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October  2011, 5(4): 711-746. doi: 10.3934/jmd.2011.5.711

Ziggurats and rotation numbers

Danny Calegari 1, and  Alden Walker 2,

1. 

DPMMS, University of Cambridge, Cambridge CB3 0WA, United Kingdom
2. 

Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, United States

Received  October 2011 Revised  November 2011 Published  March 2012

We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's theorem (i.e., the conjecture of Jankins-Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces.
Keywords: 1-dimensional dynamics., rationality, circle homeomorphism, Arnol’d tongues, rigidity, ziggurat, Rotation number.
Mathematics Subject Classification: Primary: 37E45, 58D05; Secondary: 58F1.
Citation: Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711
References:
[1]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. No., 50 (1979), 11-25.  Google Scholar

[2]

D. Calegari, Dynamical forcing of circular groups, Trans. Amer. Math. Soc., 358 (2006), 3473-3491. doi: 10.1090/S0002-9947-05-03754-2.  Google Scholar

[3]

D. Calegari, Stable commutator length is rational in free groups, Jour. Amer. Math. Soc., 22 (2009), 941-961. doi: 10.1090/S0894-0347-09-00634-1.  Google Scholar

[4]

D. Calegari, Faces of the scl norm ball, Geom. Topol., 13 (2009), 1313-1336. doi: 10.2140/gt.2009.13.1313.  Google Scholar

[5]

D. Calegari, "scl," MSJ Memoirs, 20, Mathematical Society of Japan, Tokyo, 2009.  Google Scholar

[6]

D. Calegari and K. Fujiwara, Combable functions, quasimorphisms, and the central limit theorem, Erg. Theory Dyn. Sys., 30 (2010), 1343-1369. doi: 10.1017/S0143385709000662.  Google Scholar

[7]

D. Calegari and J. Louwsma, Immersed surfaces in the modular orbifold, Proc. Amer. Math. Soc., 139 (2011), 2295-2308. doi: 10.1090/S0002-9939-2011-10911-0.  Google Scholar

[8]

D. Eisenbud, U. Hirsch and W. Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv., 56 (1981), 638-660. doi: 10.1007/BF02566232.  Google Scholar

[9]

É. Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée, in "The Lefschetz Centennial Conference, Part III" (Mexico City, 1984), Contemp. Math., 58, III, Amer. Math. Soc., Providence, RI, (1987), 81-106. doi: 10.1090/conm/058.3/893858.  Google Scholar

[10]

É. Ghys, Groups acting on the circle, Enseign. Math. (2), 47 (2001), 329-407.  Google Scholar

[11]

M. Herman, Sur la conjugaison différentiable des diffeomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. No., 49 (1979), 5-233.  Google Scholar

[12]

M. Jankins and W. Neumann, Rotation numbers of products of circle homeomorphisms, Math. Ann., 271 (1985), 381-400. doi: 10.1007/BF01456075.  Google Scholar

[13]

A. Katok and B. Hasselblatt, "An Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[14]

S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.  Google Scholar

[15]

S. Matsumoto, Some remarks on foliated $S^1$ bundles, Invent. Math., 90 (1987), 343-358. doi: 10.1007/BF01388709.  Google Scholar

[16]

W. de Melo and S. van Strien, "One-Dimensional Dynamics," Ergeb. der Math. und ihrer Grenz. (3), 25, Springer-Verlag, Berlin, 1993.  Google Scholar

[17]

R. Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv., 69 (1994), 155-162. doi: 10.1007/BF02564479.  Google Scholar

[18]

F. Przytycki and M. Urbański, Conformal fractals: Ergodic theory methods, LMS Lect. Note Ser., 371, Cambridge University Press, Cambridge, 2010.  Google Scholar

[19]

G. Światek, Rational rotation numbers for maps of the circle, Comm. Math. Phys., 119 (1988), 109-128. doi: 10.1007/BF01218263.  Google Scholar

[20]

W. Thurston, Three-manifolds, foliations and circles, I,, preprint, ().   Google Scholar

[21]

M. Urbański, Parabolic Cantor sets, Fund. Math., 151 (1996), 241-277.  Google Scholar

[22]

J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup. (4), 17 (1984), 333-359.  Google Scholar

show all references

References:
[1]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. No., 50 (1979), 11-25.  Google Scholar

[2]

D. Calegari, Dynamical forcing of circular groups, Trans. Amer. Math. Soc., 358 (2006), 3473-3491. doi: 10.1090/S0002-9947-05-03754-2.  Google Scholar

[3]

D. Calegari, Stable commutator length is rational in free groups, Jour. Amer. Math. Soc., 22 (2009), 941-961. doi: 10.1090/S0894-0347-09-00634-1.  Google Scholar

[4]

D. Calegari, Faces of the scl norm ball, Geom. Topol., 13 (2009), 1313-1336. doi: 10.2140/gt.2009.13.1313.  Google Scholar

[5]

D. Calegari, "scl," MSJ Memoirs, 20, Mathematical Society of Japan, Tokyo, 2009.  Google Scholar

[6]

D. Calegari and K. Fujiwara, Combable functions, quasimorphisms, and the central limit theorem, Erg. Theory Dyn. Sys., 30 (2010), 1343-1369. doi: 10.1017/S0143385709000662.  Google Scholar

[7]

D. Calegari and J. Louwsma, Immersed surfaces in the modular orbifold, Proc. Amer. Math. Soc., 139 (2011), 2295-2308. doi: 10.1090/S0002-9939-2011-10911-0.  Google Scholar

[8]

D. Eisenbud, U. Hirsch and W. Neumann, Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helv., 56 (1981), 638-660. doi: 10.1007/BF02566232.  Google Scholar

[9]

É. Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée, in "The Lefschetz Centennial Conference, Part III" (Mexico City, 1984), Contemp. Math., 58, III, Amer. Math. Soc., Providence, RI, (1987), 81-106. doi: 10.1090/conm/058.3/893858.  Google Scholar

[10]

É. Ghys, Groups acting on the circle, Enseign. Math. (2), 47 (2001), 329-407.  Google Scholar

[11]

M. Herman, Sur la conjugaison différentiable des diffeomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. No., 49 (1979), 5-233.  Google Scholar

[12]

M. Jankins and W. Neumann, Rotation numbers of products of circle homeomorphisms, Math. Ann., 271 (1985), 381-400. doi: 10.1007/BF01456075.  Google Scholar

[13]

A. Katok and B. Hasselblatt, "An Introduction to the Modern Theory of Dynamical Systems," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[14]

S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.  Google Scholar

[15]

S. Matsumoto, Some remarks on foliated $S^1$ bundles, Invent. Math., 90 (1987), 343-358. doi: 10.1007/BF01388709.  Google Scholar

[16]

W. de Melo and S. van Strien, "One-Dimensional Dynamics," Ergeb. der Math. und ihrer Grenz. (3), 25, Springer-Verlag, Berlin, 1993.  Google Scholar

[17]

R. Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv., 69 (1994), 155-162. doi: 10.1007/BF02564479.  Google Scholar

[18]

F. Przytycki and M. Urbański, Conformal fractals: Ergodic theory methods, LMS Lect. Note Ser., 371, Cambridge University Press, Cambridge, 2010.  Google Scholar

[19]

G. Światek, Rational rotation numbers for maps of the circle, Comm. Math. Phys., 119 (1988), 109-128. doi: 10.1007/BF01218263.  Google Scholar

[20]

W. Thurston, Three-manifolds, foliations and circles, I,, preprint, ().   Google Scholar

[21]

M. Urbański, Parabolic Cantor sets, Fund. Math., 151 (1996), 241-277.  Google Scholar

[22]

J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup. (4), 17 (1984), 333-359.  Google Scholar

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