American Institute of Mathematical Sciences

July  2016, 36(7): 3993-4014. doi: 10.3934/dcds.2016.36.3993

Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard

 1 Department of Mathematics and Statistics, The University of Western Australia, Perth, Australia

Received  April 2015 Revised  November 2015 Published  March 2016

We consider open billiards in the plane satisfying the no-eclipse condition. We show that the points in the non-wandering set depend differentiably on deformations to the boundary of the billiard. We use Bowen's equation to estimate the Hausdorff dimension of the non-wandering set of the billiard. Finally we show that the Hausdorff dimension depends differentiably on sufficiently smooth deformations to the boundary of the billiard, and estimate the derivative with respect to such deformations.
Citation: Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993
References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin Cummings, Reading, MA, 1978. [2] L. F. A. Arbogast, Du Calcul Des Dérivations, LeVrault Frères, Strasbourg, 1800. [3] C. Arzelà, Sulle funzioni di linee, Mem. Accad. Sci. Bologna, 5 (1895), 55-74. [4] L. Barreira and K. Gelfert, Dimension estimates in smooth dynamics: A survey of recent results, Ergod. Theory and Dyn. Syst., 31 (2011), 641-671. doi: 10.1017/S014338571000012X. [5] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Theory and Dyn. Syst., 16 (1996), 871-927. doi: 10.1017/S0143385700010117. [6] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590. [7] R. Bowen, Hausdorff dimension of quasi-circles, Pub. Math. de l'IHÉS, 50 (1979), 11-25. [8] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, vol. 127. Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127. [9] M. Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier, 38 (1988), 113-146. doi: 10.5802/aif.1137. [10] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54. Cambridge Univ. Press, 1995. doi: 10.1017/CBO9780511809187. [11] A. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Inventiones Math., 98 (1989), 581-597. doi: 10.1007/BF01393838. [12] L. Kaup and B. Kaup, Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory, vol. 3. Walter de Gruyter, 1983. doi: 10.1515/9783110838350. [13] R. Kenny, Estimates of Hausdorff dimension for the non-wandering set of an open planar billiard, Can. J. Math., 56 (2004), 115-133. doi: 10.4153/CJM-2004-006-8. [14] A. Lopes and R. Markarian, Open billiards: Invariant and conditionally invariant probabilities on cantor sets, SIAM J. on App. Math., 56 (1996), 651-680. doi: 10.1137/S0036139995279433. [15] R. Mañé, The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces, Bol. Soc. Bras. Mat., 20 (1990), 1-24. doi: 10.1007/BF02585431. [16] H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergod. Theory and Dyn. Syst., 3 (1983), 251-260. doi: 10.1017/S0143385700001966. [17] R. L. Mishkov, Generalization of the formula of Faá di Bruno for a composite function with a vector argument, Int. J. Math. and Math. Sci., 24 (2000), 481-491. doi: 10.1155/S0161171200002970. [18] T. Morita, The symbolic representation of billiards without boundary condition, Tran. Amer. Math. Soc., 325 (1991), 819-828. doi: 10.1090/S0002-9947-1991-1013334-6. [19] W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérisque, vol. 187-188. Soc. Math. France, Montrouge, 1990. [20] Ya. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lect. Math. Uni. of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [21] P. Richardson, Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems, Diss. Univ. of North Texas, 1998. [22] V. Petkov and L. Stoyanov, Geometry of Reflecting Rays and Inverse Spectral Problems, Wiley, Chichester, 1992. [23] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 3rd ed., 1964. [24] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241. doi: 10.1007/s002200050134. [25] Ya. G. Sinai, Dynamical systems with elastic reflections, Russian Math. Surv., 25 (1970), 137-191. [26] J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J., 60 (1990), 1-57. doi: 10.1215/S0012-7094-90-06001-6. [27] L. Stoyanov, An estimate from above of the number of periodic orbits for semi-dispersed billiards, Comm. Math. Phys., 124 (1989), 217-227. doi: 10.1007/BF01219195. [28] L. Stoyanov, Exponential instability for a class of dispersing billiards, Ergod. Theory and Dyn. Syst., 19 (1999), 201-226. doi: 10.1017/S0143385799126543. [29] J. M. Varah, A lower bound for the smallest singular value of a matrix, Linear Alg. and Its Appl., 11 (1975), 3-5. doi: 10.1016/0024-3795(75)90112-3. [30] P. Walters, An Introduction to Ergodic Theory, vol. 79. Springer, New York, 1982. [31] P. Wright, Estimates of Hausdorff dimension for non-wandering sets of higher dimensional open billiards, Can. J. Math., 65 (2013), 1384-1400. doi: 10.4153/CJM-2013-030-0.

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References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin Cummings, Reading, MA, 1978. [2] L. F. A. Arbogast, Du Calcul Des Dérivations, LeVrault Frères, Strasbourg, 1800. [3] C. Arzelà, Sulle funzioni di linee, Mem. Accad. Sci. Bologna, 5 (1895), 55-74. [4] L. Barreira and K. Gelfert, Dimension estimates in smooth dynamics: A survey of recent results, Ergod. Theory and Dyn. Syst., 31 (2011), 641-671. doi: 10.1017/S014338571000012X. [5] L. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Theory and Dyn. Syst., 16 (1996), 871-927. doi: 10.1017/S0143385700010117. [6] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. doi: 10.2307/2373590. [7] R. Bowen, Hausdorff dimension of quasi-circles, Pub. Math. de l'IHÉS, 50 (1979), 11-25. [8] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, vol. 127. Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/surv/127. [9] M. Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier, 38 (1988), 113-146. doi: 10.5802/aif.1137. [10] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54. Cambridge Univ. Press, 1995. doi: 10.1017/CBO9780511809187. [11] A. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Inventiones Math., 98 (1989), 581-597. doi: 10.1007/BF01393838. [12] L. Kaup and B. Kaup, Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory, vol. 3. Walter de Gruyter, 1983. doi: 10.1515/9783110838350. [13] R. Kenny, Estimates of Hausdorff dimension for the non-wandering set of an open planar billiard, Can. J. Math., 56 (2004), 115-133. doi: 10.4153/CJM-2004-006-8. [14] A. Lopes and R. Markarian, Open billiards: Invariant and conditionally invariant probabilities on cantor sets, SIAM J. on App. Math., 56 (1996), 651-680. doi: 10.1137/S0036139995279433. [15] R. Mañé, The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces, Bol. Soc. Bras. Mat., 20 (1990), 1-24. doi: 10.1007/BF02585431. [16] H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergod. Theory and Dyn. Syst., 3 (1983), 251-260. doi: 10.1017/S0143385700001966. [17] R. L. Mishkov, Generalization of the formula of Faá di Bruno for a composite function with a vector argument, Int. J. Math. and Math. Sci., 24 (2000), 481-491. doi: 10.1155/S0161171200002970. [18] T. Morita, The symbolic representation of billiards without boundary condition, Tran. Amer. Math. Soc., 325 (1991), 819-828. doi: 10.1090/S0002-9947-1991-1013334-6. [19] W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérisque, vol. 187-188. Soc. Math. France, Montrouge, 1990. [20] Ya. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lect. Math. Uni. of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [21] P. Richardson, Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems, Diss. Univ. of North Texas, 1998. [22] V. Petkov and L. Stoyanov, Geometry of Reflecting Rays and Inverse Spectral Problems, Wiley, Chichester, 1992. [23] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 3rd ed., 1964. [24] D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241. doi: 10.1007/s002200050134. [25] Ya. G. Sinai, Dynamical systems with elastic reflections, Russian Math. Surv., 25 (1970), 137-191. [26] J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J., 60 (1990), 1-57. doi: 10.1215/S0012-7094-90-06001-6. [27] L. Stoyanov, An estimate from above of the number of periodic orbits for semi-dispersed billiards, Comm. Math. Phys., 124 (1989), 217-227. doi: 10.1007/BF01219195. [28] L. Stoyanov, Exponential instability for a class of dispersing billiards, Ergod. Theory and Dyn. Syst., 19 (1999), 201-226. doi: 10.1017/S0143385799126543. [29] J. M. Varah, A lower bound for the smallest singular value of a matrix, Linear Alg. and Its Appl., 11 (1975), 3-5. doi: 10.1016/0024-3795(75)90112-3. [30] P. Walters, An Introduction to Ergodic Theory, vol. 79. Springer, New York, 1982. [31] P. Wright, Estimates of Hausdorff dimension for non-wandering sets of higher dimensional open billiards, Can. J. Math., 65 (2013), 1384-1400. doi: 10.4153/CJM-2013-030-0.
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