The set of non-uniquely ergodic d -IETs has Hausdorff codimension 1/2
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The set of non-uniquely ergodic d-IETs has Hausdorff codimension 1/2 Jon Chaika1,2 · Howard Masur1,2 Dedicated to the memories of William Veech and Jean-Christophe Yoccoz.
Received: 29 January 2018 / Accepted: 7 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We show that the set of not uniquely ergodic d-IETs with permutation in the Rauzy class of the hyperelliptic permutation has Hausdorff dimension d − 23 [in the (d − 1)-dimension space of d-IETs] for d ≥ 5. For d = 4 this was shown by Athreya–Chaika and for d ∈ {2, 3} the set is known to have dimension d −2. This provides lower bounds on the Hausdorff dimension of non-weakly mixing IETs and, with input from Al-Saqban et al. (Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension, 2017. arXiv:1711.10542), identifies the Hausdorff dimension of non-weakly mixing IETs with permutation (d, d − 1, . . . , 2, 1) when d is odd. Contents 1 2 3
Introduction . . . . . . . . . . . . . . . . . . . . . 1.1 History . . . . . . . . . . . . . . . . . . . . . Plan of paper and background material and notation 2.1 Preliminaries on Rauzy induction . . . . . . . Hausdorff dimension . . . . . . . . . . . . . . . .
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B Jon Chaika
[email protected] Howard Masur [email protected]
1
Department of Mathematics, University of Utah, 155 S 1400 E Room 233, Salt Lake City, UT 84112, USA
2
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Room 208C, Chicago, IL 60637, USA
123
J. Chaika, H. Masur 4
Paths and matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Choice of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Non-unique ergodicity . . . . . . . . . . . . . . . . . . . . . . . . 5 Distortion and probabilistic results . . . . . . . . . . . . . . . . . . . . . 6 Remaining on left hand side, remaining on right hand side . . . . . . . . 6.1 Proof of Proposition 6.4 . . . . . . . . . . . . . . . . . . . . . . . . 7 Input and output singular direction . . . . . . . . . . . . . . . . . . . . . 7.1 Symplectic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Largest singular input and output vectors of M T . . . . . . . . . . . 7.3 Small singular input and output directions of M and choice of planes 7.4 Diameters and choice of planes . . . . . . . . . . . . . . . . . . . . 8 Geometry of slices on LHS and illumination . . . . . . . . . . . . . . . . 9 Repeating to illuminate . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Restriction on left side . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Proof of Proposition 10.3 . . . . . . . . . . . . . . . . . . . . . . . 10.2 Proof of Proposition 10.4 . . . . . . . . . . . . . . . . . . . . . . . 11 Transition, freedom and restriction on the right hand side . . . .
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