Rich with examples and applications, this textbook provides a coherent and self-contained introduction to ergodic theory, suitable for a variety of one- or two-semester courses. The authors' clear and fluent exposition helps the reader to grasp quickly the most important ideas of the theory, and their use of concrete examples illustrates these ideas and puts the results into perspective. The book requires few prerequisites, with background material supplied in the appendix. The first four chapters cover elementary material suitable for undergraduate students invariance, recurrence and ergodicity as well as some of the main examples. The authors then gradually build up to more sophisticated topics, including correlations, equivalent systems, entropy, the variational principle and thermodynamical formalism. The 400 exercises increase in difficulty through the text and test the reader's understanding of the whole theory. Hints and solutions are provided at the end of the book.
Rich with examples, applications and over 400 exercises, this textbook provides a coherent and self-contained introduction to ergodic theory, suitable for a variety of one- or two-semester courses. It requires few prerequisites, beginning with elementary material suitable for undergraduate students and gradually building up to more sophisticated topics.
Recenzijos
'The book provides the student or researcher with an excellent reference and/or base from which to move into current research in ergodic theory. This book would make an excellent text for a graduate course on ergodic theory.' Douglas P. Dokken, Mathematical Reviews ' Viana and Oliveira have written yet another excellent textbook! It may be fruitfully used to guide a graduate course in dynamical systems, or a topics seminar at either advanced undergraduate or early graduate levels. The book is designed so that the instructor may cull a variety of courses from its contents. The authors deserve special kudos for their collection of over 400 exercises, many with hints and solutions at the end of the book. As a further bonus, if only to pique the reader's interest, a number of recent research results and open problems are sprinkled throughout the book.' Tushar Das, MAA Reivews
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Self-contained introductory textbook suitable for a variety of one- or two-semester courses. Rich with examples, applications and exercises.
| Preface |
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ix | |
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1 | (34) |
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2 | (2) |
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1.2 Poincare recurrence theorem |
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4 | (6) |
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10 | (13) |
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23 | (6) |
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1.5 Multiple recurrence theorems |
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29 | (6) |
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2 Existence of invariant measures |
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35 | (29) |
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36 | (9) |
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2.2 Proof of the existence theorem |
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45 | (4) |
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2.3 Comments in functional analysis |
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49 | (4) |
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2.4 Skew-products and natural extensions |
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53 | (5) |
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2.5 Arithmetic progressions |
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58 | (6) |
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64 | (29) |
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3.1 Ergodic theorem of von Neumann |
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65 | (5) |
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3.2 Birkhoff ergodic theorem |
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70 | (8) |
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3.3 Subadditive ergodic theorem |
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78 | (9) |
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3.4 Discrete time and continuous time |
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87 | (6) |
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93 | (49) |
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94 | (6) |
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100 | (16) |
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4.3 Properties of ergodic measures |
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116 | (4) |
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4.4 Comments in conservative dynamics |
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120 | (22) |
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142 | (15) |
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5.1 Ergodic decomposition theorem |
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142 | (8) |
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5.2 Rokhlin disintegration theorem |
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150 | (7) |
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157 | (24) |
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157 | (2) |
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159 | (3) |
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162 | (11) |
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173 | (8) |
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181 | (32) |
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182 | (8) |
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190 | (10) |
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200 | (8) |
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7.4 Decay of correlations |
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208 | (5) |
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213 | (29) |
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214 | (2) |
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216 | (6) |
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222 | (3) |
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225 | (8) |
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8.5 Lebesgue spaces and ergodic isomorphism |
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233 | (9) |
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242 | (59) |
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9.1 Definition of entropy |
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243 | (11) |
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9.2 Theorem of Kolmogorov-Sinai |
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254 | (8) |
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262 | (6) |
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268 | (6) |
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9.5 Entropy and equivalence |
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274 | (12) |
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9.6 Entropy and ergodic decomposition |
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286 | (8) |
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9.7 Jacobians and the Rokhlin formula |
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294 | (7) |
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301 | (51) |
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302 | (11) |
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313 | (12) |
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325 | (13) |
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10.4 Variational principle |
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338 | (7) |
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345 | (7) |
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352 | (28) |
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11.1 Expanding maps on manifolds |
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353 | (9) |
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11.2 Dynamics of expanding maps |
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362 | (12) |
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11.3 Entropy and periodic points |
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374 | (6) |
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12 Thermodynamic formalism |
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380 | (50) |
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381 | (18) |
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399 | (3) |
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12.3 Decay of correlations |
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402 | (15) |
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12.4 Dimension of conformal repellers |
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417 | (13) |
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Appendix A Topics in measure theory, topology and analysis |
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430 | (52) |
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430 | (13) |
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A.2 Integration in measure spaces |
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443 | (10) |
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A.3 Measures in metric spaces |
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453 | (7) |
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A.4 Differentiable manifolds |
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460 | (9) |
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469 | (4) |
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473 | (4) |
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477 | (5) |
| Hints or solutions for selected exercises |
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482 | (22) |
| References |
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504 | (7) |
| Index of notation |
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511 | (4) |
| Index |
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515 | |
Marcelo Viana is Professor of Mathematics at the Instituto Nacional de Matemįtica Pura e Aplicada (IMPA), Rio de Janeiro, and a leading research expert in ergodic theory and dynamical systems. He has served in several academic organizations, such as the International Mathematical Union (Vice President, 20112014), the Brazilian Mathematical Society (President, 20132015), the Latin American Mathematical Union (Scientific Coordinator, 20012008) and the newly founded Mathematical Council of the Americas. He is also a member of the academies of science of Brazil, Portugal, Chile and the Developing World, and he has received several academic distinctions, including the Grand Croix of Scientific Merit, granted by the President of Brazil, in 2000, and the Ramanujan Prize of ICTP and IMU in 2005. He was an Invited Speaker at the International Congress of Mathematicians in Zurich (1994), a Plenary Speaker at the International Congress of Mathematical Physics (1994), and a Plenary Speaker at the ICM in Berlin (1998). To date, he has supervised thirty-two doctoral theses. Currently, he leads the organization of the ICM 2018 in Rio de Janeiro, and he is also involved in initiatives to improve mathematical education in his country. Krerley Oliveira is Associate Professor at the Universidade Federal de Alagoas (UFAL), Brazil, where he founded the graduate program in mathematics and the State of Alagoas Math Olympiad program, where he promotes young talent for mathematics. He was a medalist at the Brazilian Mathematical Olympiad (1996) and twice at the Iberoamerican Mathematical Olympiad for university students (1999 and 2000). He was elected an Affiliate Member of the Brazilian Academy of Sciences (20072012) and his research is focused on dynamical systems and ergodic theory.
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