Table of Contents



Chapter 0. Introduction ...............................................  1  


Chapter I. Circle Diffeomorphisms .....................................  14
 1. The Combinatorial Theory of Poincare' .............................  16
 2. The Topological Theory of Denjoy ..................................  36 
    
        2.a  The Denjoy Inequality ....................................  47
     
        2.b  Ergodicity ...............................................  48
     
  3. Smooth Conjugacy Results ..........................................  50
  4. Families of Circle Diffeomorphisms: Arnol'd tongues ...............  68
  5. Counter-Examples to Smooth Linearizability ........................  73
  6. Frequency of Smooth Linearizability in Families ...................  76
  7. Some Historical Comments and Further Remarks ......................  78


Chapter II. The Combinatories of One-Dimensional Endomorphisms ........  81
  1. The Theorem of Sarkovskii .........................................  82
  2. Covering Maps of the Circle as Dynamical Systems ..................  88
  3. The Kneading Theory and Combinatorial Equivalence .................  92
      
       3.a  Examples ..................................................... 107
       3.b  Hofbauer's Tower Construction ................................ 108
       
  4. Full Families and Realization of Maps ............................. 115
  5. Families of Maps and Renormalization .............................. 138
  6. Piecewise Monotone Maps can be Modelled by Polynomial Maps ........ 153
  7. The Topological Entropy ........................................... 163
  8. The Piecewise Linear Model ........................................ 171
  9. Continuity of the Topological Entropy ............................. 186
10. Monotonicity of the Kneqding Invariant 
    for the Quadratic FAmily .......................................... 194
11. Some Historical Comments and Further Remarks ...................... 198


Chapter III. Structural Stabillity and Hyperbolicity .................. 201
  1. The Dynamics of Rational Mappings ................................. 202
  2. Structural Stability and Hyperbolicity ............................ 216
  3. Hyperbolicity in Maps with Negative Schwarzian Derivative ......... 230
  4. The Strucutre of the Non-Wandering Set ............................ 235
  5. Hyperbolicity in Smooth Maps ...................................... 247
  6. Misiurewicz Maps are Almost Hyperbolic ............................ 257
  7. Some Further Remarks and Open Questions ........................... 264


Chapter IV. The Structure of Smooth Maps .............................. 267
  1. The Cross-Ratio: the Minimum and Koebe Principle .................. 271

       1.a  Some Facts about the Schwarzian Derivative ................... 282

 2. Distortion of Cross-Ratios ........................................ 285   

      2.a  The Zygmund Conditions ....................................... 291

 3. Koebe Principles on Iterates ...................................... 295
 4. Some Simplifications and the Induction Assumption ................. 302
 5. The Pullback of Space: the Koebe/Contraction Principle ............ 305
 6. Disjointness of Orbits of Intervals ............................... 308
 7. Wandering Intervals Accumulate on Turning Points .................. 312
 8. Topological Properties of a Unimodal Pullback ..................... 315
 9. The Non-Existence of Wandering Intervals .......................... 319
10. Finiteness of Attractors .......................................... 321
11. Some Further Remarks and Open Questions ........................... 325


Chapter V. Ergodic Properties and Invariant Measures .................. 327
  1. Ergodicity, Attracotrs and Bowen-Ruelle-Sinai Measures ............ 329
  2. Invariant Measures for Markov Maps ................................ 351
  3. Constructing Invariant Measures by Inducing ....................... 362
  4. Constructing Invariant Measures by Pulling Back ................... 375
  5. Transitive Maps Without Finite Continuous Measures ................ 393
  6. Frequency of Maps with Positive Liapounov Exponents
      in Families and Jakobson's Theorem ................................ 401
  7. Some Further Remarks and Open Questions ........................... 433


Chapter VI. Renormalization ........................................... 437
  1. The Renormalization Operator ...................................... 438
  2. The Real Bounds ................................................... 453
  3. Bounded Geometry .................................................. 461
  4. The PullBack Argument ............................................. 465
  5. The Complex Bounds ................................................ 483
  6. Riemann Surface Laminations ....................................... 499
  7. The Almost Geodesic Principle ..................................... 524
  8. Renormalization is Contracting .................................... 533
  9. Universality of the Attracting Cantor Set ......................... 546
 10. Some Further Remarks and Open Questions ........................... 553


Chapter VII. Appendix ................................................. 555
  1. Some Terminology in Dynamical Systems ............................. 555
  2. Some Background in Topology ....................................... 556
  3. Some Results from Analysis and Measure Theory ..................... 558
  4. Some results from Ergodic Theory .................................. 560
  5. Some Background in Complex Analysis ............................... 561
  6. Some Results from Functional Analysis ............................. 581


Bibliography .......................................................... 582


Subject Index ......................................................... 600

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