Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces write by Lior Fishman. This book was released on 2018. Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces available in PDF, EPUB and Kindle. 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
Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces write by Lior Fishman. This book was released on 2018-08-09. Diophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces available in PDF, EPUB and Kindle. 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
Geometry and Dynamics in Gromov Hyperbolic Metric Spaces - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Geometry and Dynamics in Gromov Hyperbolic Metric Spaces write by Tushar Das. This book was released on 2017-04-14. Geometry and Dynamics in Gromov Hyperbolic Metric Spaces available in PDF, EPUB and Kindle. 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
Ergodic Theory and Negative Curvature - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Ergodic Theory and Negative Curvature write by Boris Hasselblatt. This book was released on 2017-12-15. Ergodic Theory and Negative Curvature available in PDF, EPUB and Kindle. 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.
Elements of Dynamical Systems
Elements of Dynamical Systems - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Elements of Dynamical Systems write by Anima Nagar. This book was released on 2022-11-11. Elements of Dynamical Systems available in PDF, EPUB and Kindle. This book stems from lectures that were delivered at the three-week Advanced Instructional School on Ergodic Theory and Dynamical Systems held at the Indian Institute of Technology Delhi, from 4–23 December 2017, with the support of the National Centre for Mathematics, National Board for Higher Mathematics, Department of Atomic Energy, Government of India. The book discusses various aspects of dynamical systems. Each chapter of this book specializes in one aspect of dynamical systems and thus begins at an elementary level and goes on to cover fairly advanced material. The book helps researchers be familiar with and navigate through different parts of ergodic theory and dynamical systems.
