## 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

## Additional Information

**Mark Pollicott**- Affiliation: Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, England
- MR Author ID: 140805
- Received by editor(s): October 25, 2002
- Published electronically: April 16, 2003
- Additional Notes: I am very grateful to Howie Weiss and Luis Barreira for very useful conversations on multifractal analysis
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**355**(2003), 3241-3252 - MSC (2000): Primary 28A78, 37D35; Secondary 37D40, 55N10
- DOI: https://doi.org/10.1090/S0002-9947-03-03308-7
- MathSciNet review: 1974685