Hausdorff dimension and asymptotic cycles

Pollicott, Mark

doi:10.1090/S0002-9947-03-03308-7

Hausdorff dimension and asymptotic cycles
HTML articles powered by AMS MathViewer

by Mark Pollicott PDF

Trans. Amer. Math. Soc. 355 (2003), 3241-3252 Request permission

Abstract:

We carry out a multifractal analysis for the asymptotic cycles for compact Riemann surfaces of genus $g \geq 2$. This describes the set of unit tangent vectors for which the associated orbit has a given asymptotic cycle in homology.

References

V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
V. Bangert, Minimal measures and minimizing closed normal one-currents, Geom. Funct. Anal. 9 (1999), no. 3, 413–427. MR 1708452, DOI 10.1007/s000390050093
L. Barreira, B. Saussol, and J. Schmeling, Higher-dimensional multifractal analysis, J. Math. Pures Appl. 81 (2002), 67-91.
Thierry Bousch, Le poisson n’a pas d’arêtes, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 489–508 (French, with English and French summaries). MR 1785392, DOI 10.1016/S0246-0203(00)00132-1
Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460. MR 339281, DOI 10.2307/2373793
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989, DOI 10.1007/BFb0081279
D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), no. 3, 259–269. MR 1274115, DOI 10.1007/BF01896241
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
A. H. Fan and J. Schmeling, On fast Birkhoff averaging, Math. Proc. Cambridge Philos. Soc., to appear.
John M. Franks, Knots, links and symbolic dynamics, Ann. of Math. (2) 113 (1981), no. 3, 529–552. MR 621015, DOI 10.2307/2006996
Oliver Jenkinson, Frequency locking on the boundary of the barycentre set, Experiment. Math. 9 (2000), no. 2, 309–317. MR 1780215, DOI 10.1080/10586458.2000.10504655
Oliver Jenkinson, Rotation, entropy, and equilibrium states, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3713–3739. MR 1837256, DOI 10.1090/S0002-9947-01-02706-4
D. Massart, Ph.D. Thesis, (Lyon).
D. Massart, Stable norms of surfaces: local structure of the unit ball of rational directions, Geom. Funct. Anal. 7 (1997), no. 6, 996–1010. MR 1487751, DOI 10.1007/s000390050034
Greg McShane and Igor Rivin, Simple curves on hyperbolic tori, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 12, 1523–1528 (English, with English and French summaries). MR 1340065
Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. MR 1489237, DOI 10.7208/chicago/9780226662237.001.0001
Ya. B. Pesin and V. Sadovskaya, Multifractal analysis of conformal Axiom A flows, Comm. Math. Phys. 216 (2001), no. 2, 277–312. MR 1814848, DOI 10.1007/s002200000329
Yakov Pesin and Howard Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples, Chaos 7 (1997), no. 1, 89–106. MR 1439809, DOI 10.1063/1.166242
M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math. 15 (1973), 92–114. MR 339282, DOI 10.1007/BF02771776
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
Dennis Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255. MR 433464, DOI 10.1007/BF01390011