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

Previous Article
Non ultracontractive heat kernel bounds by Lyapunov conditions
DCDS Home
This Issue
Next Article
The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions

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

Abstract
Related Papers

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 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.

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.
[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.

[1]	Bassam Fayad, A. Windsor. A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. Journal of Modern Dynamics, 2007, 1 (1) : 107-122. doi: 10.3934/jmd.2007.1.107
[2]	Krzysztof Frączek, Mariusz Lemańczyk. A class of mixing special flows over two--dimensional rotations. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4823-4829. doi: 10.3934/dcds.2015.35.4823
[3]	Krzysztof Frączek, Mariusz Lemańczyk. Ratner's property and mild mixing for special flows over two-dimensional rotations. Journal of Modern Dynamics, 2010, 4 (4) : 609-635. doi: 10.3934/jmd.2010.4.609
[4]	E.B. Pitman, C.C. Nichita, A.K. Patra, A.C. Bauer, M. Bursik, A. Webb. A model of granular flows over an erodible surface. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 589-599. doi: 10.3934/dcdsb.2003.3.589
[5]	V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263
[6]	Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691
[7]	Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427
[8]	Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[9]	Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146
[10]	Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations and Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193
[11]	Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022
[12]	Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403
[13]	François Ledrappier, Omri Sarig. Unique ergodicity for non-uniquely ergodic horocycle flows. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 411-433. doi: 10.3934/dcds.2006.16.411
[14]	Aneta Wróblewska-Kamińska. Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2565-2592. doi: 10.3934/dcds.2013.33.2565
[15]	Ítalo Melo, Sergio Romaña. Contributions to the study of Anosov geodesic flows in non-compact manifolds. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5149-5171. doi: 10.3934/dcds.2020223
[16]	César J. Niche. Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 617-630. doi: 10.3934/dcds.2006.14.617
[17]	José M. Amigó, Isabelle Catto, Ángel Giménez, José Valero. Attractors for a non-linear parabolic equation modelling suspension flows. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 205-231. doi: 10.3934/dcdsb.2009.11.205
[18]	Alex Mahalov, Mohamed Moustaoui, Basil Nicolaenko. Three-dimensional instabilities in non-parallel shear stratified flows. Kinetic and Related Models, 2009, 2 (1) : 215-229. doi: 10.3934/krm.2009.2.215
[19]	Bryce Weaver. Growth rate of periodic orbits for geodesic flows over surfaces with radially symmetric focusing caps. Journal of Modern Dynamics, 2014, 8 (2) : 139-176. doi: 10.3934/jmd.2014.8.139
[20]	Russell Johnson, Mahesh G. Nerurkar. On $SL(2, R)$ valued cocycles of Hölder class with zero exponent over Kronecker flows. Communications on Pure and Applied Analysis, 2011, 10 (3) : 873-884. doi: 10.3934/cpaa.2011.10.873

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Przemysław Berk Krzysztof Frączek
on Google Scholar
Przemysław Berk Krzysztof Frączek

[Back to Top]