Intro -- Preface -- Furstenberg Fractals -- 1 Introduction -- 2 Furstenberg Fractals -- 3 The Fractal Constructions -- 4 Density of Non-Recurrent Points -- 5 Isometries and Furstenberg Fractals -- Idris Assani and Kimberly Presser A Survey of the Return Times Theorem -- 1 Origins -- 1.1 Averages along Subsequences -- 1.2 Weighted Averages -- 1.3 Wiener-Wintner Results -- 2 Development -- 2.1 The BFKO Proof of Bourgain's Return Times Theorem -- 2.2 Extensions of the Return Times Theorem -- 2.3 Unique Ergodicity and the Return Times Theorem -- 2.4 A Joinings Proof of the Return Times Theorem -- 3 The MultitermReturn Times Theorem -- 3.1 Definitions -- 4 Characteristic Factors -- 4.1 Characteristic Factors and the Return Times Theorem -- 5 Breaking the Duality -- 5.1 Hilbert Transforms -- 5.2 The (??1, ??1) Case -- 6 Other Notes on the Return Times Theorem -- 6.1 The Sigma-Finite Case -- 6.2 Recent Extensions -- 6.3 Wiener-WintnerDynamical Functions -- 7 Conclusion -- Characterizations of Distal and Equicontinuous Extensions -- Averages Along the Squares on the Torus -- 1 Introduction and Statement of the Main Results -- 2 Preliminary Results and Notation -- 3 Proofs of the Main Results -- Stepped Hyperplane and Extension of the Three Distance Theorem -- 1 Introduction -- 2 Kwapisz's Result for Translation -- 3 Continued Fraction Expansions -- 3.1 Brun's Algorithm -- 3.2 Strong Convergence -- 4 Proof of Theorem1.1 -- 5 Appendix: Proof of Theorem2.4 and Stepped Hyperplane -- Remarks on Step Cocycles over Rotations, Centralizers and Coboundaries -- 1 Introduction -- 2 Preliminaries on Cocycles -- 2.1 Cocycles and Group Extension of Dynamical Systems -- 2.2 Essential Values, Nonregular Cocycle -- 2.3 Z2-Actions and Centralizer -- 2.4 Case of an Irrational Rotation -- 3 Coboundary Equations for Irrational Rotations.

3.1 Classical Results, Expansion in Basis qna -- 3.2 Linear and Multiplicative Equations -- 4 Applications -- 4.1 Non-Ergodic Cocycles with Ergodic Compact Quotients -- 4.2 Examples of Nontrivial and Trivial Centralizer -- 4.3 Example of a Nontrivial Conjugacy in a Group Family -- 5 Appendix: Proof of Theorem3.3 -- Hamilton's Theorem for Smooth Lie Group Actions -- 1 Introduction -- 2 Preliminaries -- 2.1 Fréchet Spaces and Tame Operators -- 2.2 Hamilton's Nash-Moser Theoremfor Exact Sequences -- 2.3 Cohomology -- 3 An Application of Hamilton's Nash-Moser Theoremfor Exact Sequences to Lie Group Actions -- 3.1 The Set-Up -- 3.2 Tamely Split First Cohomology -- 3.3 Existence of Tame Splitting for the Complex -- 3.4 A Perturbation Result -- 3.5 A Variation of Theorem 3.6 -- 4 Possible Applications -- Mixing Automorphisms which are Markov Quasi-Equivalent but not Weakly Isomorphic -- 1 Introduction -- 2 Gaussian Automorphisms and Gaussian Cocycles -- 3 Coalescence of Two-Sided Cocycle Extensions -- 4 Main Result -- On the Strong Convolution Singularity Property -- 1 Introduction -- 2 Definitions -- 2.1 Spectral Theory -- 2.2 Joinings -- 2.3 Special Flows -- 2.4 Continued Fractions -- 3 Tools -- 4 Smooth Flows on Surfaces -- 5 Results -- 5.1 New Tools - The Main Proposition -- 5.2 New Tools - Technical Details -- 5.3 Application -- Fractal Geometry of Non-Uniformly Hyperbolic Horseshoes -- 1 Part I - A Survey on Homoclinic/HeteroclinicBifurcations -- 1.1 Transverse Homoclinic Orbits and Smale's Horseshoes -- 1.2 Homoclinic Tangencies and Newhouse Phenomena -- 1.3 Homoclinic Bifurcations Associated to Thin Horseshoes -- 1.4 Homoclinic Bifurcations Associated to Fat Horseshoes and StableTangencies -- 1.5 Heteroclinic Bifurcations of Slightly Fat Horseshoes after J. Palis and J.-C. Yoccoz -- 1.6 A Global View on Palis-Yoccoz Induction Scheme.

2 Part II - A Research Announcement on Non-Uniformly Hyperbolic Horseshoes -- 2.1 Hausdorff Dimension of the Stable Sets of Non-Uniformly Hyperbolic Horseshoes -- 2.2 Final Comments on Further Results -- Adic Flows, Transversal Flows, and Horocycle Flows -- 1 Introduction -- 2 Adic Flows -- 2.1 Ergodic Properties of Adic Flows -- 3 Application to Horocycle Flows -- 3.1 The Compact Case -- Uniform Rigidity Sequences for Topologically WeaklyMixing Homeomorphisms -- 1 Introduction -- 2 Uniform Rigidity Sequences -- 2.1 Proof of Theorem1.2.

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