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March  2015, 35(3): 829-855. doi: 10.3934/dcds.2015.35.829

On special flows over IETs that are not isomorphic to their inverses

Przemysław Berk 1, and  Krzysztof Frączek 1,

1. 

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

Received  March 2014 Revised  June 2014 Published  October 2014

In this paper we give a criterion for a special flow to be not isomorphic to its inverse which is a refine of a result in [6]. We apply this criterion to special flows $T^f$ built over ergodic interval exchange transformations $T:[0,1)\to[0,1)$ (IETs) and under piecewise absolutely continuous roof functions $f:[0,1)\to\mathbb{R}_+$. We show that for almost every IET $T$ if $f$ is absolutely continuous over exchanged intervals and has non-zero sum of jumps then the special flow $T^f$ is not isomorphic to its inverse. The same conclusion is valid for a typical piecewise constant roof function.
Keywords: joinings, Non-reversibility of flows, special flows over IETs..
Mathematics Subject Classification: Primary: 37A10, 37E3.
Citation: Przemysław Berk, Krzysztof Frączek. On special flows over IETs that are not isomorphic to their inverses. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 829-855. doi: 10.3934/dcds.2015.35.829
References:
[1]

H. Anzai, On an example of a measure preserving transformation which is not conjugate to its inverse, Proc. Japan Acad., 27 (1951), 517-522. doi: 10.3792/pja/1195571227.  Google Scholar

[2]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[3]

A. I. Danilenko and V. V. Ryzhikov, On self-similarities of ergodic flows, Proc. Lond. Math. Soc., 104 (2012), 431-454. doi: 10.1112/plms/pdr032.  Google Scholar

[4]

A. del Junco, Disjointness of measure-preserving transformations, minimal self-joinings and category, in Ergodic theory and dynamical systems, I (College Park, Md., 1979-80), Progr. Math., Birkhäuser, 10 (1981), 81-89. doi: 10.1007/978-1-4899-6696-4_3.  Google Scholar

[5]

K. Frączek, Density of mild mixing property for vertical flows of Abelian differentials, Proc. Amer. Math. Soc., 137 (2009), 4129-4142. doi: 10.1090/S0002-9939-09-10025-4.  Google Scholar

[6]

K. Frączek, J. Kułaga and M. Lemańczyk, Non-reversibility and self-joinings of higher orders for ergodic flows, J. Anal. Math., 122 (2014), 163-227. doi: 10.1007/s11854-014-0007-8.  Google Scholar

[7]

K. Frączek and M. Lemańczyk, On the self-similarity problem for ergodic flows, Proc. Lond. Math. Soc., 99 (2009), 658-696. doi: 10.1112/plms/pdp013.  Google Scholar

[8]

K. Frączek and M. Lemańczyk, A class of special flows over irrational rotations which is disjoint from mixing flows, Ergodic Theory Dynam. Systems, 24 (2004), 1083-1095. doi: 10.1017/S0143385704000112.  Google Scholar

[9]

K. Frączek and M. Lemańczyk, On disjointness properties of some smooth flows, Fund. Math., 185 (2005), 117-142. doi: 10.4064/fm185-2-2.  Google Scholar

[10]

E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[11]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2), 43 (1942), 332-350. doi: 10.2307/1968872.  Google Scholar

[12]

A. Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310. doi: 10.1007/BF02760655.  Google Scholar

[13]

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

[14]

J. King, Joining-rank and the structure of finite rank mixing transformations, J. Anal. Math., 51 (1988), 182-227. doi: 10.1007/BF02791123.  Google Scholar

[15]

J. Kułaga, On the self-similarity problem for smooth flows on orientable surfaces, Ergodic Theory Dynam. Systems, 32 (2012), 1615-1660. doi: 10.1017/S0143385711000459.  Google Scholar

[16]

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

[17]

V. V. Ryzhikov, Partial multiple mixing on subsequences can distinguish between automorphisms $T$ and $T^{-1}$, Math. Notes, 74 (2003), 841-847. doi: 10.1023/B:MATN.0000009020.82284.54.  Google Scholar

[18]

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

[19]

W. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations, Ergodic theory and dynamical systems, I (College Park, Md., 1979-80), Progr. Math., Birkháuser, 10 (1981), 113-193. doi: 10.1007/978-1-4899-6696-4_5.  Google Scholar

[20]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.  Google Scholar

[21]

W. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties, Amer. J. Math., 106 (1984), 1331-1359. doi: 10.2307/2374396.  Google Scholar

[22]

M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100.  Google Scholar

show all references

References:
[1]

H. Anzai, On an example of a measure preserving transformation which is not conjugate to its inverse, Proc. Japan Acad., 27 (1951), 517-522. doi: 10.3792/pja/1195571227.  Google Scholar

[2]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[3]

A. I. Danilenko and V. V. Ryzhikov, On self-similarities of ergodic flows, Proc. Lond. Math. Soc., 104 (2012), 431-454. doi: 10.1112/plms/pdr032.  Google Scholar

[4]

A. del Junco, Disjointness of measure-preserving transformations, minimal self-joinings and category, in Ergodic theory and dynamical systems, I (College Park, Md., 1979-80), Progr. Math., Birkhäuser, 10 (1981), 81-89. doi: 10.1007/978-1-4899-6696-4_3.  Google Scholar

[5]

K. Frączek, Density of mild mixing property for vertical flows of Abelian differentials, Proc. Amer. Math. Soc., 137 (2009), 4129-4142. doi: 10.1090/S0002-9939-09-10025-4.  Google Scholar

[6]

K. Frączek, J. Kułaga and M. Lemańczyk, Non-reversibility and self-joinings of higher orders for ergodic flows, J. Anal. Math., 122 (2014), 163-227. doi: 10.1007/s11854-014-0007-8.  Google Scholar

[7]

K. Frączek and M. Lemańczyk, On the self-similarity problem for ergodic flows, Proc. Lond. Math. Soc., 99 (2009), 658-696. doi: 10.1112/plms/pdp013.  Google Scholar

[8]

K. Frączek and M. Lemańczyk, A class of special flows over irrational rotations which is disjoint from mixing flows, Ergodic Theory Dynam. Systems, 24 (2004), 1083-1095. doi: 10.1017/S0143385704000112.  Google Scholar

[9]

K. Frączek and M. Lemańczyk, On disjointness properties of some smooth flows, Fund. Math., 185 (2005), 117-142. doi: 10.4064/fm185-2-2.  Google Scholar

[10]

E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[11]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2), 43 (1942), 332-350. doi: 10.2307/1968872.  Google Scholar

[12]

A. Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310. doi: 10.1007/BF02760655.  Google Scholar

[13]

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

[14]

J. King, Joining-rank and the structure of finite rank mixing transformations, J. Anal. Math., 51 (1988), 182-227. doi: 10.1007/BF02791123.  Google Scholar

[15]

J. Kułaga, On the self-similarity problem for smooth flows on orientable surfaces, Ergodic Theory Dynam. Systems, 32 (2012), 1615-1660. doi: 10.1017/S0143385711000459.  Google Scholar

[16]

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

[17]

V. V. Ryzhikov, Partial multiple mixing on subsequences can distinguish between automorphisms $T$ and $T^{-1}$, Math. Notes, 74 (2003), 841-847. doi: 10.1023/B:MATN.0000009020.82284.54.  Google Scholar

[18]

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

[19]

W. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations, Ergodic theory and dynamical systems, I (College Park, Md., 1979-80), Progr. Math., Birkháuser, 10 (1981), 113-193. doi: 10.1007/978-1-4899-6696-4_5.  Google Scholar

[20]

W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.  Google Scholar

[21]

W. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties, Amer. J. Math., 106 (1984), 1331-1359. doi: 10.2307/2374396.  Google Scholar

[22]

M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100.  Google Scholar

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