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

Ziggurats and rotation numbers

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.
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.

[2]

D. Calegari, Dynamical forcing of circular groups,, Trans. Amer. Math. Soc., 358 (2006), 3473. 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. doi: 10.1090/S0894-0347-09-00634-1.

[4]

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

[5]

D. Calegari, "scl,", MSJ Memoirs, 20 (2009).

[6]

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

[7]

D. Calegari and J. Louwsma, Immersed surfaces in the modular orbifold,, Proc. Amer. Math. Soc., 139 (2011), 2295. 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. doi: 10.1007/BF02566232.

[9]

É. Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée,, in, 58 (1987), 81. doi: 10.1090/conm/058.3/893858.

[10]

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

[11]

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

[12]

M. Jankins and W. Neumann, Rotation numbers of products of circle homeomorphisms,, Math. Ann., 271 (1985), 381. 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 (1995).

[14]

S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics, (1992).

[15]

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

[16]

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

[17]

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

[18]

F. Przytycki and M. Urbański, Conformal fractals: Ergodic theory methods,, LMS Lect. Note Ser., 371 (2010).

[19]

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

[20]

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

[21]

M. Urbański, Parabolic Cantor sets,, Fund. Math., 151 (1996), 241.

[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.

show all references

References:
[1]

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

[2]

D. Calegari, Dynamical forcing of circular groups,, Trans. Amer. Math. Soc., 358 (2006), 3473. 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. doi: 10.1090/S0894-0347-09-00634-1.

[4]

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

[5]

D. Calegari, "scl,", MSJ Memoirs, 20 (2009).

[6]

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

[7]

D. Calegari and J. Louwsma, Immersed surfaces in the modular orbifold,, Proc. Amer. Math. Soc., 139 (2011), 2295. 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. doi: 10.1007/BF02566232.

[9]

É. Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée,, in, 58 (1987), 81. doi: 10.1090/conm/058.3/893858.

[10]

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

[11]

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

[12]

M. Jankins and W. Neumann, Rotation numbers of products of circle homeomorphisms,, Math. Ann., 271 (1985), 381. 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 (1995).

[14]

S. Katok, "Fuchsian Groups,", Chicago Lectures in Mathematics, (1992).

[15]

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

[16]

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

[17]

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

[18]

F. Przytycki and M. Urbański, Conformal fractals: Ergodic theory methods,, LMS Lect. Note Ser., 371 (2010).

[19]

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

[20]

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

[21]

M. Urbański, Parabolic Cantor sets,, Fund. Math., 151 (1996), 241.

[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.

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