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Regularity of Conjugacies of Linearizable Generalized Interval Exchange Transformations
Selim Ghazouani1,2, Corinna Ulcigrai1
1Institute of Mathematics, University of Zuerich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
Communications in Mathematical Physics
|January 22, 2025
Summary
We study the regularity of conjugacies for generalized interval exchange transformations (GIETs). We show that under specific conditions, these conjugacies are smoother than previously known, improving rigidity results for interval exchange maps.
Area of Science:
- Dynamical Systems
- Ergodic Theory
- Geometric Rigidity
Background:
- Generalized Interval Exchange Transformations (GIETs) are a class of dynamical systems.
- Understanding the regularity of their conjugacies is crucial for rigidity theory.
Purpose of the Study:
- To investigate the regularity of diffeomorphisms conjugating GIETs to standard Interval Exchange Maps (IETs).
- To establish improved rigidity results for IETs.
Main Methods:
- Utilizing a renormalization operator derived from accelerated Rauzy-Veech induction.
- Applying a full measure condition on linearized IETs.
- Analyzing exponential convergence rates under renormalization.
Main Results:
- Demonstrated that under specific conditions, the conjugacy h is smooth.
- Showed that for almost every irreducible IET, the topological conjugacy h is a diffeomorphism.
- Improved rigidity results from to for genus two IETs.
Conclusions:
- The study provides a significant advancement in understanding the regularity of conjugacies for GIETs.
- The findings contribute to the broader field of geometric rigidity in dynamical systems.
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