Ziggurats and rotation numbers

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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

Abstract
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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.
[2]	D. Calegari, Dynamical forcing of circular groups, Trans. Amer. Math. Soc., 358 (2006), 3473-3491. doi: 10.1090/S0002-9947-05-03754-2.
[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.
[4]	D. Calegari, Faces of the scl norm ball, Geom. Topol., 13 (2009), 1313-1336. doi: 10.2140/gt.2009.13.1313.
[5]	D. Calegari, "scl," MSJ Memoirs, 20, Mathematical Society of Japan, Tokyo, 2009.
[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.
[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.
[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.
[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.
[10]	É. Ghys, Groups acting on the circle, Enseign. Math. (2), 47 (2001), 329-407.
[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.
[12]	M. Jankins and W. Neumann, Rotation numbers of products of circle homeomorphisms, Math. Ann., 271 (1985), 381-400. doi: 10.1007/BF01456075.
[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.
[14]	S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.
[15]	S. Matsumoto, Some remarks on foliated $S^1$ bundles, Invent. Math., 90 (1987), 343-358. doi: 10.1007/BF01388709.
[16]	W. de Melo and S. van Strien, "One-Dimensional Dynamics," Ergeb. der Math. und ihrer Grenz. (3), 25, Springer-Verlag, Berlin, 1993.
[17]	R. Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv., 69 (1994), 155-162. doi: 10.1007/BF02564479.
[18]	F. Przytycki and M. Urbański, Conformal fractals: Ergodic theory methods, LMS Lect. Note Ser., 371, Cambridge University Press, Cambridge, 2010.
[19]	G. Światek, Rational rotation numbers for maps of the circle, Comm. Math. Phys., 119 (1988), 109-128. doi: 10.1007/BF01218263.
[20]	W. Thurston, Three-manifolds, foliations and circles, I, preprint, arXiv:math/9712268.
[21]	M. Urbański, Parabolic Cantor sets, Fund. Math., 151 (1996), 241-277.
[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.

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References:

[1]	R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. No., 50 (1979), 11-25.
[2]	D. Calegari, Dynamical forcing of circular groups, Trans. Amer. Math. Soc., 358 (2006), 3473-3491. doi: 10.1090/S0002-9947-05-03754-2.
[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.
[4]	D. Calegari, Faces of the scl norm ball, Geom. Topol., 13 (2009), 1313-1336. doi: 10.2140/gt.2009.13.1313.
[5]	D. Calegari, "scl," MSJ Memoirs, 20, Mathematical Society of Japan, Tokyo, 2009.
[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.
[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.
[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.
[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.
[10]	É. Ghys, Groups acting on the circle, Enseign. Math. (2), 47 (2001), 329-407.
[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.
[12]	M. Jankins and W. Neumann, Rotation numbers of products of circle homeomorphisms, Math. Ann., 271 (1985), 381-400. doi: 10.1007/BF01456075.
[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.
[14]	S. Katok, "Fuchsian Groups," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.
[15]	S. Matsumoto, Some remarks on foliated $S^1$ bundles, Invent. Math., 90 (1987), 343-358. doi: 10.1007/BF01388709.
[16]	W. de Melo and S. van Strien, "One-Dimensional Dynamics," Ergeb. der Math. und ihrer Grenz. (3), 25, Springer-Verlag, Berlin, 1993.
[17]	R. Naimi, Foliations transverse to fibers of Seifert manifolds, Comment. Math. Helv., 69 (1994), 155-162. doi: 10.1007/BF02564479.
[18]	F. Przytycki and M. Urbański, Conformal fractals: Ergodic theory methods, LMS Lect. Note Ser., 371, Cambridge University Press, Cambridge, 2010.
[19]	G. Światek, Rational rotation numbers for maps of the circle, Comm. Math. Phys., 119 (1988), 109-128. doi: 10.1007/BF01218263.
[20]	W. Thurston, Three-manifolds, foliations and circles, I, preprint, arXiv:math/9712268.
[21]	M. Urbański, Parabolic Cantor sets, Fund. Math., 151 (1996), 241-277.
[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.

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