August 2017, 11: 249-262. doi: 10.3934/jmd.2017011

Most interval exchanges have no roots

Department of Mathematics, Rice University, MS 136, P.O. Box 1892, Houston, TX 77251-1892, USA

Received  April 30, 2016 Revised  January 24, 2017 Published  March 2017

Let $T$ be an $m$-interval exchange transformation. By the rank of $T$ we mean the dimension of the $\mathbb{Q}$-vector space spanned by the lengths of the exchanged intervals. We prove that if $T$ is minimal and the rank of $T$ is greater than $1+\lfloor m/2 \rfloor$, then $T$ cannot be written as a power of another interval exchange. We also demonstrate that this estimate on the rank cannot be improved.

In the case that $T$ is a minimal 3-interval exchange transformation, we prove a stronger result: $T$ cannot be written as a power of another interval exchange if and only if $T$ satisfies Keane's infinite distinct orbit condition. In the course of proving this result, we give a classification (up to conjugacy) of those minimal interval exchange transformations whose discontinuities all belong to a single orbit.

Citation: Daniel Bernazzani. Most interval exchanges have no roots. Journal of Modern Dynamics, 2017, 11: 249-262. doi: 10.3934/jmd.2017011
References:
[1]

A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math., 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637.

[2]

M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J., 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.

[3]

M. Boshernitzan, Rank two interval exchange transformations, Ergodic Theory Dynam. Systems, 8 (1988), 379-394. doi: 10.1017/S0143385700004521.

[4]

M. Boshernitzan, Subgroup of interval exchanges generated by torsion elements and rotations, Proc. Amer. Math. Soc., 144 (2016), 2565-2573. doi: 10.1090/proc/12958.

[5]

J. Chaika, Every ergodic transformation is disjoint from almost every interval exchange transformation, Ann. of Math., 175 (2012), 237-253. doi: 10.4007/annals.2012.175.1.6.

[6]

F. DahmaniK. Fujiwara and V. Guirardel, Free groups of interval exchange transformations are rare, Groups Geom. Dyn., 7 (2013), 883-910. doi: 10.1017/etds.2016.32.

[7]

K. Juschenko, N. Matte Bon, N. Monod and M. de la Salle, Extensive amenability and an application to interval exchanges, arXiv: 1503.04977, (2016). doi: 10.4171/GGD/209.

[8]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[9]

M. Keane, Non-ergodic interval exchange maps, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.

[10]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 168-200. doi: 10.2307/1971341.

[11]

C. Novak, Discontinuity growth of interval exchange maps, J. Mod. Dyn., 3 (2009), 379-405. doi: 10.3934/jmd.2009.3.379.

[12]

C. Novak, Continuous interval exchange actions, Algebr. Geom. Topol., 10 (2010), 1609-1625. doi: 10.2140/agt.2010.10.1609.

[13]

C. Novak, Interval exchanges that do not embed in free groups, Groups Geom. Dyn., 6 (2012), 755-763. doi: 10.4171/GGD/173.

[14]

G. Rauzy, Echanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.

[15]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115 (1982), 201-242. doi: 10.2307/1971391.

[16]

W. A. Veech, Interval exchange transformations, J. Analyse Math., 33 (1978), 222-272. doi: 10.1007/BF02790174.

[17]

M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. doi: 10.5209/rev_REMA.2006.v19.n1.16621.

[18]

Y. Vorobets, Notes on the commutator group of the group of interval exchange transformations, arXiv: 1109.1352, (2011).

show all references

References:
[1]

A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math., 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637.

[2]

M. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J., 52 (1985), 723-752. doi: 10.1215/S0012-7094-85-05238-X.

[3]

M. Boshernitzan, Rank two interval exchange transformations, Ergodic Theory Dynam. Systems, 8 (1988), 379-394. doi: 10.1017/S0143385700004521.

[4]

M. Boshernitzan, Subgroup of interval exchanges generated by torsion elements and rotations, Proc. Amer. Math. Soc., 144 (2016), 2565-2573. doi: 10.1090/proc/12958.

[5]

J. Chaika, Every ergodic transformation is disjoint from almost every interval exchange transformation, Ann. of Math., 175 (2012), 237-253. doi: 10.4007/annals.2012.175.1.6.

[6]

F. DahmaniK. Fujiwara and V. Guirardel, Free groups of interval exchange transformations are rare, Groups Geom. Dyn., 7 (2013), 883-910. doi: 10.1017/etds.2016.32.

[7]

K. Juschenko, N. Matte Bon, N. Monod and M. de la Salle, Extensive amenability and an application to interval exchanges, arXiv: 1503.04977, (2016). doi: 10.4171/GGD/209.

[8]

M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981.

[9]

M. Keane, Non-ergodic interval exchange maps, Israel J. Math., 26 (1977), 188-196. doi: 10.1007/BF03007668.

[10]

H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 168-200. doi: 10.2307/1971341.

[11]

C. Novak, Discontinuity growth of interval exchange maps, J. Mod. Dyn., 3 (2009), 379-405. doi: 10.3934/jmd.2009.3.379.

[12]

C. Novak, Continuous interval exchange actions, Algebr. Geom. Topol., 10 (2010), 1609-1625. doi: 10.2140/agt.2010.10.1609.

[13]

C. Novak, Interval exchanges that do not embed in free groups, Groups Geom. Dyn., 6 (2012), 755-763. doi: 10.4171/GGD/173.

[14]

G. Rauzy, Echanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.

[15]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math., 115 (1982), 201-242. doi: 10.2307/1971391.

[16]

W. A. Veech, Interval exchange transformations, J. Analyse Math., 33 (1978), 222-272. doi: 10.1007/BF02790174.

[17]

M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. doi: 10.5209/rev_REMA.2006.v19.n1.16621.

[18]

Y. Vorobets, Notes on the commutator group of the group of interval exchange transformations, arXiv: 1109.1352, (2011).

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