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson-Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.
Diophantine Approximation And The Geometry Of Limit Sets In Gromov Hyperbolic Metric Spaces
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Author : Lior Fishman
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-08-09
Diophantine Approximation And The Geometry Of Limit Sets In Gromov Hyperbolic Metric Spaces written by Lior Fishman and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-08-09 with Mathematics categories.
In this paper, the authors provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes a long line of results by many authors, from Patterson's classic 1976 paper to more recent results of Hersonsky and Paulin (2002, 2004, 2007). The authors consider concrete examples of situations which have not been considered before. These include geometrically infinite Kleinian groups, geometrically finite Kleinian groups where the approximating point is not a fixed point of any element of the group, and groups acting on infinite-dimensional hyperbolic space. Moreover, in addition to providing much greater generality than any prior work of which the authors are aware, the results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of Bishop and Jones (1997) to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero Patterson–Sullivan measure unless the group is quasiconvex-cocompact. The latter is an application of a Diophantine theorem.
Geometry And Dynamics In Gromov Hyperbolic Metric Spaces
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Author : Tushar Das
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-04-14
Geometry And Dynamics In Gromov Hyperbolic Metric Spaces written by Tushar Das and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-04-14 with Mathematics categories.
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.
Ergodic Theory And Negative Curvature
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Author : Boris Hasselblatt
language : en
Publisher: Springer
Release Date : 2017-12-15
Ergodic Theory And Negative Curvature written by Boris Hasselblatt and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-12-15 with Mathematics categories.
Focussing on the mathematics related to the recent proof of ergodicity of the (Weil–Petersson) geodesic flow on a nonpositively curved space whose points are negatively curved metrics on surfaces, this book provides a broad introduction to an important current area of research. It offers original textbook-level material suitable for introductory or advanced courses as well as deep insights into the state of the art of the field, making it useful as a reference and for self-study. The first chapters introduce hyperbolic dynamics, ergodic theory and geodesic and horocycle flows, and include an English translation of Hadamard's original proof of the Stable-Manifold Theorem. An outline of the strategy, motivation and context behind the ergodicity proof is followed by a careful exposition of it (using the Hopf argument) and of the pertinent context of Teichmüller theory. Finally, some complementary lectures describe the deep connections between geodesic flows in negative curvature and Diophantine approximation.
Dynamics And Analytic Number Theory
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Author : Dzmitry Badziahin
language : en
Publisher: Cambridge University Press
Release Date : 2016-11-10
Dynamics And Analytic Number Theory written by Dzmitry Badziahin and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-11-10 with Mathematics categories.
Presents current research in various topics, including homogeneous dynamics, Diophantine approximation and combinatorics.
On The Geometric Side Of The Arthur Trace Formula For The Symplectic Group Of Rank 2
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Author : Werner Hoffmann
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-10-03
On The Geometric Side Of The Arthur Trace Formula For The Symplectic Group Of Rank 2 written by Werner Hoffmann and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-10-03 with Mathematics categories.
The authors study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group and the split symplectic group of rank over any algebraic number field. In particular, they express the global coefficients of unipotent orbital integrals in terms of Dedekind zeta functions, Hecke -functions, and the Shintani zeta function for the space of binary quadratic forms.
Moufang Sets And Structurable Division Algebras
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Author : Lien Boelaert
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-06-10
Moufang Sets And Structurable Division Algebras written by Lien Boelaert and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-06-10 with Mathematics categories.
A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. The authors extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, they show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field k of characteristic different from 2 and 3 arises from a structurable division algebra. The authors also obtain explicit formulas for the root groups, the τ-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups.
Geometric Pressure For Multimodal Maps Of The Interval
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Author : Feliks Przytycki
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-06-10
Geometric Pressure For Multimodal Maps Of The Interval written by Feliks Przytycki and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-06-10 with Mathematics categories.
This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. The authors work in a setting of generalized multimodal maps, that is, smooth maps f of a finite union of compact intervals Iˆ in R into R with non-flat critical points, such that on its maximal forward invariant set K the map f is topologically transitive and has positive topological entropy. They prove that several notions of non-uniform hyperbolicity of f|K are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in K hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). They prove that several definitions of geometric pressure P(t), that is pressure for the map f|K and the potential −tlog|f′|, give the same value (including pressure on periodic orbits, “tree” pressure, variational pressures and conformal pressure). Finally they prove that, provided all periodic orbits in K are hyperbolic repelling, the function P(t) is real analytic for t between the “condensation” and “freezing” parameters and that for each such t there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.
On Mesoscopic Equilibrium For Linear Statistics In Dyson S Brownian Motion
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Author : Maurice Duits
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-10-03
On Mesoscopic Equilibrium For Linear Statistics In Dyson S Brownian Motion written by Maurice Duits and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-10-03 with Mathematics categories.
In this paper the authors study mesoscopic fluctuations for Dyson's Brownian motion with β=2 . Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from a generic configuration of initial points, the time that is needed for the GUE statistics to become dominant depends on the scale we look at: The microscopic correlations arrive at the equilibrium regime sooner than the macrosopic correlations. The authors investigate the transition on the intermediate, i.e. mesoscopic, scales. The time scales that they consider are such that the system is already in microscopic equilibrium (sine-universality for the local correlations), but have not yet reached equilibrium at the macrosopic scale. The authors describe the transition to equilibrium on all mesoscopic scales by means of Central Limit Theorems for linear statistics with sufficiently smooth test functions. They consider two situations: deterministic initial points and randomly chosen initial points. In the random situation, they obtain a transition from the classical Central Limit Theorem for independent random variables to the one for the GUE.
Bellman Function For Extremal Problems In Bmo Ii Evolution
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Author : Paata Ivanisvili
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-10-03
Bellman Function For Extremal Problems In Bmo Ii Evolution written by Paata Ivanisvili and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-10-03 with Mathematics categories.
In a previous study, the authors built the Bellman function for integral functionals on the space. The present paper provides a development of the subject. They abandon the majority of unwanted restrictions on the function that generates the functional. It is the new evolutional approach that allows the authors to treat the problem in its natural setting. What is more, these new considerations lighten dynamical aspects of the Bellman function, in particular, the evolution of its picture.