On the Hausdorff dimension of minimal interval exchange transformations with flips. (English) Zbl 1473.37050
The authors consider linear upper and lower bounds for the Hausdorff dimension of the set of minimal
interval exchange transformations with flips (fIETs). Special emphasis is given to transformations without periodic points, and
to the set of non-uniquely ergodic minimal
fIETs.
An fIET is a piecewise isometry of an interval to itself with a finite number of jump discontinuities, reversing the orientation of at least one of the intervals of continuity. fIETs generalize the notion of orientation-preserving interval exchange transformations (IETs). Each IET is the first return map to a transversal for a measured foliation on a surface.
The main results are:
(1) The Hausdorff dimension of the set \(MF_n\) satisfies \(n-2 \leq H \dim (MF_n) < n-1.\)
(2) For each \(n > 4\) the Hausdorff dimension of the set of non-uniquely ergodic minimal \(n\)-fIETs satisfies \(n-3 \leq H \dim (NUE_n).\)
3) For \(n = 5\) there holds \(H \dim (NUE_n) \geq \frac 5 2.\) This follows from a result in [J. S. Athreya and J. Chaika, Geom. Topol. 19, No. 6, 3537–3563 (2015; Zbl 1353.37066)].
An fIET is a piecewise isometry of an interval to itself with a finite number of jump discontinuities, reversing the orientation of at least one of the intervals of continuity. fIETs generalize the notion of orientation-preserving interval exchange transformations (IETs). Each IET is the first return map to a transversal for a measured foliation on a surface.
The main results are:
(1) The Hausdorff dimension of the set \(MF_n\) satisfies \(n-2 \leq H \dim (MF_n) < n-1.\)
(2) For each \(n > 4\) the Hausdorff dimension of the set of non-uniquely ergodic minimal \(n\)-fIETs satisfies \(n-3 \leq H \dim (NUE_n).\)
3) For \(n = 5\) there holds \(H \dim (NUE_n) \geq \frac 5 2.\) This follows from a result in [J. S. Athreya and J. Chaika, Geom. Topol. 19, No. 6, 3537–3563 (2015; Zbl 1353.37066)].
Reviewer: Yuang-Ling Ye (Guangzhou)
MSC:
| 37E05 | Dynamical systems involving maps of the interval |
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37C45 | Dimension theory of smooth dynamical systems |
Citations:
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