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

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.
Citation: Przemysław Berk, Krzysztof Frączek. On special flows over IETs that are not isomorphic to their inverses. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.3792/pja/1195571227.  Google Scholar

[2]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory,, Springer-Verlag, (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.  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, 10 (1981), 1979.  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.  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.  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.  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.  doi: 10.1017/S0143385704000112.  Google Scholar

[9]

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

[10]

E. Glasner, Ergodic Theory Via Joinings,, Mathematical Surveys and Monographs, (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.  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.  doi: 10.1007/BF02760655.  Google Scholar

[13]

M. Keane, Interval exchange transformations,, Math. Z., 141 (1975), 25.  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.  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.  doi: 10.1017/S0143385711000459.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

W. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations,, Ergodic theory and dynamical systems, 10 (1981), 1979.  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.  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.  doi: 10.2307/2374396.  Google Scholar

[22]

M. Viana, Ergodic theory of interval exchange maps,, Rev. Mat. Complut., 19 (2006), 7.   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.  doi: 10.3792/pja/1195571227.  Google Scholar

[2]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory,, Springer-Verlag, (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.  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, 10 (1981), 1979.  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.  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.  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.  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.  doi: 10.1017/S0143385704000112.  Google Scholar

[9]

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

[10]

E. Glasner, Ergodic Theory Via Joinings,, Mathematical Surveys and Monographs, (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.  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.  doi: 10.1007/BF02760655.  Google Scholar

[13]

M. Keane, Interval exchange transformations,, Math. Z., 141 (1975), 25.  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.  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.  doi: 10.1017/S0143385711000459.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

W. Veech, Projective Swiss cheeses and uniquely ergodic interval exchange transformations,, Ergodic theory and dynamical systems, 10 (1981), 1979.  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.  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.  doi: 10.2307/2374396.  Google Scholar

[22]

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

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