# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. [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. [12] A. Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310. doi: 10.1007/BF02760655. [13] M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981. [14] J. King, Joining-rank and the structure of finite rank mixing transformations, J. Anal. Math., 51 (1988), 182-227. doi: 10.1007/BF02791123. [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. [16] G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. [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. [18] W. Veech, Interval exchange transformations, J. Anal. Math., 33 (1978), 222-272. doi: 10.1007/BF02790174. [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. [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. [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. [22] M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. [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. [12] A. Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310. doi: 10.1007/BF02760655. [13] M. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31. doi: 10.1007/BF01236981. [14] J. King, Joining-rank and the structure of finite rank mixing transformations, J. Anal. Math., 51 (1988), 182-227. doi: 10.1007/BF02791123. [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. [16] G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328. [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. [18] W. Veech, Interval exchange transformations, J. Anal. Math., 33 (1978), 222-272. doi: 10.1007/BF02790174. [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. [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. [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. [22] M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100.
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