# American Institute of Mathematical Sciences

January  2013, 7(1): 45-74. doi: 10.3934/jmd.2013.7.45

## On bounded cocycles of isometries over minimal dynamics

 1 Departamento deMatemática, UNAB, República 220, 2 piso, Santiago, Chile 2 Departamento de Matemática y C.C., USACH, Alameda 3363, Estación Central, Santiago, Chile 3 Facultad deMatemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile

Received  June 2012 Revised  January 2013 Published  May 2013

We show the following geometric generalization of a classical theorem of W. H. Gottschalk and G. A. Hedlund: a skew action induced by a cocycle of (affine) isometries of a Hilbert space over a minimal dynamical system has a continuous invariant section if and only if the cocycle is bounded. Equivalently, the associated twisted cohomological equation has a continuous solution if and only if the cocycle is bounded. We interpret this as a version of the Bruhat-Tits Center Lemma in the space of continuous functions. Our result also holds when the fiber is a proper CAT(0) space. One of the applications concerns matrix cocycles. Using the action of $\mathrm{GL} (n,\mathbb{R})$ on the (nonpositively curved) space of positively definite matrices, we show that every bounded linear cocycle over a minimal dynamical system is cohomologous to a cocycle taking values in the orthogonal group.
Citation: Daniel Coronel, Andrés Navas, Mario Ponce. On bounded cocycles of isometries over minimal dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 45-74. doi: 10.3934/jmd.2013.7.45
##### References:
 [1] G. Atkinson, A class of transitive cylinder transformations, J. London Math. Soc. (2), 17 (1978), 263-270. doi: 10.1112/jlms/s2-17.2.263. [2] U. Bader, T. Gelander and N. Monod, A fixed point theorem for $L^1$ spaces, Inventiones Mathematicae, 189 (2012), 143-148. doi: 10.1007/s00222-011-0363-2. [3] U. Bader, A. Furman, T. Gelander and N. Monod, Property (T) and rigidity for actions on Banach spaces, Acta Math., 198 (2007), 57-105. doi: 10.1007/s11511-007-0013-0. [4] R. Baire, "Leçons sur les Fonctions Discontinues," Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1995. [5] A. Ballmann, "Lectures on Spaces of Nonpositive Curvature," DMV Seminar, 25, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9240-7. [6] S. Banach, "Théorie des Opérations Linéaires," Monografie Matematyczne, Volume 1, Warszawa, 1932. [7] M. R. Bridson and A. Haefliger, "Metric Spaces of Non-Positive Curvature," Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999. [8] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math., 41 (1972), 5-251. [9] D. Coronel, A. Navas and M. Ponce, On the dynamics of non-reducible cylindrical vortices, J. Lond. Math. Soc. (2), 85 (2012), 789-808. doi: 10.1112/jlms/jdr068. [10] W. H. Gottschalk and G. A. Hedlund, "Topological Dynamics," American Mathematical Society Colloquium Publications, Vol. 36, Amer. Math. Soc., Providence, R. I., 1955. [11] M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233. [12] M. Jerison, The space of bounded maps into a Banach space, Annals of Math. (2), 52 (1950), 309-327. doi: 10.2307/1969472. [13] V. Kaimanovich, Double ergodicity of the Poisson boundary and applications to bounded cohomology, Geom. and Functional Analysis (GAFA), 13 (2003), 852-861. doi: 10.1007/s00039-003-0433-8. [14] B. Kalinin, Livšic theorem for matrix cocycles, Annals of Math. (2), 173 (2011), 1025-1042. doi: 10.4007/annals.2011.173.2.11. [15] B. Kalinin and V. Sadovskaya, Linear cocycles over hyperbolic systems and criteria of conformality, J. Mod. Dyn., 4 (2010), 419-441. doi: 10.3934/jmd.2010.4.419. [16] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. [17] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, in collaboration with E. A. Robinson, Jr., in "Smooth Ergodic Theory and its Applications" (Seattle, WA, 1999), Proceedings of the Symposia in Pure Mathematics, 69, Amer. Math. Soc., Providence, RI, (2001), 107-173. [18] I. Kornfeld and M. Lin, Coboundaries of irreducible Markov operators on $C(K)$, Israel J. of Mathematics, 97 (1997), 189-202. doi: 10.1007/BF02774036. [19] S. Lang, "Fundamentals of Differential Geometry," Graduate Texts in Mathematics, 191, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8. [20] V. Markovic, Quasisymmetric groups, J. Amer. Math. Soc., 19 (2006), 673-715. doi: 10.1090/S0894-0347-06-00518-2. [21] S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval-exchange maps, J. Amer. Math. Soc., 18 (2005), 823-872. doi: 10.1090/S0894-0347-05-00490-X. [22] J. Moulin Ollagnier and D. Pinchon, A note about Hedlund's theorem, in "Dynamical Systems," Vol. II-Warsaw, Astérisque, No. 50, Soc. Math. France, Paris, (1977), 311-313. [23] R. McCutcheon, The Gottschalk-Hedlund Theorem, Am. Math. Monthly, 106 (1999), 670-672. doi: 10.2307/2589497. [24] I. Namioka and E. Asplund, A geometric proof of Ryll-Nardzewski's fixed point theorem, Bull. Amer. Math. Soc., 73 (1967), 443-445. doi: 10.1090/S0002-9904-1967-11779-8. [25] A. Navas, Three remarks on one-dimensional bi-Lipschitz conjugacies,, unpublished note, (). [26] A. Navas, "Groups of Circle Diffeomorphisms," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2011. [27] J. C. Oxtoby, "Measure and Category. A Survey of the Analogies Between Topological and Measure Spaces," Second edition, Graduate Texts in Mathematics, 2, Springer-Verlag, New York-Berlin, 1980. [28] M. Ponce, Local dynamics for fibred holomorphic transformations, Nonlinearity, 20 (2007), 2939-2955. doi: 10.1088/0951-7715/20/12/011. [29] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590. [30] A. Quas, Rigidity of continuous coboundaries, Bull. London Math. Soc., 29 (1997), 595-600. doi: 10.1112/S0024609396002810. [31] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in "Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference" (State Univ. New York, Stony Brook, N.Y., 1978), Annals of Math. Studies, 97, Princeton Univ. Press, Princeton, N.J., (1981), 465-496. [32] P. Tukia, On quasiconformal groups, Journal d'Analyse Math., 46 (1986), 318-346. doi: 10.1007/BF02796595. [33] P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. [34] J.-C. Yoccoz, Some questions and remarks about $SL(2,\mathbbR)$ cocycles, in "Modern Dynamical Systems and Applications," Cambridge Univ. Press, Cambridge, (2004), 447-458.

show all references

##### References:
 [1] G. Atkinson, A class of transitive cylinder transformations, J. London Math. Soc. (2), 17 (1978), 263-270. doi: 10.1112/jlms/s2-17.2.263. [2] U. Bader, T. Gelander and N. Monod, A fixed point theorem for $L^1$ spaces, Inventiones Mathematicae, 189 (2012), 143-148. doi: 10.1007/s00222-011-0363-2. [3] U. Bader, A. Furman, T. Gelander and N. Monod, Property (T) and rigidity for actions on Banach spaces, Acta Math., 198 (2007), 57-105. doi: 10.1007/s11511-007-0013-0. [4] R. Baire, "Leçons sur les Fonctions Discontinues," Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1995. [5] A. Ballmann, "Lectures on Spaces of Nonpositive Curvature," DMV Seminar, 25, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9240-7. [6] S. Banach, "Théorie des Opérations Linéaires," Monografie Matematyczne, Volume 1, Warszawa, 1932. [7] M. R. Bridson and A. Haefliger, "Metric Spaces of Non-Positive Curvature," Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999. [8] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math., 41 (1972), 5-251. [9] D. Coronel, A. Navas and M. Ponce, On the dynamics of non-reducible cylindrical vortices, J. Lond. Math. Soc. (2), 85 (2012), 789-808. doi: 10.1112/jlms/jdr068. [10] W. H. Gottschalk and G. A. Hedlund, "Topological Dynamics," American Mathematical Society Colloquium Publications, Vol. 36, Amer. Math. Soc., Providence, R. I., 1955. [11] M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5-233. [12] M. Jerison, The space of bounded maps into a Banach space, Annals of Math. (2), 52 (1950), 309-327. doi: 10.2307/1969472. [13] V. Kaimanovich, Double ergodicity of the Poisson boundary and applications to bounded cohomology, Geom. and Functional Analysis (GAFA), 13 (2003), 852-861. doi: 10.1007/s00039-003-0433-8. [14] B. Kalinin, Livšic theorem for matrix cocycles, Annals of Math. (2), 173 (2011), 1025-1042. doi: 10.4007/annals.2011.173.2.11. [15] B. Kalinin and V. Sadovskaya, Linear cocycles over hyperbolic systems and criteria of conformality, J. Mod. Dyn., 4 (2010), 419-441. doi: 10.3934/jmd.2010.4.419. [16] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. [17] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, in collaboration with E. A. Robinson, Jr., in "Smooth Ergodic Theory and its Applications" (Seattle, WA, 1999), Proceedings of the Symposia in Pure Mathematics, 69, Amer. Math. Soc., Providence, RI, (2001), 107-173. [18] I. Kornfeld and M. Lin, Coboundaries of irreducible Markov operators on $C(K)$, Israel J. of Mathematics, 97 (1997), 189-202. doi: 10.1007/BF02774036. [19] S. Lang, "Fundamentals of Differential Geometry," Graduate Texts in Mathematics, 191, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-0541-8. [20] V. Markovic, Quasisymmetric groups, J. Amer. Math. Soc., 19 (2006), 673-715. doi: 10.1090/S0894-0347-06-00518-2. [21] S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval-exchange maps, J. Amer. Math. Soc., 18 (2005), 823-872. doi: 10.1090/S0894-0347-05-00490-X. [22] J. Moulin Ollagnier and D. Pinchon, A note about Hedlund's theorem, in "Dynamical Systems," Vol. II-Warsaw, Astérisque, No. 50, Soc. Math. France, Paris, (1977), 311-313. [23] R. McCutcheon, The Gottschalk-Hedlund Theorem, Am. Math. Monthly, 106 (1999), 670-672. doi: 10.2307/2589497. [24] I. Namioka and E. Asplund, A geometric proof of Ryll-Nardzewski's fixed point theorem, Bull. Amer. Math. Soc., 73 (1967), 443-445. doi: 10.1090/S0002-9904-1967-11779-8. [25] A. Navas, Three remarks on one-dimensional bi-Lipschitz conjugacies,, unpublished note, (). [26] A. Navas, "Groups of Circle Diffeomorphisms," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2011. [27] J. C. Oxtoby, "Measure and Category. A Survey of the Analogies Between Topological and Measure Spaces," Second edition, Graduate Texts in Mathematics, 2, Springer-Verlag, New York-Berlin, 1980. [28] M. Ponce, Local dynamics for fibred holomorphic transformations, Nonlinearity, 20 (2007), 2939-2955. doi: 10.1088/0951-7715/20/12/011. [29] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590. [30] A. Quas, Rigidity of continuous coboundaries, Bull. London Math. Soc., 29 (1997), 595-600. doi: 10.1112/S0024609396002810. [31] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in "Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference" (State Univ. New York, Stony Brook, N.Y., 1978), Annals of Math. Studies, 97, Princeton Univ. Press, Princeton, N.J., (1981), 465-496. [32] P. Tukia, On quasiconformal groups, Journal d'Analyse Math., 46 (1986), 318-346. doi: 10.1007/BF02796595. [33] P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. [34] J.-C. Yoccoz, Some questions and remarks about $SL(2,\mathbbR)$ cocycles, in "Modern Dynamical Systems and Applications," Cambridge Univ. Press, Cambridge, (2004), 447-458.
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