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  2019, 15: 41-93. doi: 10.3934/jmd.2019013

Global rigidity of conjugations for locally non-discrete subgroups of $ {\rm {Diff}}^{\omega} (S^1) $

Anas Eskif and  Julio C. Rebelo

Institut de Mathématiques de Toulouse, Université de Toulouse, 118 Route de Narbonne F-31062, Toulouse, France

Received  June 03, 2015 Revised  January 03, 2018 Published  February 2019

We prove a global topological rigidity theorem for locally $ C^2 $-non-discrete subgroups of $ {\rm {Diff}}^{\omega} (S^1) $.

Keywords: Locally non-discrete group, topological rigidity, stationary measures.
Mathematics Subject Classification: Primary: 37C85; Secondary: 22F05, 37E10.
Citation: Anas Eskif, Julio C. Rebelo. Global rigidity of conjugations for locally non-discrete subgroups of $ {\rm {Diff}}^{\omega} (S^1) $. Journal of Modern Dynamics, 2019, 15: 41-93. doi: 10.3934/jmd.2019013
References:
[1]

S. Alvarez, D. Filimonov, V. Kleptsyn, D. Malicet, C. Meniño, A. Navas and M. Triestino, Groups with infinitely many ends acting analytically on the circle, preprint, 2018, arXiv: 1506.03839.

[2]

V. Antonov, Model of processes of cyclic evolution type. Synchronisation by a random signal, Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron., 2 (1984), 67-76. 

[3]

V. Arnold, Small denominators I. Mappings of the circle onto itself, Translations of the American Mathematical Society (series 2), 46 (1965), 213-284. 

[4]

I. Baker, Fractional iteration near a fixpoint of multiplier 1, J. Australian Math. Soc., 4 (1964), 143-148.  doi: 10.1017/S144678870002334X.

[5]

R. Bartle, The Elements of Integration and Lebesgue measure, Wiley Classics Library, 1995. doi: 10.1002/9781118164471.

[6]

A. Candel and L. Conlon, Foliations. Ⅰ, Ⅱ, Graduate Studies in Mathematics, 23, 60. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/060.

[7]

C. Connell and R. Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, GAGA, 17 (2007), 707-769.  doi: 10.1007/s00039-007-0608-9.

[8]

B. Deroin, The Poisson boundary of a locally discrete group of diffeomorphisms of the circle, Ergodic Theory and Dynamical Systems, 33 (2013), 400-415.  doi: 10.1017/S0143385711001155.

[9]

B. Deroin, V. Kleptsyn and A. Navas, Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math., 199 (2007), 199-262.  doi: 10.1007/s11511-007-0020-1.

[10]

B. Deroin, V. Kleptsyn and A. Navas, Towards the solution of some fundamental questions concerning group actions on the circle and codimension one foliations, preprint, 2016, arXiv: 1312.4133v3.

[11]

B. Deroin, D. Filimonov, V. Kleptsyn and A. Navas, A paradigm for codimension 1 foliations, to appear in Advanced Studies in Pure Mathematics.

[12]

J. Écalle, Les fonctions résurgentes, Publ. Math. Orsay, Vol 1: 81-05, Vol 2: 81-06, Vol 3: 85-05, 1981, 1985.

[13]

Y. Eliashberg and W. Thurston, Confoliations, University Lecture Series, 13, Amer. Math. Soc., Providence, RI, 1998.

[14]

P. Elizarov, Y. Il'yashenko, A. Scherbakov and S. Voronin, Finitely generated groups of germs of one-dimensional conformal mappings and invariants for complex singular points of analytic foliations of the complex plane, Adv. in Soviet Math. 14 (1993) 57–105.

[15]

D. Filimonov and V. Kleptsyn, Structure of groups of circle diffeomorphisms with the property of fixing nonexpandable points, Funct. Anal. Appl., 46 (2012), 191-209.  doi: 10.1007/s10688-012-0025-1.

[16]

H. Furstenberg, Random walks and discrete subgroups of Lie groups, Advances in Probability and Related Topics 1, Dekker, New York 1 (1971), 1–63.

[17]

E. Ghys, Sur les groupes engendrés par des difféomorphismes proches de l'identité, Bol. Soc. Bras. Mat., 24 (1993), 137-178.  doi: 10.1007/BF01237675.

[18]

E. Ghys, Rigidité Différentiable des Groupes Fuchsiens, Publ. Math. I.H.E.S., 78 (1993), 163-185. 

[19]

E. Ghys, Groups acting on the circle, Enseign. Math., 47 (2001), 329-407. 

[20]

E. Ghys and P. de la Harpe, Sur les Groupes Hyperboliques d'aprés Mikhael Gromov, (Editors), Birkhäuser, Boston, 1990.

[21]

E. Ghys and V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv., 62 (1987), 185-239.  doi: 10.1007/BF02564445.

[22]

E. Ghys and T. Tsuboi, Différentiabilité des conjugaisons entre systémes dynamiques de dimension 1, Ann. Inst. Fourier (Grenoble), 38 (1988), 215-244.  doi: 10.5802/aif.1131.

[23]

G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, part B, Braunschweig, Friedr. Vieweg, 1987. doi: 10.1007/978-3-322-90161-3.

[24]

V. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2), 152 (2000), 659-692. doi: 10.2307/2661351.

[25]

V. Kleptsyn and M. Nal'ski, Convergence of orbits in random dynamical systems on the circle, Funct. Anal. Appl., 38 (2004), 267-282. 

[26]

J. Moser, On commuting circle maps and simultaneous Diophantine approximations, Math. Z., 205 (1990), 105-121.  doi: 10.1007/BF02571227.

[27]

I. Nakai, Separatrix for non solvable dynamics on $ {\mathbb C},0$, Ann. Inst. Fourier, 44 (1994), 569-599.  doi: 10.5802/aif.1410.

[28]

I. Nakai, A rigidity theorem for transverse dynamics of real analytic foliations of co-dimension one, (Complex analytic methods in dynamical systems), Astérisque, 222 (1994), 327-343. 

[29] A. Navas, Groups of Circle Diffeomorphisms, Chicago Lectures in Mathematics, University of Chicago Press, 2011.  doi: 10.7208/chicago/9780226569505.001.0001.
[30]

J. C. Rebelo, Ergodicity and rigidity for certain subgroups of $ {\rm Diff}^{\omega} (S^1)$, Ann. Sci. l'ENS (4), 32 (1999), 433–453. doi: 10.1016/S0012-9593(99)80019-6.

[31]

J. C. Rebelo, A theorem of measurable rigidity in $ {\rm Diff}^{\omega} (S^1)$, Ergodic Theory and Dynamical Systems, 21 (2001), 1525-1561.  doi: 10.1017/S0143385701001742.

[32]

J. C. Rebelo, Subgroups of $ {\rm Diff} ^{\infty}_+ (S^1)$ acting transitively on unordered 4-tuples, Transactions of the American Mathematical Society, 356 (2004), 4543-4557.  doi: 10.1090/S0002-9947-04-03466-X.

[33]

J. C. Rebelo, On the higher ergodic theory of certain non-discrete actions, Mosc. Math. J., 14 (2014), 385-423.  doi: 10.17323/1609-4514-2014-14-2-385-423.

[34]

J. C. Rebelo, On the structure of quasi-invariant measures for non-discrete subgroups of Diffω(S1), Proc. Lond. Math. Soc. (3), 107 (2013), 932-964.  doi: 10.1112/plms/pdt002.

[35]

A. A. Shcherbakov, On the density of an orbit of a pseudogroup of conformal mappings and a generalization of the Hudai-Verenov theorem, Vestnik Movskovskogo Universiteta Mathematika, 31 (1982), 10-15. 

[36]

M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory and Dynamical Systems, 5 (1985), 285-289.  doi: 10.1017/S014338570000290X.

[37]

S. Sternberg, Local Cn transformations of the real line, Duke Math. J., 24 (1957), 97-102.  doi: 10.1215/S0012-7094-57-02415-8.

[38]

D. Sullivan, Discrete conformal groups and measurable dynamics, Bulletin of the AMS (New Series), 6 (1982), 57-73.  doi: 10.1090/S0273-0979-1982-14966-7.

[39]

A. Vershik, Dynamic theory of growth in groups: Entropy, boundaries, examples, Russian Math. Surveys, 55 (2000), 667-733.  doi: 10.1070/rm2000v055n04ABEH000314.

[40]

J.-C. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Petits diviseurs en dimension 1, Astérisque, 231 (1995), 89-242. 

show all references

References:
[1]

S. Alvarez, D. Filimonov, V. Kleptsyn, D. Malicet, C. Meniño, A. Navas and M. Triestino, Groups with infinitely many ends acting analytically on the circle, preprint, 2018, arXiv: 1506.03839.

[2]

V. Antonov, Model of processes of cyclic evolution type. Synchronisation by a random signal, Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron., 2 (1984), 67-76. 

[3]

V. Arnold, Small denominators I. Mappings of the circle onto itself, Translations of the American Mathematical Society (series 2), 46 (1965), 213-284. 

[4]

I. Baker, Fractional iteration near a fixpoint of multiplier 1, J. Australian Math. Soc., 4 (1964), 143-148.  doi: 10.1017/S144678870002334X.

[5]

R. Bartle, The Elements of Integration and Lebesgue measure, Wiley Classics Library, 1995. doi: 10.1002/9781118164471.

[6]

A. Candel and L. Conlon, Foliations. Ⅰ, Ⅱ, Graduate Studies in Mathematics, 23, 60. American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/060.

[7]

C. Connell and R. Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, GAGA, 17 (2007), 707-769.  doi: 10.1007/s00039-007-0608-9.

[8]

B. Deroin, The Poisson boundary of a locally discrete group of diffeomorphisms of the circle, Ergodic Theory and Dynamical Systems, 33 (2013), 400-415.  doi: 10.1017/S0143385711001155.

[9]

B. Deroin, V. Kleptsyn and A. Navas, Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math., 199 (2007), 199-262.  doi: 10.1007/s11511-007-0020-1.

[10]

B. Deroin, V. Kleptsyn and A. Navas, Towards the solution of some fundamental questions concerning group actions on the circle and codimension one foliations, preprint, 2016, arXiv: 1312.4133v3.

[11]

B. Deroin, D. Filimonov, V. Kleptsyn and A. Navas, A paradigm for codimension 1 foliations, to appear in Advanced Studies in Pure Mathematics.

[12]

J. Écalle, Les fonctions résurgentes, Publ. Math. Orsay, Vol 1: 81-05, Vol 2: 81-06, Vol 3: 85-05, 1981, 1985.

[13]

Y. Eliashberg and W. Thurston, Confoliations, University Lecture Series, 13, Amer. Math. Soc., Providence, RI, 1998.

[14]

P. Elizarov, Y. Il'yashenko, A. Scherbakov and S. Voronin, Finitely generated groups of germs of one-dimensional conformal mappings and invariants for complex singular points of analytic foliations of the complex plane, Adv. in Soviet Math. 14 (1993) 57–105.

[15]

D. Filimonov and V. Kleptsyn, Structure of groups of circle diffeomorphisms with the property of fixing nonexpandable points, Funct. Anal. Appl., 46 (2012), 191-209.  doi: 10.1007/s10688-012-0025-1.

[16]

H. Furstenberg, Random walks and discrete subgroups of Lie groups, Advances in Probability and Related Topics 1, Dekker, New York 1 (1971), 1–63.

[17]

E. Ghys, Sur les groupes engendrés par des difféomorphismes proches de l'identité, Bol. Soc. Bras. Mat., 24 (1993), 137-178.  doi: 10.1007/BF01237675.

[18]

E. Ghys, Rigidité Différentiable des Groupes Fuchsiens, Publ. Math. I.H.E.S., 78 (1993), 163-185. 

[19]

E. Ghys, Groups acting on the circle, Enseign. Math., 47 (2001), 329-407. 

[20]

E. Ghys and P. de la Harpe, Sur les Groupes Hyperboliques d'aprés Mikhael Gromov, (Editors), Birkhäuser, Boston, 1990.

[21]

E. Ghys and V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv., 62 (1987), 185-239.  doi: 10.1007/BF02564445.

[22]

E. Ghys and T. Tsuboi, Différentiabilité des conjugaisons entre systémes dynamiques de dimension 1, Ann. Inst. Fourier (Grenoble), 38 (1988), 215-244.  doi: 10.5802/aif.1131.

[23]

G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, part B, Braunschweig, Friedr. Vieweg, 1987. doi: 10.1007/978-3-322-90161-3.

[24]

V. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2), 152 (2000), 659-692. doi: 10.2307/2661351.

[25]

V. Kleptsyn and M. Nal'ski, Convergence of orbits in random dynamical systems on the circle, Funct. Anal. Appl., 38 (2004), 267-282. 

[26]

J. Moser, On commuting circle maps and simultaneous Diophantine approximations, Math. Z., 205 (1990), 105-121.  doi: 10.1007/BF02571227.

[27]

I. Nakai, Separatrix for non solvable dynamics on $ {\mathbb C},0$, Ann. Inst. Fourier, 44 (1994), 569-599.  doi: 10.5802/aif.1410.

[28]

I. Nakai, A rigidity theorem for transverse dynamics of real analytic foliations of co-dimension one, (Complex analytic methods in dynamical systems), Astérisque, 222 (1994), 327-343. 

[29] A. Navas, Groups of Circle Diffeomorphisms, Chicago Lectures in Mathematics, University of Chicago Press, 2011.  doi: 10.7208/chicago/9780226569505.001.0001.
[30]

J. C. Rebelo, Ergodicity and rigidity for certain subgroups of $ {\rm Diff}^{\omega} (S^1)$, Ann. Sci. l'ENS (4), 32 (1999), 433–453. doi: 10.1016/S0012-9593(99)80019-6.

[31]

J. C. Rebelo, A theorem of measurable rigidity in $ {\rm Diff}^{\omega} (S^1)$, Ergodic Theory and Dynamical Systems, 21 (2001), 1525-1561.  doi: 10.1017/S0143385701001742.

[32]

J. C. Rebelo, Subgroups of $ {\rm Diff} ^{\infty}_+ (S^1)$ acting transitively on unordered 4-tuples, Transactions of the American Mathematical Society, 356 (2004), 4543-4557.  doi: 10.1090/S0002-9947-04-03466-X.

[33]

J. C. Rebelo, On the higher ergodic theory of certain non-discrete actions, Mosc. Math. J., 14 (2014), 385-423.  doi: 10.17323/1609-4514-2014-14-2-385-423.

[34]

J. C. Rebelo, On the structure of quasi-invariant measures for non-discrete subgroups of Diffω(S1), Proc. Lond. Math. Soc. (3), 107 (2013), 932-964.  doi: 10.1112/plms/pdt002.

[35]

A. A. Shcherbakov, On the density of an orbit of a pseudogroup of conformal mappings and a generalization of the Hudai-Verenov theorem, Vestnik Movskovskogo Universiteta Mathematika, 31 (1982), 10-15. 

[36]

M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory and Dynamical Systems, 5 (1985), 285-289.  doi: 10.1017/S014338570000290X.

[37]

S. Sternberg, Local Cn transformations of the real line, Duke Math. J., 24 (1957), 97-102.  doi: 10.1215/S0012-7094-57-02415-8.

[38]

D. Sullivan, Discrete conformal groups and measurable dynamics, Bulletin of the AMS (New Series), 6 (1982), 57-73.  doi: 10.1090/S0273-0979-1982-14966-7.

[39]

A. Vershik, Dynamic theory of growth in groups: Entropy, boundaries, examples, Russian Math. Surveys, 55 (2000), 667-733.  doi: 10.1070/rm2000v055n04ABEH000314.

[40]

J.-C. Yoccoz, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Petits diviseurs en dimension 1, Astérisque, 231 (1995), 89-242. 

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