Entropy of systolically extremal surfaces and asymptotic bounds

Ergodic Theory and Dynamical Systems

Volumn 25, Issue 4, 2005, Pages 1209-1220

  Katz, Mikhail G ^a   Sabourau, Stéphane ^b,c  

^a BAR ILAN UNIVERSITY   (Israel)

^b UNIVERSITÉ DE TOURS   (France)

^c SAINT JOSEPH S UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23444451555     PISSN: 01433857     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0143385704001014     Document Type: Article

References (39)

1. [Am04] B. Ammann. Dirac eigenvalue estimates on two-tori. J. Geom. Phys. 51 (2004), 372-386.
2. [Ba93] I. Babenko. Asymptotic invariants of smooth manifolds. RUSS. Acad. Sci. Izv. Math. 41 (1993), 1-38.
3. [Ba03] I. Babenko. Nature topologique des systoles unidimensionnelles. Preprint 03-07, 2003. Available at: http://www.math.univ-montp2.fr/lsprepub/ prepub.html
4. 2. Sémin. Théor. Spectr. Géom. Grenoble 22 (2004), 25-52.
5. [Bal04] F. Balacheff. Sur des problèmes de la géomeé trie systolique. Seémin. Théor. Spectr. Géom. Grenoble 22 (2004), 71-82.
6. [BB04] I. Babenko and F. Balacheff. Géométrie systolique des sommes connexes et des revêtements cycliques. Math. Ann. to appear.
7. [BCG95] G. Besson, G. Courtois and S. Gallot. Volume et entropie minimale des espaces localement symétriques. Invent. Math. 103 (1991), 417-445.
8. [BCIK04] V. Bangert, C. Croke, S. Ivanov and M. Katz. Boundary case of equality in optimal Loewner-type inequalities. Trans. Amer. Math. Soc., to appear. See arXiv:math.DG/0406008.
9. [BCIK05] V. Bangert, C. Croke, S. Ivanov and M. Katz. Filling area conjecture and ovalless real hyperelliptic surfaces. Geom. Funct. Anal., to appear. See arXiv:math.DG/0405583.
10. [Be8O] M. Berger. Une borne inférieure pour le volume d'une variété riemannienne en fonction du rayon d'injectivité. Ann. Inst. Fourier 30 (1980), 259-265.
11. [BK03] V. Bangert and M. Katz. Stable systolic inequalities and cohomology products. Comm. PureAppl. Math. 56 (2003), 979-997.
12. [BK04] V. Bangert and M. Katz. An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm. Comm. Anal. Geom. 12 (2004), 703-732.
13. [BM02] M. Burger, T. Gelander, A. Lubotzky and S. Mozes. Counting hyperbolic manifolds. Geom. Funct. Anal. 12(6) (2002), 1161-1173.
14. [BS02] B. Bollobás and E. Szemeredi. Girth of sparse graphs. J. Graph Theory 39 (2002), 194-200.
15. [BS94] P. Buser and P. Sarnak. On the period matrix of a Riemann surface of large genus. With an appendix by J. H. Conway and N. J. A. Sloane. Invent. Math. 117 (1994), 27-56.
16. [CK03] C. Croke and M. Katz. Universal volume bounds in Riemannian manifolds. Lectures on Geometry and Topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck (Harvard University, 3-5 May 2002) (Surveys in Differential Geometry, VIII). Ed. S. T. Yau. International Press, Somerville, MA, 2003, pp. 109-137.
17. [Cr80] C. Croke. Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. École Norm. Sup. 13 (1980), 519-535.
18. [Cr88] C. Croke. An isoembolic pinching theorem. Invent. Math. 92 (1988), 385-387.
19. [CT03] O. Cornea, G. Lupton, J. Oprea and D. Tanré. Lusternik-Schnirelmann Category (Mathematical Surveys and Monographs, 103). American Mathematical Society, Providence, RI, 2003.
20. [Du01] O. Durumeric. Growth of fundamental groups and isoembolic volume and diameter. Proc. Amer. Math. Soc. 130 (2001), 585-590.
21. [Ge04] T. Gelander. Homotopy type and volume of locally symmetric manifolds. Duke Math. J. 124(3) (2004), 459-515.
22. [Gr81] M. Gromov. Volume and bounded cohomology. Publ. Math. Inst. Hautes Études Sci. 56 (1981), 213-307.
23. [Gr83] M. Gromov. Filling Riemannian manifolds. J. Diff. Geom. 18 (1983), 1-147.
24. [Gr96] M. Gromov. Systoles and intersystolic inequalities. Actes de la Table Ronde de Géométrie Différentielle (Luminy 1992) (Séminaires et Congrés, 1). Soc. Math. France, Institut Henri Poincaré, Paris, 1996, pp. 291-362.
25. [Gr99] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces (Progress in Mathematics, 152). Birkháuser, Boston, MA, 1999.
26. [IK04] S. Ivanov and M. Katz. Generalized degree and optimal Loewner-type inequalities. Israel J. Math. 141 (2004), 221-233.
27. [Kato83] A. Katok. Entropy and closed geodesics. Ergod. Th. & Dynam. Sys. 2 (1983), 339-365.
28. [Katz03] M. Katz. Four-manifold systoles and surjectivity of period map. Comment. Math. Helv. 78 (2003), 772-786.
29. [KH95] A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. With a supplementary chapter by Katok and Leonardo Mendoza (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.
30. [KL04] M. Katz and C. Lescop. Filling area conjecture, optimal systolic inequalities, and the fiber class in abelian covers. Proceedings of Conference and Workshop in Memory of R. Brooks, held at the Technion (Israel Mathematical Conference Proceedings, Contemporary Mathematics). American Mathematical Society, Providence, RI, 2004, to appear. See arXiv:math.DG/0412011.
31. [Ko87] S. Kodani. On two-dimensional isosystolic inequalities. Kodai Math. J. 10 (1987), 314-327.
32. [KR04] M. Katz and Y. Rudyak. Lusternik-Schnirelmann category and systolic category of low dimensional manifolds. Comm. Pure Appl. Math., to appear. See arXiv:math.DG/0410456.
33. [KS04] M. Katz and S. Sabourau. Hyperelliptic surfaces are Loewner. Proc. Amer. Math. Soc., to appear. See arXiv:math.DG/0407009.
34. [KS05] M. Katz and S. Sabourau. An optimal systolic inequality for CAT(0) metrics in genus two. Preprint. See arXiv:math.DG/0501017.
35. [Ma79] A. Manning. Topological entropy for geodesic flows. Ann. Math. (2) 110 (1979), 567-573.
36. [Sa04] S. Sabourau. Systoles des surfaces plates singulières de genre deux. Math. Z. 247(4) (2004), 693-709.
37. [Sa05] S. Sabourau. Entropy and systoles on surfaces. Preprint.
38. [Sa06] S. Sabourau. Systolic volume and minimal entropy of aspherical manifolds. Preprint.
39. [Ya88] T. Yamaguchi. Homotopy type finiteness theorems for certain precompact families of Riemannian manifolds. Proc. Amer. Math. Soc. 102 (1988), 660-666.