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.  Google Scholar

[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.  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), 57-105. doi: 10.1007/s11511-007-0013-0.  Google Scholar

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R. Baire, "Leçons sur les Fonctions Discontinues," Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1995.  Google Scholar

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A. Ballmann, "Lectures on Spaces of Nonpositive Curvature," DMV Seminar, 25, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9240-7.  Google Scholar

[6]

S. Banach, "Théorie des Opérations Linéaires," Monografie Matematyczne, Volume 1, Warszawa, 1932. Google Scholar

[7]

M. R. Bridson and A. Haefliger, "Metric Spaces of Non-Positive Curvature," Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999.  Google Scholar

[8]

F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math., 41 (1972), 5-251.  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), 789-808. doi: 10.1112/jlms/jdr068.  Google Scholar

[10]

W. H. Gottschalk and G. A. Hedlund, "Topological Dynamics," American Mathematical Society Colloquium Publications, Vol. 36, Amer. Math. Soc., Providence, R. I., 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), 5-233.  Google Scholar

[12]

M. Jerison, The space of bounded maps into a Banach space, Annals of Math. (2), 52 (1950), 309-327. 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), 852-861. doi: 10.1007/s00039-003-0433-8.  Google Scholar

[14]

B. Kalinin, Livšic theorem for matrix cocycles, Annals of Math. (2), 173 (2011), 1025-1042. 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), 419-441. 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, Cambridge University Press, Cambridge, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

V. Markovic, Quasisymmetric groups, J. Amer. Math. Soc., 19 (2006), 673-715. 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), 823-872. doi: 10.1090/S0894-0347-05-00490-X.  Google Scholar

[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.  Google Scholar

[23]

R. McCutcheon, The Gottschalk-Hedlund Theorem, Am. Math. Monthly, 106 (1999), 670-672. 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), 443-445. 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, University of Chicago Press, Chicago, IL, 2011.  Google Scholar

[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.  Google Scholar

[28]

M. Ponce, Local dynamics for fibred holomorphic transformations, Nonlinearity, 20 (2007), 2939-2955. 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), 559-608. doi: 10.1007/BF01221590.  Google Scholar

[30]

A. Quas, Rigidity of continuous coboundaries, Bull. London Math. Soc., 29 (1997), 595-600. doi: 10.1112/S0024609396002810.  Google Scholar

[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.  Google Scholar

[32]

P. Tukia, On quasiconformal groups, Journal d'Analyse Math., 46 (1986), 318-346. doi: 10.1007/BF02796595.  Google Scholar

[33]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  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), 57-105. doi: 10.1007/s11511-007-0013-0.  Google Scholar

[4]

R. Baire, "Leçons sur les Fonctions Discontinues," Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1995.  Google Scholar

[5]

A. Ballmann, "Lectures on Spaces of Nonpositive Curvature," DMV Seminar, 25, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9240-7.  Google Scholar

[6]

S. Banach, "Théorie des Opérations Linéaires," Monografie Matematyczne, Volume 1, Warszawa, 1932. Google Scholar

[7]

M. R. Bridson and A. Haefliger, "Metric Spaces of Non-Positive Curvature," Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999.  Google Scholar

[8]

F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math., 41 (1972), 5-251.  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), 789-808. doi: 10.1112/jlms/jdr068.  Google Scholar

[10]

W. H. Gottschalk and G. A. Hedlund, "Topological Dynamics," American Mathematical Society Colloquium Publications, Vol. 36, Amer. Math. Soc., Providence, R. I., 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), 5-233.  Google Scholar

[12]

M. Jerison, The space of bounded maps into a Banach space, Annals of Math. (2), 52 (1950), 309-327. 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), 852-861. doi: 10.1007/s00039-003-0433-8.  Google Scholar

[14]

B. Kalinin, Livšic theorem for matrix cocycles, Annals of Math. (2), 173 (2011), 1025-1042. 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), 419-441. 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, Cambridge University Press, Cambridge, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

V. Markovic, Quasisymmetric groups, J. Amer. Math. Soc., 19 (2006), 673-715. 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), 823-872. doi: 10.1090/S0894-0347-05-00490-X.  Google Scholar

[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.  Google Scholar

[23]

R. McCutcheon, The Gottschalk-Hedlund Theorem, Am. Math. Monthly, 106 (1999), 670-672. 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), 443-445. 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, University of Chicago Press, Chicago, IL, 2011.  Google Scholar

[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.  Google Scholar

[28]

M. Ponce, Local dynamics for fibred holomorphic transformations, Nonlinearity, 20 (2007), 2939-2955. 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), 559-608. doi: 10.1007/BF01221590.  Google Scholar

[30]

A. Quas, Rigidity of continuous coboundaries, Bull. London Math. Soc., 29 (1997), 595-600. doi: 10.1112/S0024609396002810.  Google Scholar

[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.  Google Scholar

[32]

P. Tukia, On quasiconformal groups, Journal d'Analyse Math., 46 (1986), 318-346. doi: 10.1007/BF02796595.  Google Scholar

[33]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[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.  Google Scholar

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