# 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:
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##### References:
 [1] G. Atkinson, A class of transitive cylinder transformations,, J. London Math. Soc. (2), 17 (1978), 263. doi: 10.1112/jlms/s2-17.2.263. Google Scholar [2] U. Bader, T. Gelander and N. Monod, A fixed point theorem for $L^1$ spaces,, Inventiones Mathematicae, 189 (2012), 143. doi: 10.1007/s00222-011-0363-2. Google Scholar [3] U. Bader, A. Furman, T. Gelander and N. Monod, Property (T) and rigidity for actions on Banach spaces,, Acta Math., 198 (2007), 57. doi: 10.1007/s11511-007-0013-0. Google Scholar [4] R. Baire, "Leçons sur les Fonctions Discontinues,", Les Grands Classiques Gauthier-Villars, (1995). Google Scholar [5] A. Ballmann, "Lectures on Spaces of Nonpositive Curvature,", DMV Seminar, 25 (1995). doi: 10.1007/978-3-0348-9240-7. Google Scholar [6] S. Banach, "Théorie des Opérations Linéaires,", Monografie Matematyczne, 1 (1932). Google Scholar [7] M. R. Bridson and A. Haefliger, "Metric Spaces of Non-Positive Curvature,", Grundlehren der Mathematischen Wissenschaften, 319 (1999). Google Scholar [8] F. Bruhat and J. Tits, Groupes réductifs sur un corps local,, Inst. Hautes Études Sci. Publ. Math., 41 (1972), 5. Google Scholar [9] D. Coronel, A. Navas and M. Ponce, On the dynamics of non-reducible cylindrical vortices,, J. Lond. Math. Soc. (2), 85 (2012), 789. doi: 10.1112/jlms/jdr068. Google Scholar [10] W. H. Gottschalk and G. A. Hedlund, "Topological Dynamics,", American Mathematical Society Colloquium Publications, (1955). Google Scholar [11] M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5. Google Scholar [12] M. Jerison, The space of bounded maps into a Banach space,, Annals of Math. (2), 52 (1950), 309. doi: 10.2307/1969472. Google Scholar [13] V. Kaimanovich, Double ergodicity of the Poisson boundary and applications to bounded cohomology,, Geom. and Functional Analysis (GAFA), 13 (2003), 852. doi: 10.1007/s00039-003-0433-8. Google Scholar [14] B. Kalinin, Livšic theorem for matrix cocycles,, Annals of Math. (2), 173 (2011), 1025. doi: 10.4007/annals.2011.173.2.11. Google Scholar [15] B. Kalinin and V. Sadovskaya, Linear cocycles over hyperbolic systems and criteria of conformality,, J. Mod. Dyn., 4 (2010), 419. doi: 10.3934/jmd.2010.4.419. Google Scholar [16] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Encyclopedia of Mathematics and its Applications, 54 (1995). Google Scholar [17] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in collaboration with E. A. Robinson, 69 (2001), 107. Google Scholar [18] I. Kornfeld and M. Lin, Coboundaries of irreducible Markov operators on $C(K)$,, Israel J. of Mathematics, 97 (1997), 189. doi: 10.1007/BF02774036. Google Scholar [19] S. Lang, "Fundamentals of Differential Geometry,", Graduate Texts in Mathematics, 191 (1999). doi: 10.1007/978-1-4612-0541-8. Google Scholar [20] V. Markovic, Quasisymmetric groups,, J. Amer. Math. Soc., 19 (2006), 673. doi: 10.1090/S0894-0347-06-00518-2. Google Scholar [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. doi: 10.1090/S0894-0347-05-00490-X. Google Scholar [22] J. Moulin Ollagnier and D. Pinchon, A note about Hedlund's theorem,, in, (1977), 311. Google Scholar [23] R. McCutcheon, The Gottschalk-Hedlund Theorem,, Am. Math. Monthly, 106 (1999), 670. doi: 10.2307/2589497. Google Scholar [24] I. Namioka and E. Asplund, A geometric proof of Ryll-Nardzewski's fixed point theorem,, Bull. Amer. Math. Soc., 73 (1967), 443. doi: 10.1090/S0002-9904-1967-11779-8. Google Scholar [25] A. Navas, Three remarks on one-dimensional bi-Lipschitz conjugacies,, unpublished note, (). Google Scholar [26] A. Navas, "Groups of Circle Diffeomorphisms,", Chicago Lectures in Mathematics, (2011). Google Scholar [27] J. C. Oxtoby, "Measure and Category. A Survey of the Analogies Between Topological and Measure Spaces,", Second edition, 2 (1980). Google Scholar [28] M. Ponce, Local dynamics for fibred holomorphic transformations,, Nonlinearity, 20 (2007), 2939. doi: 10.1088/0951-7715/20/12/011. Google Scholar [29] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems,, Math. Z., 202 (1989), 559. doi: 10.1007/BF01221590. Google Scholar [30] A. Quas, Rigidity of continuous coboundaries,, Bull. London Math. Soc., 29 (1997), 595. doi: 10.1112/S0024609396002810. Google Scholar [31] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions,, in, 97 (1981), 465. Google Scholar [32] P. Tukia, On quasiconformal groups,, Journal d'Analyse Math., 46 (1986), 318. doi: 10.1007/BF02796595. Google Scholar [33] P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982). Google Scholar [34] J.-C. Yoccoz, Some questions and remarks about $SL(2,\mathbbR)$ cocycles,, in, (2004), 447. Google Scholar
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