Abstract
We consider volume-preserving flows \({(\Phi^f_t)_{t\in\mathbb{R}}}\) on \({S \times \mathbb{R}}\) , where S is a compact connected surface of genus g ≥ 2 and \({(\Phi^f_t)_{t\in\mathbb{R}}}\) has the form \({\Phi^f_t(x, y) = (\phi_{t}x, y + \int_0^{t}f(\phi_{s}x)\,ds)}\) where \({(\phi_t)_{t\in\mathbb{R}}}\) is a locally Hamiltonian flow of hyperbolic periodic type on S and f is a smooth real valued function on S. We investigate ergodic properties of these infinite measure-preserving flows and prove that if f belongs to a space of finite codimension in \({\fancyscript{C}^{2+\epsilon}(S)}\) , then the following dynamical dichotomy holds: if there is a fixed point of \({(\phi_t)_{t\in\mathbb{R}}}\) on which f does not vanish, then \({(\Phi^f_t)_{t\in\mathbb{R}}}\) is ergodic, otherwise, if f vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension \({(\Phi^0_t)_{t\in\mathbb{R}}}\) . The proof of this result exploits the reduction of \({(\Phi^f_t)_{t\in\mathbb{R}}}\) to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of \({(\phi_t)_{t\in\mathbb{R}}}\) on which f does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.
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References
Aaronson, J.: An introduction to infinite ergodic theory. Mathematical Surveys and Monographs, vol. 50. AMS, Providence (1997)
Aaronson, J., Lemańczyk, M., Mauduit, C., Nakada, H.: Koksma’s inequality and group extensions of Kronecker transformations. In: Algorithms, Fractals, and Dynamics (Okayama/Kyoto, 1992), pp. 27–50. Plenum, New York (1995)
Arnold V.I.: Topological and ergodic properties of closed 1-forms with incommensurable periods. Funct. Anal. Appl. 25, 81–90 (1991)
Conze J.-P.: Ergodicité d’une transformation cylindrique. Bull. Soc. Math. France 108, 441–456 (1980)
Conze J.-P., Frączek K.: Cocycles over interval exchange transformations and multivalued Hamiltonian flows. Adv. Math. 226, 4373–4428 (2011)
Conze, J.-P., Gutkin, E.: On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces. arXiv:1008.0136
Fathi A., Lauderbach F., Poenaru V.: Travaux de Thurston sur les surfaces. Séminaire Orsay (Astérisque 66–67). Société, Paris (1979)
Fayad B., Lemańczyk M.: On the ergodicity of cylindrical transformations given by the logarithm. Mosc. Math. J. 6, 771–772 (2006)
Frączek K.: On ergodicity of some cylinder flows. Fund. Math. 163, 117–130 (2000)
Frączek K., Lemańczyk M.: On symmetric logarithm and some old examples in smooth ergodic theory. Fund. Math. 180, 241–255 (2003)
Forni G.: Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. Math. 146(2), 295–344 (1997)
Forni G.: Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. Math. 155(2), 1–103 (2002)
Gutkin E.: Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces. Regul. Chaotic Dyn. 15, 482–503 (2010)
Hooper, P., Hubert, P., Weiss, B.: Dynamics on the infinite staircase surface. Discrete Contin. Dyn. Syst. (to appear)
Hooper, P., Weiss, B.: Generalized staircases: recurrence and symmetry. Annales de L’Institut Fourier (to appear). arXiv:0905.3736
Host, B., Méla, J.-F., Parreau, F.: Analyse harmonique des mesures. Astérisque No. 135–136 (1986)
Hubert P., Lelievre S., Troubetzkoy S.: The Ehrenfest wind-tree model: periodic directions, recurrence, diffusion. J. Reine Angew. Math. 656, 223–244 (2011)
Hubert, P., Weiss, B.: Ergodicity for infinite periodic translation surfaces. Preprint (2010). http://www.cmi.univ-mrs.fr/~hubert/articles/hub-weiss2.pdf
Katok A.: Invariant measures of flows on orientable surfaces. Soviet Math. Dokl. 14, 1104–1108 (1973)
Katok A.: Interval exchange transformations and some special flows are not mixing. Isr. J. Math. 35, 301–310 (1980)
Keane M.: Interval exchange transformations. Math. Z. 141, 25–31 (1975)
Kočergin A.V.: On the absence of mixing in special flows over the rotation of a circle and in flows on a two-dimensional torus. Dokl. Akad. Nauk SSSR 205, 949–952 (1972)
Kočergin A.V.: Mixing in special flows over a shifting of segments and in smooth flows on surfaces. Mat. Sb. 96(138), 471–502 (1975)
Lemańczyk M., Parreau F., Volný D.: Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces. Trans. Am. Math. Soc. 348, 4919–4938 (1996)
Maier A.G.: Trajectories on closed orientable surfaces. Mat. Sb. 12(54), 71–84 (1943)
Marmi S., Moussa P., Yoccoz J.-C.: The cohomological equation for Roth-type interval exchange maps. J. Am. Math. Soc. 18, 823–872 (2005)
Marmi, S., Moussa, P., Yoccoz, J.-C.: Linearization of generalized interval exchange maps. arXiv:1003.1191
Masur H.: Interval exchange transformations and measured foliations. Ann. Math. 115(2), 169–200 (1982)
Novikov S.: The Hamiltonian formalism and a multivalued analogue of Morse theory. Math. Surveys 37, 1–56 (1982)
Oren I.: Ergodicity of cylinder flows arising from irregularities of distribution. Isr. J. Math. 44, 127–138 (1983)
Pask D.: Skew products over irrational rotation. Isr. J. Math. 69, 65–74 (1990)
Pask D.: Ergodicity of certain cylinder flows. Isr. J. Math. 76, 129–152 (1991)
Rauzy G.: Échanges d’intervalles et transformations induites. Acta Arith. 34, 315–328 (1979)
Scheglov D.: Absence of mixing for smooth flows on genus two surfaces. J. Mod. Dyn. 3, 13–34 (2009)
Schmidt, K.: Cocycle of Ergodic Transformation Groups. Lect. Notes in Math., vol. 1. Macmillan (1977)
Sinai Ya.G., Ulcigrai C.: Weak mixing in interval exchange transformations of periodic type. Lett. Math. Phys. 74, 111–133 (2005)
Thurston W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. (N.S.) 19, 417–431 (1988)
Ulcigrai C.: Mixing for suspension flows over interval exchange transformations. Ergodic Theory Dyn. Syst. 27, 991–1035 (2007)
Ulcigrai C.: Weak mixing for logarithmic flows over interval exchange transformations. J. Mod. Dyn. 3, 35–49 (2009)
Ulcigrai C.: Absence of mixing in area-preserving flows on surfaces. Ann. Math. 173(2), 1743–1778 (2011)
Veech W.A.: Interval exchange transformations. J. Analyse Math. 33, 222–272 (1978)
Veech, W.A.: Projective Swiss cheeses and uniquely ergodic interval exchange transformations. In: Progress in Mathematics, vol. I. Ergodic Theory and Dynamical Systems, I, pp. 113–193. Birkhäuser, Boston (1981)
Veech W.A.: Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 115(2), 201–242 (1982)
Veech W.A.: The metric theory of interval exchange transformations I. Generic spectral properties. Am. J. Math. 106, 1331–1358 (1984)
Viana M.: Ergodic theory of interval exchange maps. Rev. Mat. Complut. 19, 7–100 (2006)
Viana, M.: Dynamics of Interval Exchange Transformations and Teichmüller Flows. Lecture Notes. http://w3.impa.br/~viana/out/ietf.pdf
Yoccoz, J.-C.: Continued fraction algorithms for interval exchange maps: an introduction. In: Frontiers in Number Theory, Physics, and Geometry. I, pp. 401–435. Springer, Berlin (2006)
Yoccoz, J.-C.: Interval Exchange Maps and Translation Surfaces. Lecture Notes. http://www.college-de-france.fr/media/equ_dif/UPL15305_PisaLecturesJCY2007.pdf
Zorich A.: Deviation for interval exchange transformations. Ergodic Theory Dyn. Syst. 17, 1477–1499 (1997)
Zorich A.: How do the leaves of a closed 1-form wind around a surface?. Am. Math. Soc. Transl. Ser. 197(2), 135–178 (1999)
Zorich, A.: Flat surfaces. In: Frontiers in Number Theory, Physics, and Geometry. I, pp. 437–583 Springer, Berlin (2006)
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Research partially supported by a NCN grant in years 2012–2015. C. Ulcigrai is currently supported by an RCUK Fellowship and the EPSRC First Grant EP/I019030/1.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Frączek, K., Ulcigrai, C. Ergodic properties of infinite extensions of area-preserving flows. Math. Ann. 354, 1289–1367 (2012). https://doi.org/10.1007/s00208-011-0764-y
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DOI: https://doi.org/10.1007/s00208-011-0764-y