# American Institute of Mathematical Sciences

October  2015, 35(10): 5055-5082. doi: 10.3934/dcds.2015.35.5055

## Computing Mather's $\beta$-function for Birkhoff billiards

 1 Department of Mathematics, Università degli Studi Roma Tor Vergata, Viale della Ricerca Scientifica 1, 00133 Rome, Italy

Received  September 2013 Revised  February 2014 Published  April 2015

This article is concerned with the study of Mather's $\beta$-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it can be related to the maximal perimeter of periodic orbits with a given rotation number, the so-called Marked length spectrum. After having recalled its main properties and its relevance to the study of the billiard dynamics, we stress its connections to some intriguing open questions: Birkhoff conjecture and the isospectral rigidity of convex billiards. Both these problems, in fact, can be conveniently translated into questions on this function. This motivates our investigation aiming at understanding its main features and properties. In particular, we provide an explicit representation of the coefficients of its (formal) Taylor expansion at zero, only in terms of the curvature of the boundary. In the case of integrable billiards, this result provides a representation formula for the $\beta$-function near $0$. Moreover, we apply and check these results in the case of circular and elliptic billiards.
Citation: Alfonso Sorrentino. Computing Mather's $\beta$-function for Birkhoff billiards. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5055-5082. doi: 10.3934/dcds.2015.35.5055
##### References:

show all references

##### References:
 [1] Alexandre Rocha, Mário Jorge Dias Carneiro. A dynamical condition for differentiability of Mather's average action. Journal of Geometric Mechanics, 2014, 6 (4) : 549-566. doi: 10.3934/jgm.2014.6.549 [2] Michael Khanevsky. Hofer's length spectrum of symplectic surfaces. Journal of Modern Dynamics, 2015, 9: 219-235. doi: 10.3934/jmd.2015.9.219 [3] Jianlu Zhang. Suspension of the billiard maps in the Lazutkin's coordinate. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2227-2242. doi: 10.3934/dcds.2017096 [4] Dmitry Dolgopyat, Dmitry Jakobson. On small gaps in the length spectrum. Journal of Modern Dynamics, 2016, 10: 339-352. doi: 10.3934/jmd.2016.10.339 [5] Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228 [6] Richard Evan Schwartz. Unbounded orbits for outer billiards I. Journal of Modern Dynamics, 2007, 1 (3) : 371-424. doi: 10.3934/jmd.2007.1.371 [7] Vladimir Dragović, Milena Radnović. Pseudo-integrable billiards and arithmetic dynamics. Journal of Modern Dynamics, 2014, 8 (1) : 109-132. doi: 10.3934/jmd.2014.8.109 [8] Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112 [9] Kaizhi Wang. Action minimizing stochastic invariant measures for a class of Lagrangian systems. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1211-1223. doi: 10.3934/cpaa.2008.7.1211 [10] Daniel Genin. Research announcement: Boundedness of orbits for trapezoidal outer billiards. Electronic Research Announcements, 2008, 15: 71-78. doi: 10.3934/era.2008.15.71 [11] Richard Evan Schwartz. Research announcement: unbounded orbits for outer billiards. Electronic Research Announcements, 2007, 14: 1-6. doi: 10.3934/era.2007.14.1 [12] Misha Bialy. Maximizing orbits for higher-dimensional convex billiards. Journal of Modern Dynamics, 2009, 3 (1) : 51-59. doi: 10.3934/jmd.2009.3.51 [13] Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109 [14] Gary Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 671-689. doi: 10.3934/dcds.2007.17.671 [15] Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013 [16] B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835 [17] Michel L. Lapidus, Robert G. Niemeyer. Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3719-3740. doi: 10.3934/dcds.2013.33.3719 [18] Dmitri Scheglov. Growth of periodic orbits and generalized diagonals for typical triangular billiards. Journal of Modern Dynamics, 2013, 7 (1) : 31-44. doi: 10.3934/jmd.2013.7.31 [19] Vincent Millot, Yannick Sire, Hui Yu. Minimizing fractional harmonic maps on the real line in the supercritical regime. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6195-6214. doi: 10.3934/dcds.2018266 [20] Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587

2018 Impact Factor: 1.143