October  2020, 40(10): 6061-6088. doi: 10.3934/dcds.2020259

Integrability of moduli and regularity of denjoy counterexamples

1. 

School of Mathematics, Korea Institute for Advanced Study (KIAS), Seoul, 02455, Korea

2. 

Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA

* Corresponding author: Thomas Koberda

Received  April 2020 Revised  May 2020 Published  June 2020

Fund Project: The first author is supported by a KIAS Individual Grant (MG073601) at Korea Institute for Advanced Study and by the National Research Foundation (2018R1A2B6004003). The second author is partially supported by an Alfred P. Sloan Foundation Research Fellowship, and by NSF Grant DMS-1711488

We study the regularity of exceptional actions of groups by $ C^{1, \alpha} $ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $ \alpha $. Let $ G $ be a finitely generated group admitting a $ C^{1, \alpha} $ action $ \rho $ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $ G $ has spherical growth bounded by $ c n^{d-1} $ and if the function $ 1/\alpha^d $ is integrable near zero, then under some mild technical assumptions on $ \alpha $, there is a sequence of exceptional $ C^{1, \alpha} $ actions of $ G $ which converge to $ \rho $ in the $ C^1 $ topology. As a consequence for a single diffeomorphism, we obtain that if the function $ 1/\alpha $ is integrable near zero, then there exists a $ C^{1, \alpha} $ exceptional diffeomorphism of the circle. This corollary accounts for all previously known moduli of continuity for derivatives of exceptional diffeomorphisms. We also obtain a partial converse to our main result. For finitely generated free abelian groups, the existence of an exceptional action, together with some natural hypotheses on the derivatives of group elements, puts integrability restrictions on the modulus $ \alpha $. These results are related to a long-standing question of D. McDuff concerning the length spectrum of exceptional $ C^1 $ diffeomorphisms of the circle.

Citation: Sang-hyun Kim, Thomas Koberda. Integrability of moduli and regularity of denjoy counterexamples. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 6061-6088. doi: 10.3934/dcds.2020259
References:
[1]

K. Athanassopoulos, Denjoy $C^1$ diffeomorphisms of the circle and McDuff's question, Expo. Math., 33 (2015), 48-66.  doi: 10.1016/j.exmath.2013.12.005.  Google Scholar

[2]

P. Bohl, Über die Hinsichtlich der Unabhängigen und Abhängigen Variabeln Periodische Differentialgleichung Erster Ordnung, Acta Math., 40 (1916), 321-336.  doi: 10.1007/BF02418549.  Google Scholar

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E. Breuillard and E. Le Donne, On the rate of convergence to the asymptotic cone for nilpotent groups and sub{F}insler geometry, Proc. Natl. Acad. Sci. USA, 110 (2013), 19220-19226.  doi: 10.1073/pnas.1203854109.  Google Scholar

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M. BucherR. Frigerio and T. Hartnick, A note on semi-conjugacy for circle actions, Enseign. Math., 62 (2016), 317-360.  doi: 10.4171/LEM/62-3/4-1.  Google Scholar

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D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math., 152 (2003), 149-204.  doi: 10.1007/s00222-002-0271-6.  Google Scholar

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J. Chang, S. Kim and T. Koberda, Algebraic structure of diffeomorphism groups of one-manifolds, preprint, arXiv: 1904.08793. Google Scholar

[7]

D. CoronelA. Navas and M. Ponce, On bounded cocycles of isometries over minimal dynamics, J. Mod. Dyn., 7 (2013), 45-74.  doi: 10.3934/jmd.2013.7.45.  Google Scholar

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P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.  Google Scholar

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A. Denjoy, Sur la continuité des fonctions analytiques singulières, Bull. Soc. Math. France, 60 (1932), 27-105.  doi: 10.24033/bsmf.1183.  Google Scholar

[10]

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

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B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups, Ergodic Theory Dynam. Systems, 23 (2003), 1467-1484.  doi: 10.1017/S0143385702001712.  Google Scholar

[12]

É. Ghys, Actions de réseaux sur le cercle, Invent. Math., 137 (1999), 199-231.  doi: 10.1007/s002220050329.  Google Scholar

[13]

G. R. Hall, A $C^{\infty }$ Denjoy counterexample, Ergodic Theory Dynam. Systems, 1 (1981), 261-272.  doi: 10.1017/S0143385700001243.  Google Scholar

[14]

M.-R. Herman, Sur la conjugaison différentiable des diffeomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5–233.  Google Scholar

[15]

J. Hu and D. P. Sullivan, Topological conjugacy of circle diffeomorphisms, Ergodic Theory Dynam. Systems, 17 (1997), 173-186.  doi: 10.1017/S0143385797061002.  Google Scholar

[16]

E. Jorquera, A universal nilpotent group of $C^1$ diffeomorphisms of the interval, Topology Appl., 159 (2012), 2115-2126.  doi: 10.1016/j.topol.2012.02.003.  Google Scholar

[17]

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. doi: 10.1017/CBO9780511809187.  Google Scholar

[18]

S.-H. Kim, T. Koberda and M. Mj, Flexibility of Group Actions on the Circle, Lecture Notes in Mathematics, 2231, Springer, Cham, 2019. doi: 10.1007/978-3-030-02855-8.  Google Scholar

[19]

A. Mann, How Groups Grow, London Mathematical Society Lecture Note Series, 395, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139095129.  Google Scholar

[20]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623813.  Google Scholar

[21]

D. McDuff, $C^{1}$-minimal subsets of the circle, Ann. Inst. Fourier (Grenoble), 31 (1981), 177-193.  doi: 10.5802/aif.822.  Google Scholar

[22]

A. V. Medvedev, On a concave differentiable majorant of a modulus of continuity, Real Anal. Exchange, 27 (2001/02), 123-129.  doi: 10.2307/44154112.  Google Scholar

[23]

A. Navas, Growth of groups and diffeomorphisms of the interval, Geom. Funct. Anal., 18 (2008), 988-1028.  doi: 10.1007/s00039-008-0667-6.  Google Scholar

[24]

A. Navas, Groups of Circle Diffeomorphisms, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2011. doi: 10.7208/chicago/9780226569505.001.0001.  Google Scholar

[25]

M. Stoll, On the asymptotics of the growth of 2-step nilpotent groups, J. London Math. Soc. (2), 58 (1998), 38–48. doi: 10.1112/S0024610798006371.  Google Scholar

[26]

D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, in American Mathematical Society Centennial Publications Vol. 3, Amer. Math. Soc., Providence, RI, 1992,417–466.  Google Scholar

[27]

T. Tsuboi, Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle, J. Math. Soc. Japan, 47 (1995), 1-30.  doi: 10.2969/jmsj/04710001.  Google Scholar

show all references

References:
[1]

K. Athanassopoulos, Denjoy $C^1$ diffeomorphisms of the circle and McDuff's question, Expo. Math., 33 (2015), 48-66.  doi: 10.1016/j.exmath.2013.12.005.  Google Scholar

[2]

P. Bohl, Über die Hinsichtlich der Unabhängigen und Abhängigen Variabeln Periodische Differentialgleichung Erster Ordnung, Acta Math., 40 (1916), 321-336.  doi: 10.1007/BF02418549.  Google Scholar

[3]

E. Breuillard and E. Le Donne, On the rate of convergence to the asymptotic cone for nilpotent groups and sub{F}insler geometry, Proc. Natl. Acad. Sci. USA, 110 (2013), 19220-19226.  doi: 10.1073/pnas.1203854109.  Google Scholar

[4]

M. BucherR. Frigerio and T. Hartnick, A note on semi-conjugacy for circle actions, Enseign. Math., 62 (2016), 317-360.  doi: 10.4171/LEM/62-3/4-1.  Google Scholar

[5]

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math., 152 (2003), 149-204.  doi: 10.1007/s00222-002-0271-6.  Google Scholar

[6]

J. Chang, S. Kim and T. Koberda, Algebraic structure of diffeomorphism groups of one-manifolds, preprint, arXiv: 1904.08793. Google Scholar

[7]

D. CoronelA. Navas and M. Ponce, On bounded cocycles of isometries over minimal dynamics, J. Mod. Dyn., 7 (2013), 45-74.  doi: 10.3934/jmd.2013.7.45.  Google Scholar

[8]

P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.  Google Scholar

[9]

A. Denjoy, Sur la continuité des fonctions analytiques singulières, Bull. Soc. Math. France, 60 (1932), 27-105.  doi: 10.24033/bsmf.1183.  Google Scholar

[10]

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

[11]

B. Farb and J. Franks, Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups, Ergodic Theory Dynam. Systems, 23 (2003), 1467-1484.  doi: 10.1017/S0143385702001712.  Google Scholar

[12]

É. Ghys, Actions de réseaux sur le cercle, Invent. Math., 137 (1999), 199-231.  doi: 10.1007/s002220050329.  Google Scholar

[13]

G. R. Hall, A $C^{\infty }$ Denjoy counterexample, Ergodic Theory Dynam. Systems, 1 (1981), 261-272.  doi: 10.1017/S0143385700001243.  Google Scholar

[14]

M.-R. Herman, Sur la conjugaison différentiable des diffeomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5–233.  Google Scholar

[15]

J. Hu and D. P. Sullivan, Topological conjugacy of circle diffeomorphisms, Ergodic Theory Dynam. Systems, 17 (1997), 173-186.  doi: 10.1017/S0143385797061002.  Google Scholar

[16]

E. Jorquera, A universal nilpotent group of $C^1$ diffeomorphisms of the interval, Topology Appl., 159 (2012), 2115-2126.  doi: 10.1016/j.topol.2012.02.003.  Google Scholar

[17]

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. doi: 10.1017/CBO9780511809187.  Google Scholar

[18]

S.-H. Kim, T. Koberda and M. Mj, Flexibility of Group Actions on the Circle, Lecture Notes in Mathematics, 2231, Springer, Cham, 2019. doi: 10.1007/978-3-030-02855-8.  Google Scholar

[19]

A. Mann, How Groups Grow, London Mathematical Society Lecture Note Series, 395, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139095129.  Google Scholar

[20]

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623813.  Google Scholar

[21]

D. McDuff, $C^{1}$-minimal subsets of the circle, Ann. Inst. Fourier (Grenoble), 31 (1981), 177-193.  doi: 10.5802/aif.822.  Google Scholar

[22]

A. V. Medvedev, On a concave differentiable majorant of a modulus of continuity, Real Anal. Exchange, 27 (2001/02), 123-129.  doi: 10.2307/44154112.  Google Scholar

[23]

A. Navas, Growth of groups and diffeomorphisms of the interval, Geom. Funct. Anal., 18 (2008), 988-1028.  doi: 10.1007/s00039-008-0667-6.  Google Scholar

[24]

A. Navas, Groups of Circle Diffeomorphisms, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2011. doi: 10.7208/chicago/9780226569505.001.0001.  Google Scholar

[25]

M. Stoll, On the asymptotics of the growth of 2-step nilpotent groups, J. London Math. Soc. (2), 58 (1998), 38–48. doi: 10.1112/S0024610798006371.  Google Scholar

[26]

D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, in American Mathematical Society Centennial Publications Vol. 3, Amer. Math. Soc., Providence, RI, 1992,417–466.  Google Scholar

[27]

T. Tsuboi, Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle, J. Math. Soc. Japan, 47 (1995), 1-30.  doi: 10.2969/jmsj/04710001.  Google Scholar

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