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Continuity of the SRB entropy of convex projective structures

Part of: Real and complex geometry

Published online by Cambridge University Press:  04 June 2020

PATRICK FOULON  and
INKANG KIM
PATRICK FOULON
Affiliation:
Aix-Marseille Université, CNRS, Société Mathématique de France, CIRM (Centre International de Rencontres Mathématiques), Marseille, France UMR 822, 163 avenue de Luminy, 13288 Marseille cedex 9, France (e-mail: [email protected])
INKANG KIM
Affiliation:
School of Mathematics, KIAS, Heogiro 85, Dongdaemen-gu, Seoul 02455, Korea (e-mail: [email protected])
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Abstract

The space of convex projective structures has been well studied with respect to the topological entropy. But, to better understand the geometry of the structure, we study the entropy of the Sinai–Ruelle–Bowen measure and show that it is a continuous function on the space of strictly convex real projective structures.

Keywords

real projective structureSinai–Ruelle–Bowen measureentropy

MSC classification

Primary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations
Secondary: 57S25: Groups acting on specific manifolds
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 8 , August 2021 , pp. 2369 - 2381
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Abramov, L.. On the entropy of a flow. Amer. Math. Soc. Transl. 49 (1966), 167–170.Google Scholar
Anosov, D. and Sinai, Y.. Some smooth ergodic systems. Russian Math. Surveys 22 (1967), 103–167.Google Scholar
Barreira, L. and Pesin, Y. B.. Nonuniform Hyperbolicity. Cambridge University Press, New York, 2007.Google Scholar
Benoist, Y.. Convexes divisibles I. Algebraic Groups and Arithmetic (Tata Institute of Fundamental Research Studies in Mathematics, 17). Tata Institute, Bombay, 2004, pp. 339–374.Google Scholar
Benoist, Y. and Hulin, D.. Cubic differentials and finite volume convex projective surfaces. J. Differential Geom. 98 (2014), 1–19.Google Scholar
Benzécri, J.-P.. Sur les variétés localement affines et localement projectives. Bull. Soc. Math. France 88 (1960), 229–332.Google Scholar
Bergeron, N. and Gelander, T.. A note on local rigidity. Geom. Dedicata 107 (2004), 111–131.Google Scholar
Bonahon, F. and Dreyer, G.. Parameterizing Hitchin components. Duke Math. J. 163(15) (2014), 2935–2975.Google Scholar
Bonahon, F. and Kim, I.. The Goldman and Fock–Goncharov coordinates for convex projective structures on surfaces. Geom. Dedicata 192 (2018), 43–55.Google Scholar
Bowen, R. and Ruelle, D.. The ergodic theory of axiom A flows. Invent. Math. 29 (1975), 181–202.Google Scholar
Calabi, E.. Complete affine hyperspheres I. Sympos. Math. 10 (1972), 19–38.Google Scholar
Cheng, S. and Yau, S.. On the regularity of the Monge–Ampére equation det(∂ 2 u/∂x i ∂sx u) = F (x, u). Comm. Pure Appl. Math. 30 (1977), 41–68.Google Scholar
Choi, S. and Goldman, W.. Convex real projective structures on closed surfaces are closed. Proc. Amer. Math. Soc. 118(2) (1993), 657–661.Google Scholar
Contreras, G.. Regularity of topological and metric entropy of hyperbolic flows. Math. Z. 210 (1992), 97–111.Google Scholar
Crampon, M.. Entropies of strictly convex projective manifolds. J. Mod. Dyn. 3 (2009), 511–547.Google Scholar
Crampon, M.. Dynamics and entropies of Hilbert metrics. Thèse, Université de Strasbourg, 2011.Google Scholar
Crampon, M.. The boundary of a divisible convex set. Publ. Mat. Urug. 14 (2013), 105–119.Google Scholar
Crampon, M.. Lyapunov exponents in Hilbert geometry. Ergod. Th. & Dynam. Sys. 34 (2014), 501–533.Google Scholar
Fock, V. and Goncharov, A.. Moduli spaces of convex projective structures on surfaces. Adv. Math. 208(1) (2007), 249–273.Google Scholar
Foulon, P.. Géométrie des équations différentielle du second ordre. Ann. Inst. Henri Poincaré 45 (1986), 1–28.Google Scholar
Foulon, P.. Estimation de l’entropie des systèmes lagrangiens sans points conjugués. Ann. Inst. Henri Poincaré Phys. Théor. 57(2) (1992), 117–146. With an appendix, ‘About Finsler geometry’.Google Scholar
Goldman, W.. Convex real projective structures on compact surfaces. J. Differential Geom. 31 (1990), 791–845.Google Scholar
Hitchin, N.. Lie groups and Teichmüller space. Topology 31 (1992), 449–473.Google Scholar
Kim, I. and Zhang, G.. Kähler metric on the space of convex real projective structures on surface. J. Differential Geom. 106 (2017), 127–137.Google Scholar
Kuiper, N.. On convex locally projective spaces. Convegno Internazionale di Geometria Differentiale. Cremonese, New York, 1954, pp. 200–213.Google Scholar
Labourie, F.. Anosov flows, surface groups and curves in projective space. Invent. Math. 165 (2006), 51–114.Google Scholar
Labourie, F.. Flat projective structures on surfaces and cubic holomorphic differentials. Pure Appl. Math. Q. 3 (2007), 1057–1099.Google Scholar
Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Ann. of Math. (2) 122 (1985), 509–574.Google Scholar
Loftin, J.. Affine spheres and convex RP n manifolds. Amer. J. Math. 123(2) (2001), 255–274.Google Scholar
Oseledec, V. I.. A multiplicative ergodic theorem. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
Parry, W.. Synchronisation of canonical measures for hyperbolic attractors. Comm. Math. Phys. 106 (1986), 267–275.Google Scholar
Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Brasil. Mat. 9 (1978), 83–87.Google Scholar