Transversely Cantor laminations as inverse limits

Alcalde Cuesta, Fernando; Lozano Rojo, Álvaro; Stadler, Marta

doi:10.1090/S0002-9939-2010-10665-2

Transversely Cantor laminations as inverse limits
HTML articles powered by AMS MathViewer

by Fernando Alcalde Cuesta, Álvaro Lozano Rojo and Marta Macho Stadler PDF

Proc. Amer. Math. Soc. 139 (2011), 2615-2630 Request permission

Abstract:

We demonstrate that any minimal transversely Cantor compact lamination of dimension $p$ and class $C^1$ without holonomy is an inverse limit of compact branched manifolds of dimension $p$. To prove this result, we extend the triangulation theorem for $C^1$ manifolds to transversely Cantor $C^1$ laminations. In fact, we give a simple proof of this classical theorem based on the existence of $C^1$-compatible differentiable structures of class $C^\infty$.

References

Fernando Alcalde Cuesta, Alvaro Lozano Rojo, and Marta Macho Stadler, Dynamique transverse de la lamination de Ghys-Kenyon, Astérisque 323 (2009), 1–16 (French, with English and French summaries). MR 2647961
Fernando Alcalde Cuesta, Pablo González Sequeiros, and Álvaro Lozano Rojo, Affability of Euclidean tilings, C. R. Math. Acad. Sci. Paris 347 (2009), no. 15-16, 947–952 (English, with English and French summaries). MR 2542900, DOI 10.1016/j.crma.2009.06.011
Jean Bellissard, Riccardo Benedetti, and Jean-Marc Gambaudo, Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys. 261 (2006), no. 1, 1–41. MR 2193205, DOI 10.1007/s00220-005-1445-z
Riccardo Benedetti and Jean-Marc Gambaudo, On the dynamics of $\Bbb G$-solenoids. Applications to Delone sets, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 673–691. MR 1992658, DOI 10.1017/S0143385702001578
Miguel Bermúdez and Gilbert Hector, Laminations hyperfinies et revêtements, Ergodic Theory Dynam. Systems 26 (2006), no. 2, 305–339 (French, with English and French summaries). MR 2218763, DOI 10.1017/S0143385705000519
E. Blanc. Examples of mixed minimal foliated spaces. Unpublished paper, 2002.
Stewart S. Cairns, On the triangulation of regular loci, Ann. of Math. (2) 35 (1934), no. 3, 579–587. MR 1503181, DOI 10.2307/1968752
Stewart S. Cairns, A simple triangulation method for smooth manifolds, Bull. Amer. Math. Soc. 67 (1961), 389–390. MR 149491, DOI 10.1090/S0002-9904-1961-10631-9
Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868, DOI 10.1090/gsm/023
Étienne Ghys, Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix, xi, 49–95 (French, with English and French summaries). MR 1760843
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085
André Haefliger, Foliations and compactly generated pseudogroups, Foliations: geometry and dynamics (Warsaw, 2000) World Sci. Publ., River Edge, NJ, 2002, pp. 275–295. MR 1882774, DOI 10.1142/9789812778246_{0}013
James L. Heitsch and Connor Lazarov, Homotopy invariance of foliation Betti numbers, Invent. Math. 104 (1991), no. 2, 321–347. MR 1098613, DOI 10.1007/BF01245079
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
Á. Lozano Rojo. Dinámica transversa de laminaciones definidas por grafos repetitivos. Ph.D. Thesis, UPV/EHU, 2008.
Á. Lozano Rojo. An example of a non-uniquely ergodic lamination. To appear in Ergodic Theory Dynam. Systems. DOI:10.1017/S0143385710000064
Calvin C. Moore and Claude L. Schochet, Global analysis on foliated spaces, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 9, Cambridge University Press, New York, 2006. MR 2202625
James R. Munkres, Elementary differential topology, Revised edition, Annals of Mathematics Studies, No. 54, Princeton University Press, Princeton, N.J., 1966. Lectures given at Massachusetts Institute of Technology, Fall, 1961. MR 0198479
James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128
Anthony Phillips and Dennis Sullivan, Geometry of leaves, Topology 20 (1981), no. 2, 209–218. MR 605659, DOI 10.1016/0040-9383(81)90039-2
Georges de Rham, Variétés différentiables. Formes, courants, formes harmoniques, Publications de l’Institut de Mathématique de l’Université de Nancago, III, Hermann, Paris, 1973 (French). Troisième édition revue et augmentée. MR 0346830
William Thurston, The theory of foliations of codimension greater than one, Comment. Math. Helv. 49 (1974), 214–231. MR 370619, DOI 10.1007/BF02566730
W. P. Thurston. The Geometry and Topology of Three-Manifolds. Electronic version, 2002. http://www.msri.org/publications/books/gt3m/
A. M. Veršik, A theorem on periodical Markov approximation in ergodic theory, Ergodic theory and related topics (Vitte, 1981) Math. Res., vol. 12, Akademie-Verlag, Berlin, 1982, pp. 195–206. MR 730788
André Weil, Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119–145 (French). MR 50280, DOI 10.1007/BF02564296
R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 169–203. MR 348794
J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math. (2) 41 (1940), 809–824. MR 2545, DOI 10.2307/1968861
Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148