T. Eisner, B. Farkas, M. Haase, R. Nagel Operator Theoretic Aspects of Ergodic Theory July 21, 2012 Springer Contents Preface ............................................................ ix 1 What is Ergodic Theory? ....................................... 1 2 Topological Dynamical Systems .................................. 7 2.1 Some Basic Examples . .7 2.2 Basic Constructions . 11 2.3 Topological Transitivity . 17 2.4 Transitivity of Subshifts . 21 Exercises . 23 3 Minimality and Recurrence ..................................... 27 3.1 Minimality . 28 3.2 Topological Recurrence . 30 3.3 Recurrence in Extensions . 32 Exercises . 35 4 The C∗-algebra C(K) and the Koopman Operator . 37 4.1 Continuous Functions on Compact Spaces . 38 4.2 The Space C(K) as a Commutative C∗-Algebra . 43 4.3 The Koopman Operator . 45 4.4 The Theorem of Gelfand–Naimark . 49 Supplement: Proof of the Gelfand–Naimark Theorem . 50 Exercises . 56 5 Measure-Preserving Systems .................................... 59 5.1 Examples . 61 5.2 Measures on Compact Spaces . 66 5.3 Haar Measure and Rotations . 70 Exercises . 71 v vi Contents 6 Recurrence and Ergodicity ...................................... 75 6.1 The Measure Algebra and Invertible MDS . 75 6.2 Recurrence . 77 6.3 Ergodicity . 82 Supplement: Induced Transformation and Kakutani–Rokhlin Lemma . 85 Exercises . 88 7 The Banach Lattice Lp and the Koopman Operator . 91 7.1 Banach Lattices and Positive Operators . 92 7.2 The Space Lp(X) as a Banach Lattice . 97 7.3 The Koopman Operator and Ergodicity . 99 Supplement: Interplay between Lattice and Algebra Structure . 102 Exercises . 104 8 The Mean Ergodic Theorem .....................................107 8.1 Von Neumann’s Mean Ergodic Theorem . 108 8.2 The Fixed Space and Ergodicity . 111 8.3 Perron’s Theorem and the Ergodicity of Markov Shifts . 114 8.4 Mean Ergodic Operators . 116 Supplement: Operations with Mean Ergodic Operators . 120 Exercises . 123 9 Mixing Dynamical Systems ......................................125 9.1 Strong Mixing . 126 9.2 Weakly Mixing Systems . 131 9.3 Weak Mixing of All Orders . 138 Exercises . 143 10 Mean Ergodic Operators on C(K) ................................147 10.1 Invariant Measures . 147 10.2 Uniquely and Strictly Ergodic Systems . 150 10.3 Mean Ergodicity of Group Rotations . 152 10.4 Furstenberg’s Theorem on Group Extensions . 154 10.5 Application: Equidistribution . 157 Exercises . 160 11 The Pointwise Ergodic Theorem .................................163 11.1 Pointwise Ergodic Operators . 164 11.2 Banach’s Principle and Maximal Inequalities . 166 11.3 Applications . 170 Exercises . 173 Contents vii 12 Isomorphisms and Topological Models . 175 12.1 Point Isomorphisms and Factor Maps . 175 12.2 Algebra Isomorphisms of Measure Preserving Systems . 179 12.3 Topological Models . 183 12.4 The Stone Representation . 186 12.5 Mean Ergodic Subalgebras . 188 Exercises . 191 13 Markov Operators .............................................193 13.1 Examples and Basic Properties . 194 13.2 Embeddings, Factor Maps and Isomorphisms . 196 13.3 Markov Projections . 199 13.4 Factors and their Models . 203 Supplement: Operator Theory on Hilbert Space . 206 Exercises . 210 14 Compact Semigroups and Groups ................................211 14.1 Compact Semigroups . 211 14.2 Compact Groups . 217 14.3 The Character Group. 219 14.4 The Pontryagin Duality Theorem . 223 14.5 Application: Ergodic Rotations and Kronecker’s Theorem . 225 Exercises . 227 15 Topological Dynamics Revisited ..................................229 15.1 The Cech–Stoneˇ Compactification . 229 15.2 The Space of Ultrafilters . 229 15.3 Multiple Recurrence . 229 15.4 Applications to Ramsey Theory. 229 Exercises . 229 16 The Jacobs–de Leeuw–Glicksberg Decomposition . 231 16.1 Weakly Compact Operator Semigroups . 232 16.2 The Jacobs–de Leeuw–Glicksberg Decomposition . 237 16.3 The Almost Weakly Stable Part . 240 16.4 Compact Group Actions on Banach Spaces . 244 16.5 The Reversible Part Revisited . 250 Supplement: Unitary Representations of Compact Groups . 252 Exercises . 257 17 Dynamical Systems with Discrete Spectrum . 261 17.1 Operators with Discrete Spectrum . 261 17.2 Monothetic Groups . 263 17.3 Dynamical Systems with Discrete Spectrum . 265 17.4 Examples . 269 Exercises . 272 viii Contents 18 A Glimpse at Arithmetic Progressions ............................275 18.1 The Furstenberg Correspondence Principle . ..