Elements of dynamical systems : lecture notes from NCM School / / Anima Nagar, Riddhi Shah, Shrihari Sridharan, editors

• Record Nr.: UNISA996499870603316
• Pubbl/distr/stampa: Singapore : , : Springer, , [2022]; ©2022
• ISBN: 981-16-7962-2
• Descrizione fisica: 1 online resource (190 pages)
• Collana: Texts and readings in mathematics ; ; Volume 79
• Disciplina: 515.352
• Soggetti: Differentiable dynamical systems; Sistemes dinàmics diferenciables; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Intro -- Preface -- Contents -- Real Dynamics -- 1 Introduction and Preliminaries from Topological Dynamics -- 1.1 Introduction -- 1.2 Preliminaries -- 2 Attracting Fixed Point -- 2.1 Banach's Contraction Mapping Theorem -- 2.2 Various Versions of Attraction -- 2.3 Examples -- 3 Topological Transitivity -- 3.1 Five Views of Topological Transitivity -- 3.2 Five Proofs of Topological Transitivity of the Tent Map -- 4 Three Ingredients of Chaos -- 4.1 T, DP and SDIC -- 4.2 T, DP &amp -- NOT SDIC -- 4.3 T, NOT DP &amp -- SDIC -- 4.4 T, NOT DP &amp -- NOT SDIC -- 4.5 NOT T, DP &amp -- SDIC -- 4.6 NOT T, DP &amp -- NOT SDIC -- 4.7 NOT T, NOT DP &amp -- SDIC -- 4.8 NOT T, NOT DP &amp -- NOT SDIC -- 5 Chaos For Interval Maps -- 5.1 For Interval Maps Transitivity Implies Chaos -- 5.2 T &amp -- DP -3mu SDIC -- 6 Some Consequences of Intermediate Value Theorem in Dynamics -- 6.1 Immediate Applications -- 6.2 Sarkovskii's Theorem: A Statement -- 6.3 Digraphs of Cycles -- 6.4 Use of digraphs in the proof of Sarkovskii's theorem -- 6.5 Doubling periods -- 6.6 Use of doubling periods in the converse of Sarkovskii's Theorem -- 7 Proofs of Some Theorems Used -- 8 Notes &amp -- Exercises -- References -- Topological Dynamics -- 1 G-Spaces -- 2 Minimal Systems -- 3 Multiple Recurrence and Van Der Waerden's Theorem -- 4 Enveloping Semigroups -- 5 Proximal and Distal -- 6 Topological Transitivity and ; Mixing -- 7 Summary -- References -- Basic Ergodic Theory -- 1 Introduction -- 2 Measure Theoretic Preliminaries -- 2.1 Measurable Functions and Transformations -- 2.2 Hausdorff Measures -- 3 Recurrence and Ergodic Theorems -- 3.1 Recurrence -- 3.2 Birkhoff Ergodic Theorem and the Notion of Ergodicity -- 4 Geodesic Flows on Closed Surfaces -- 4.1 Isometries and Geodesics of mathbbH2 -- 4.2 Hopf's Proof of Ergodicity -- References -- Symbolic Dynamics -- 1 Introduction.; 2 Basic Concepts -- 3 Entropy -- 4 Computations of Entropy -- 4.1 Entropy of Shifts -- 4.2 Entropy of Translations -- 5 Tilings -- 6 3-Dot Shifts -- References -- Complex Dynamics -- 1 Introduction -- 2 Some Preliminaries from Complex Analysis and Motivation -- 3 Normal Families and Dichotomy of mathbbP1 -- 4 Rational Maps with Empty Fatou Set -- 5 Some Properties of the Julia Set -- 6 Local Analysis Near a Fixed Point -- 7 Brolin's Theorem -- 8 What Happens in Higher Dimensions? -- References -- Topics in Homogeneous Dynamics and Number Theory -- 1 Introduction -- 1.1 Homogeneous Dynamics -- 1.2 Diophantine Approximation -- 2 On the Distribution of Approximates -- 2.1 The EST Distribution -- 2.2 Spiraling of Approximates -- 3 Diophantine Approximation in Number Fields -- 4 A Projective Duffin Schaeffer Theorem -- 5 The Hyperbolic Picture -- References -- On Certain Unusual Large Subsets Arising as Winning Sets of Some Games -- 1 Introduction -- 2 Schmidt's ( α,β)-Game -- 3 Largeness of Winning Sets -- 4 Large Sets Involved in Diophantine Approximation -- 5 Large Sets in Geometry and Dynamics -- 5.1 Winning Sets in mathbbRd -- 5.2 Toral Automorphisms -- 5.3 Hyperbolic Geometry -- 6 Further Generalisations and Applications -- 6.1 Strong and Absolute Winning Sets -- 6.2 Winning Sets on Lie Groups -- 6.3 Badly Approximable Numbers in Closed Subsets -- References.