| Mathematical symbols |
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ix | |
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1 Number-theoretical dynamical systems |
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1 | (63) |
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1.1 Continued fractions and Diophantine approximation |
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1 | (11) |
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1.1.1 Continued fractions |
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1 | (6) |
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1.1.2 Elementary Diophantine approximation: Hurwitz's Theorem and badly approximable numbers |
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7 | (5) |
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1.2 Topological Dynamical Systems |
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12 | (18) |
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15 | (1) |
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16 | (3) |
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1.2.3 A return to the Gauss map |
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19 | (2) |
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1.2.4 Elementary metrical Diophantine analysis |
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21 | (5) |
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1.2.5 Markov partitions for interval maps |
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26 | (4) |
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1.3 The Farey map: definition and topological properties |
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30 | (10) |
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30 | (4) |
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1.3.2 Topological properties of the Farey map |
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34 | (6) |
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40 | (18) |
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40 | (4) |
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44 | (4) |
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1.4.3 Topological properties of Fα |
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48 | (3) |
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1.4.4 Expanding and expansive partitions |
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51 | (2) |
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1.4.5 Metrical Diophantine-like results for the α-Luroth expansion |
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53 | (5) |
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1.5 Notes and historical remarks |
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58 | (3) |
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58 | (1) |
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1.5.2 The classical Luroth series |
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59 | (2) |
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61 | (3) |
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64 | (58) |
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64 | (5) |
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2.1.1 Invariant measures for the Gauss and α-Luroth system |
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66 | (3) |
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2.2 Recurrence and conservativity |
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69 | (6) |
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2.3 The transfer operator |
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75 | (13) |
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2.3.1 Jacobians and the change of variable formula |
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75 | (1) |
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2.3.2 Obtaining invariant measures via the transfer operator |
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76 | (3) |
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2.3.3 The Ruelle operator |
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79 | (3) |
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2.3.4 Invariant measures for F, Fα, G and Lα |
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82 | (4) |
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2.3.5 Invariant measures via the jump transformation |
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86 | (2) |
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2.4 Ergodicity and exactness |
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88 | (27) |
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2.4.1 Ergodicity of the systems G and Lα |
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92 | (4) |
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2.4.2 Ergodic theorems for probability spaces and consequences for the Gauss and α-Luroth systems |
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96 | (7) |
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2.4.3 Ergodic theorems for infinite measures |
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103 | (2) |
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105 | (4) |
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2.4.5 Uniqueness of the invariant measures for F and Fα |
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109 | (3) |
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2.4.6 Proof of Hopf's Ratio Ergodic Theorem |
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112 | (3) |
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115 | (5) |
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120 | (2) |
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3 Renewal theory and α-sum-level sets |
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122 | (15) |
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122 | (1) |
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3.2 Sum-level sets for the α-Luroth expansion |
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123 | (12) |
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3.2.1 Classical renewal results |
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124 | (8) |
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3.2.2 Renewal theory applied to the α-sum-level sets |
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132 | (3) |
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135 | (2) |
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4 Infinite ergodic theory |
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137 | (22) |
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4.1 The functional analytic perspective and the Chacon--Ornstein Ergodic Theorem |
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137 | (10) |
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4.2 Pointwise dual ergodicity |
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147 | (4) |
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4.3 ψ-mixing, Darling--Kac sets and pointwise dual ergodicity |
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151 | (6) |
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157 | (2) |
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5 Applications of infinite ergodic theory |
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159 | (24) |
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5.1 Sum-level sets for the continued fraction expansion, first investigations |
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159 | (2) |
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5.2 ψ-mixing for the Gauss map and the Gauss problem |
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161 | (6) |
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5.3 Pointwise dual ergodicity for the Farey map |
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167 | (1) |
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5.4 Uniform and uniformly returning sets |
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168 | (5) |
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5.5 Finer asymptotics of Lebesgue measure of sum-level sets |
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173 | (4) |
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5.6 Uniform distribution of the even Stern--Brocot sequence |
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177 | (4) |
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181 | (2) |
| Bibliography |
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183 | (5) |
| Index |
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188 | |
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