Given a closed, orientable surface of constant negative curvature and genus $g\ge 2$, we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen–Series boundary maps. Each such map is defined for a particular fundamental polygon for the surface and a particular multi-parameter.

We survey two strikingly different recent results by the authors: topological entropy is constant in this entire family (“rigidity”), while measure-theoretic entropy varies within Teichmüller space, taking all values (“flexibility”) between zero and a maximum. This maximum is achieved on the surface that admits a regular fundamental $(8g-4)$-gon. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof—valid for a large set of multi-parameters—uses the realization of geodesic flow as a special flow over the natural extension of the boundary map. We obtain explicit formulas for both entropies. We also present some new details pertaining to specific multi-parameters and particular polygons.

Received:

Revised:

Published online:

DOI:
10.5802/mrr.10

Revised:

Published online:

Classification:
37D40, 37E10

Keywords: Fuchsian groups, boundary maps, entropy, topological entropy, flexibility, rigidity

Keywords: Fuchsian groups, boundary maps, entropy, topological entropy, flexibility, rigidity

Author's affiliations:

Adam Abrams ^{1};
Svetlana Katok ^{2};
Ilie Ugarcovici ^{3}

License: CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{MRR_2022__3__1_0, author = {Adam Abrams and Svetlana Katok and Ilie Ugarcovici}, title = {Rigidity and flexibility of entropies of boundary maps associated to {Fuchsian} groups}, journal = {Mathematics Research Reports}, pages = {1--19}, publisher = {MathOA foundation}, volume = {3}, year = {2022}, doi = {10.5802/mrr.10}, language = {en}, url = {https://mrr.centre-mersenne.org/articles/10.5802/mrr.10/} }

TY - JOUR AU - Adam Abrams AU - Svetlana Katok AU - Ilie Ugarcovici TI - Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups JO - Mathematics Research Reports PY - 2022 SP - 1 EP - 19 VL - 3 PB - MathOA foundation UR - https://mrr.centre-mersenne.org/articles/10.5802/mrr.10/ DO - 10.5802/mrr.10 LA - en ID - MRR_2022__3__1_0 ER -

%0 Journal Article %A Adam Abrams %A Svetlana Katok %A Ilie Ugarcovici %T Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups %J Mathematics Research Reports %D 2022 %P 1-19 %V 3 %I MathOA foundation %U https://mrr.centre-mersenne.org/articles/10.5802/mrr.10/ %R 10.5802/mrr.10 %G en %F MRR_2022__3__1_0

Adam Abrams; Svetlana Katok; Ilie Ugarcovici. Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups. Mathematics Research Reports, Volume 3 (2022), pp. 1-19. doi : 10.5802/mrr.10. https://mrr.centre-mersenne.org/articles/10.5802/mrr.10/

[1] Extremal parameters and their duals for boundary maps associated to Fuchsian groups, Illinois J. Math., Volume 65 (2021) no. 1, pp. 153-179 | DOI | MR | Zbl

[2] Adler and Flatto revisited: cross-sections for geodesic flow on compact surfaces of constant negative curvature, Studia Math., Volume 246 (2019) no. 2, pp. 167-202 | DOI | MR | Zbl

[3] Rigidity of topological entropy of boundary maps associated to Fuchsian groups (to appear in A Vision For Dynamics: The Legacy of Anatole Katok, Cambridge University Press. http://arxiv.org/abs/2101.10271)

[4] Flexibility of measure-theoretic entropy of boundary maps associated to Fuchsian groups, Ergodic Theory Dynam. Systems, Volume 42 (2022) no. 2, pp. 389-401 | DOI | MR | Zbl

[5] Topological entropy, Trans. Amer. Math. Soc., Volume 114 (1965), pp. 309-319 | DOI | MR | Zbl

[6] Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. (N.S.), Volume 25 (1991) no. 2, pp. 229-334 | DOI | MR | Zbl

[7] Combinatorial Dynamics and Entropy in Dimension One, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000, xvi+415 pages | DOI | MR

[8] Semiconjugacy to a map of a constant slope, Discrete Contin. Dyn. Syst. Ser. B, Volume 20 (2015) no. 10, pp. 3403-3413 | DOI | MR | Zbl

[9] Dehn’s algorithm revisited, with applications to simple curves on surfaces, Combinatorial group theory and topology (Alta, Utah, 1984) (Ann. of Math. Stud.), Volume 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 451-478 | DOI | MR | Zbl

[10] The geometry of Teichmüller space via geodesic currents, Invent. Math., Volume 92 (1988) no. 1, pp. 139-162 | DOI | MR | Zbl

[11] Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., Volume 153 (1971), pp. 401-414 | DOI | MR | Zbl

[12] Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, p. 11–25 | MR | Zbl

[13] Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 153-170 | DOI | Numdam | MR | Zbl

[14] A correlation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR, Volume 190 (1970), pp. 19-22 | MR

[15] Discontinuous Groups of Isometries in the Hyperbolic Plane, De Gruyter Studies in Mathematics, 29, Walter de Gruyter & Co., Berlin, 2003, xxii+364 pages (Edited and with a preface by Asmus L. Schmidt, Biography of the authors by Bent Fuglede) | MR

[16] Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn., Volume 8 (2014) no. 3-4, pp. 271-436 | DOI | MR | Zbl

[17]

(Lecture notes for the Summer School “Géométries à courbure négative ou nulle, groupes discrets et rigidités”, Institut Fourier, Université de Grenoble, June–July 2004)[18] Fuchsian groups and ergodic theory, Trans. Amer. Math. Soc., Volume 39 (1936) no. 2, pp. 299-314 | DOI | MR | Zbl

[19] Structure of attractors for boundary maps associated to Fuchsian groups, Geom. Dedicata, Volume 191 (2017), pp. 171-198 | DOI | MR | Zbl

[20] Correction to: Structure of attractors for boundary maps associated to Fuchsian groups, Geom. Dedicata, Volume 198 (2019), pp. 189-191 | DOI | MR | Zbl

[21] Riemannsche Mannigfaltigkeiten und nichteuklidische Raumformen, IV., Sitzungsberichte Deutsche Akademie von Wissenschaften (1929), pp. 414-557 | Zbl

[22] Isoperimetric inequalities on surfaces of constant curvature, Canad. J. Math., Volume 49 (1997) no. 6, pp. 1162-1187 | DOI | MR | Zbl

[23] New parameters for Fuchsian groups of genus $2$, Proc. Amer. Math. Soc., Volume 127 (1999) no. 12, pp. 3643-3652 | DOI | MR | Zbl

[24] Entropy of piecewise monotone mappings, Studia Math., Volume 67 (1980) no. 1, pp. 45-63 | DOI | MR | Zbl

[25] Horseshoes and entropy for piecewise continuous piecewise monotone maps, From phase transitions to chaos, World Sci. Publ., River Edge, NJ, 1992, pp. 489-500 | DOI | MR

[26] Intrinsic Markov chains, Trans. Amer. Math. Soc., Volume 112 (1964), pp. 55-66 | DOI | MR | Zbl

[27] Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc., Volume 122 (1966), pp. 368-378 | DOI | MR | Zbl

[28] Riemann surfaces with shortest geodesic of maximal length, Geom. Funct. Anal., Volume 3 (1993) no. 6, pp. 564-631 | DOI | MR | Zbl

[29] Teichmüller space and fundamental domains of Fuchsian groups, Enseign. Math. (2), Volume 45 (1999) no. 1-2, pp. 169-187 | MR | Zbl

[30] The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 171-202 | DOI | Numdam | MR | Zbl

[31] Three-dimensional Geometry and Topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997, x+311 pages (Edited by Silvio Levy) | MR

[32] On discrete groups of the unit disk and their isomorphisms, Ann. Acad. Sci. Fenn. Ser. A. I. (1972) no. 504, p. 45 pp. (errata insert) | MR | Zbl

[33] An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982, ix+250 pages | DOI | MR

[34] On the work of Roy Adler in ergodic theory and dynamical systems, Symbolic dynamics and its applications (New Haven, CT, 1991) (Contemp. Math.), Volume 135, Amer. Math. Soc., Providence, RI, 1992, pp. 19-32 | DOI | MR | Zbl

[35] Surfaces and Planar Discontinuous Groups, Lecture Notes in Mathematics, 835, Springer, Berlin, 1980, x+334 pages (Translated from the German by John Stillwell) | DOI | MR

[36] Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity, Volume 11 (1998) no. 5, pp. 1263-1276 | DOI | MR | Zbl

*Cited by Sources: *