Abstract
In this short note we will describe an old problem and a new approach which casts light upon it. The old problem is to understand the nature of harmonic measures for cocompact Fuchsian groups. The new approach is to compute numerically the value of the drift and, in particular, get new results on the dimension of the measure in some new examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Avez. Théoréme de Choquet–Deny pour les groupes à croissance non exponentielle. C. R. Acad. Sci. Paris Sér. A279:2528 (1974).
B. Barany, M. Pollicott and K. Simon. Stationary measures for projective transformations. Journal of Statistical Physics148(3), 393–421 (2012).
A.F. Beardon. The geometry of discrete groups. Corrected reprint of the 1983 original. Graduate Texts in Mathematics 91. Springer-Verlag, New York (1995).
S. Blachére, P. Haissinsky and P. Mathieu. Harmonic measures versus quasiconformal measures for hyperbolic groups. Ann. Sci. Ec. Norm. Sup.44(4), 683–721 (2011).
O. Bolza. On binary sextics with linear transformations into themselves. Amer. J. Math.10 no. 1, 47–70 (1887).
J. Bourgain. On the Furstenberg measure and density of states for the Anderson–Bernoulli model at small disorder. Journal d’Analyse Mathématique117, 273–295 (2012).
V. Gadre, J. Maher and G. Tiozzo. Word length statistics for Teichmüler geodesics and singularity of harmonic measure. Comment. Math. Helv.92, no. 1, 1–36 (2017).
Schreiben Gauss an Wolfgang Bolyai, Göttingen, 2. 9. 1808. In: Franz Schmidt, Paul Stäckel (Hrsg.): Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai, B.G. Teubner, Leipzig, S. 94 (1899).
C.F. Gauß. Werke. Band VIII. (German) [Collected works. Vol. VIII] Reprint of the 1900 original. Georg Olms Verlag, Hildesheim (1973).
S. Gouëzel, F. Mathéus and F. Maucourant. Sharp lower bounds for the asymptotic entropy of symmetric random walks. Groups, Geometry, and Dynamics9, 711–735 (2015).
H. Ninnemann. Gutzwiller’s octagon and the triangular billiard T∗(2,  3,  8) as models for the quantization of chaotic systems by Selberg’s trace formula. Internat. J. Modern Phys. B9, no. 13–14, 1647–1753 (1995).
V. Kaimanovich and V. Le Prince. Matrix random products with singular harmonic measure. Geometriae Dedicata150, 257–279 (2011).
P. Kosenko. Fundamental inequality for hyperbolic Coxeter and Fuchsian groups equipped with geometric distances. arXiv preprint: arXiv:1911:00801 (2019).
P. Kosenko and G. Tiozzo. The fundamental inequality for cocompact fuchsian groups. arXiv preprint: arXiv:2012:07417 (2020).
T. Kuusalo and M. Näätänen. On arithmetic genus 2 subgroups of triangle groups. In: Extremal Riemann surfaces (San Francisco, CA, 1995), 21–28, Contemp. Math. 201, Amer. Math. Soc., Providence, RI (1997).
J. Stillwell. Mathematics and Its History. Springer, Berlin (2010).
R. Tanaka. Dimension of harmonic measures in hyperbolic spaces. Ergod. Th. and Dynam. Sys.39, 474–499 (2019).
Acknowledgements
The first author is partly supported by ERC-Advanced Grant 833802-Resonances and EPSRC grant EP/T001674/1 the second author is partly supported by EPSRC grant EP/T001674/1.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Pollicott, M., Vytnova, P. (2023). Groups, Drift and Harmonic Measures. In: Morel, JM., Teissier, B. (eds) Mathematics Going Forward . Lecture Notes in Mathematics, vol 2313. Springer, Cham. https://doi.org/10.1007/978-3-031-12244-6_21
Download citation
DOI: https://doi.org/10.1007/978-3-031-12244-6_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-12243-9
Online ISBN: 978-3-031-12244-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)