Contents 0. Introduction 0.1. Interval exchange maps 0.2. The cohomological equation 0.3. Summary of the contents 1. The continued fraction algorithm for interval exchange maps 1.1. Interval exchange maps 1.2. The continued fraction algorithm 1.3. Roth-type interval exchange maps 2. The cohomological equation 2.1. The theorem of Gottschalk and Hedlund 2.2. Special Birkhoff sums 2.3. Estimates for functions of bounded variation 2.4. Primitives of functions of bounded variation 3. Suspensions of interval exchange maps 3.1. Suspension data 3.2. Construction of a Riemann surface 3.3. Compactification of M∗ 3.4. The cohomological equation for higher smoothness 4. Proof of full measure for Roth type 4.1. The basic operation of the algorithm for suspensions 4.2. The Teichm¨uller flow 4.3. The absolutely continuous invariant measure 4.4. Integrability of log Z(1) 4.5. Conditions (b) and (c) have full measure 4.6. The main step 4.7. Condition (a) has full measure 4.8. Proof of the proposition Appendix A. Roth-type conditions in a concrete family of interval exchange maps Appendix B. A nonuniquely ergodic interval exchange map satisfying condition (a)

### The cohomological equation for Roth-type interval exchange maps

#### Abstract

Contents 0. Introduction 0.1. Interval exchange maps 0.2. The cohomological equation 0.3. Summary of the contents 1. The continued fraction algorithm for interval exchange maps 1.1. Interval exchange maps 1.2. The continued fraction algorithm 1.3. Roth-type interval exchange maps 2. The cohomological equation 2.1. The theorem of Gottschalk and Hedlund 2.2. Special Birkhoff sums 2.3. Estimates for functions of bounded variation 2.4. Primitives of functions of bounded variation 3. Suspensions of interval exchange maps 3.1. Suspension data 3.2. Construction of a Riemann surface 3.3. Compactification of M∗ 3.4. The cohomological equation for higher smoothness 4. Proof of full measure for Roth type 4.1. The basic operation of the algorithm for suspensions 4.2. The Teichm¨uller flow 4.3. The absolutely continuous invariant measure 4.4. Integrability of log Z(1) 4.5. Conditions (b) and (c) have full measure 4.6. The main step 4.7. Condition (a) has full measure 4.8. Proof of the proposition Appendix A. Roth-type conditions in a concrete family of interval exchange maps Appendix B. A nonuniquely ergodic interval exchange map satisfying condition (a)
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2005
Settore MAT/07 - Fisica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11384/513`