메뉴 건너뛰기




Volumn 8, Issue 1, 2003, Pages 1-13

Billiard dynamics: A survey with the emphasis on open problems

Author keywords

[No Author keywords available]

Indexed keywords


EID: 21944447975     PISSN: 15603547     EISSN: 14684845     Source Type: Journal    
DOI: 10.1070/rd2003v008n01abeh000222     Document Type: Short Survey
Times cited : (65)

References (84)
  • 1
    • 0039641251 scopus 로고
    • A remark on the topological entropy and invariant circles of an area preserving twist map
    • S. Angenent. A remark on the topological entropy and invariant circles of an area preserving twist map. Proceedings IMA. V. 44. 1992. P. 1-6.
    • (1992) Proceedings IMA , vol.44 , pp. 1-6
    • Angenent, S.1
  • 2
    • 4243693952 scopus 로고    scopus 로고
    • Numerical study BH on ergodic properties of triangular billiards
    • R. Artuso, G. Casati, I. Guarneri. Numerical study BH on ergodic properties of triangular billiards. Phys. Rev. E. V. 55. 1997. P. 6384-6390.
    • (1997) Phys. Rev. E , vol.55 , pp. 6384-6390
    • Artuso, R.1    Casati, G.2    Guarneri, I.3
  • 3
    • 51249169453 scopus 로고
    • Convex billiards and a theorem by E. Hopf
    • M. Bialy. Convex billiards and a theorem by E. Hopf. Math. Zeitschr. V. 214. 199.3. P. 147-154.
    • (1993) Math. Zeitschr. , vol.214 , pp. 147-154
    • Bialy, M.1
  • 4
    • 0004223966 scopus 로고
    • A.M.S., Providence
    • G. D.Birkhoff. Dynamical Systems. A.M.S., Providence. 1927. (available online; http://www.ams.org/bookstore/collseries).
    • (1927) Dynamical Systems
    • Birkhoff, G.D.1
  • 5
    • 84972546502 scopus 로고
    • A condition for minimal interval exchange maps to be uniquely ergodic
    • M. Boshernitzan. A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. V. 52. 1985. P. 733-752.
    • (1985) Duke Math. J. , vol.52 , pp. 733-752
    • Boshernitzan, M.1
  • 6
    • 85183829372 scopus 로고    scopus 로고
    • Unpublished Manuscript
    • M. Boshernitzan. Unpublished Manuscript.
    • Boshernitzan, M.1
  • 7
    • 34250395980 scopus 로고
    • On the ergodic properties of certain billiards
    • L. Bunimovich. On the ergodic properties of certain billiards. Func. Analys. & Appl. V. 8. 1974. P. 254-255.
    • (1974) Func. Analys. & Appl. , vol.8 , pp. 254-255
    • Bunimovich, L.1
  • 8
    • 0037805840 scopus 로고    scopus 로고
    • Billiards and other hyperbolic systems
    • (edited by Ya. Sinai). Springer-Verlag, New York
    • L. Bunimovich. Billiards and other hyperbolic systems, pp. 192 - 233 in Encyclopedia of Mathematical Sciences, vol. 100 (edited by Ya. Sinai). Springer-Verlag, New York. 2000.
    • (2000) Encyclopedia of Mathematical Sciences , vol.100 , pp. 192-233
    • Bunimovich, L.1
  • 9
    • 0002031610 scopus 로고
    • Markov partitions for two-dimensional hyperbolic billiards
    • L. Bunimovich. Ya. Sinai. N. Chernov. Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. V. 45. 1990. P. 105-152.
    • (1990) Russ. Math. Surv. , vol.45 , pp. 105-152
    • Bunimovich, L.1    Sinai, Ya.2    Chernov, N.3
  • 10
    • 84956084246 scopus 로고
    • Statistical properties of two-dimensional hyperbolic billiards
    • L. Bunimovich. Ya. Sinai. N. Chernov. Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. V. 46. 1991. P. 47-106.
    • (1991) Russ. Math. Surv. , vol.46 , pp. 47-106
    • Bunimovich, L.1    Sinai, Ya.2    Chernov, N.3
  • 11
    • 0000814298 scopus 로고    scopus 로고
    • Mixing property of triangular billiards
    • G. Casati, T. Prosen. Mixing property of triangular billiards. Phys. Rev. Lett. V. 83. 1999. P. 4729-4732.
    • (1999) Phys. Rev. Lett. , vol.83 , pp. 4729-4732
    • Casati, G.1    Prosen, T.2
  • 12
    • 0010107975 scopus 로고
    • Topological entropy and periodic points of two-dimensional hyperbolic billiards
    • N. Chernov. Topological entropy and periodic points of two-dimensional hyperbolic billiards. Funct. Anal. Appl. V. 25. 1991. P. 39-45.
    • (1991) Funct. Anal. Appl. , vol.25 , pp. 39-45
    • Chernov, N.1
  • 13
    • 0040360945 scopus 로고    scopus 로고
    • Entropy, Lyapunov exponents and mean-free path for billiards
    • N. Chernov. Entropy, Lyapunov exponents and mean-free path for billiards. J. Stat. Phys. V. 88. 1997. P. 1-29.
    • (1997) J. Stat. Phys. , vol.88 , pp. 1-29
    • Chernov, N.1
  • 14
    • 0033074305 scopus 로고    scopus 로고
    • Decay of correlations in dispersing billiards
    • N. Chernov. Decay of correlations in dispersing billiards. J. Stat. Phys. V. 94. 1999. P. 513-556.
    • (1999) J. Stat. Phys. , vol.94 , pp. 513-556
    • Chernov, N.1
  • 16
    • 35949005254 scopus 로고
    • Periodic trajectories EM in right-triangle billiards
    • B. Cipra, R. Hanson, E. Kolan. Periodic trajectories EM in right-triangle billiards. Phys. Rev. E. V. 52. 1995. P. 2066-2071.
    • (1995) Phys. Rev. E , vol.52 , pp. 2066-2071
    • Cipra, B.1    Hanson, R.2    Kolan, E.3
  • 17
    • 0035635339 scopus 로고    scopus 로고
    • Asymptotic formulas on flat surfaces
    • A. Eskin, H. Masur. Asymptotic formulas on flat surfaces. Erg. Theor. & Dyn. Sys. V. 21. 2001. P. 443-478.
    • (2001) Erg. Theor. & Dyn. Sys. , vol.21 , pp. 443-478
    • Eskin, A.1    Masur, H.2
  • 20
    • 0035621503 scopus 로고    scopus 로고
    • Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
    • A. Eskin, A. Okunkov. Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. V. 145. 2001. P. 59-103.
    • (2001) Invent. Math. , vol.145 , pp. 59-103
    • Eskin, A.1    Okunkov, A.2
  • 21
    • 84972506844 scopus 로고
    • Concerning the transitive properties of geodesies on a rational polyhedron
    • R. H. Fox, R. B. Kershner. Concerning the transitive properties of geodesies on a rational polyhedron. Duke Math. J. V. 2. 19.36. P. 147-150.
    • (1936) Duke Math. J. , vol.2 , pp. 147-150
    • Fox, R.H.1    Kershner, R.B.2
  • 22
    • 84956110926 scopus 로고
    • Periodic orbits in polygons: Generating mechanisms
    • G. Galperin, A. Stepin, Ya. Vorohets. Periodic orbits in polygons: generating mechanisms. Russ. Math. Surv. V. 47. 1992. P. 5-80.
    • (1992) Russ. Math. Surv. , vol.47 , pp. 5-80
    • Galperin, G.1    Stepin, A.2    Vorohets, Ya.3
  • 23
    • 0040515167 scopus 로고    scopus 로고
    • Three rods on a ring and the triangular billiard
    • S. L. Glashow, L. Mittag. Three rods on a ring and the triangular billiard. J. Stat. Phys. V. 87. 1997. P. 937-941.
    • (1997) J. Stat. Phys. , vol.87 , pp. 937-941
    • Glashow, S.L.1    Mittag, L.2
  • 24
    • 84958434552 scopus 로고
    • Billiard flows on almost integrable polyhedral surfaces
    • E. Gutkin. Billiard flows on almost integrable polyhedral surfaces. Erg. Theor. & Dyn. Sys. V. 4. 1984. P. 560-584.
    • (1984) Erg. Theor. & Dyn. Sys. , vol.4 , pp. 560-584
    • Gutkin, E.1
  • 25
    • 45949131024 scopus 로고
    • Billiards in polygons
    • E. Gutkin. Billiards in polygons. Physica D. V. 19. 1986 p. 311-333.
    • (1986) Physica D , vol.19 , pp. 311-333
    • Gutkin, E.1
  • 27
    • 0030541868 scopus 로고    scopus 로고
    • Billiards in polygons: Survey of recent results
    • E. Gutkin. Billiards in polygons: Survey of recent results. J. Stat. Phys. V. 8.3. 1996. P. 7-26.
    • (1996) J. Stat. Phys. , vol.83 , pp. 7-26
    • Gutkin, E.1
  • 28
    • 0031508438 scopus 로고    scopus 로고
    • Two applications of calculus to triangular billiards
    • E. Gutkin. Two applications of calculus to triangular billiards. Amer. Math. Monthly. V. 104. 1997. P. 618-623.
    • (1997) Amer. Math. Monthly , vol.104 , pp. 618-623
    • Gutkin, E.1
  • 29
    • 0031533730 scopus 로고    scopus 로고
    • Topological entropy of polygon exchange transformations and polygonal billiards
    • E. Gutkin, N. Haydn. Topological entropy of polygon exchange transformations and polygonal billiards. Erg. Theor. & Dyn. Sys. V. 17. 1997. P. 849-867.
    • (1997) Erg. Theor. & Dyn. Sys. , vol.17 , pp. 849-867
    • Gutkin, E.1    Haydn, N.2
  • 30
    • 0030305240 scopus 로고    scopus 로고
    • The geometry and arithmetic of translation surfaces with applications to polygonal billiards
    • E. Gutkin, C. Judge. The geometry and arithmetic of translation surfaces with applications to polygonal billiards. Math. Res. Lett. V. 3. 1996. P. 391-403.
    • (1996) Math. Res. Lett. , vol.3 , pp. 391-403
    • Gutkin, E.1    Judge, C.2
  • 31
    • 0011635694 scopus 로고    scopus 로고
    • Affine mappings of translation surfaces: Geometry and arithmetic
    • E. Gutkin, C. Judge. Affine mappings of translation surfaces: geometry and arithmetic. Duke Math. J. V. 10.3. 2000. P. 191-213.
    • (2000) Duke Math. J. , vol.103 , pp. 191-213
    • Gutkin, E.1    Judge, C.2
  • 32
    • 21844483677 scopus 로고
    • Caustics for inner and outer billiards
    • E. Gutkin, A. Katok. Caustics for inner and outer billiards. Comm. Math. Phys. V. 17.3. 1995. P. 101-133.
    • (1995) Comm. Math. Phys. , vol.173 , pp. 101-133
    • Gutkin, E.1    Katok, A.2
  • 33
    • 0033235578 scopus 로고    scopus 로고
    • Hyperbolic billiards on surfaces of constant curvature
    • B. Gutkin, E. Gutkin, U. Smilansky. Hyperbolic billiards on surfaces of constant curvature. Comm. Math. Phys. V. 208. 1999. P. 65-90.
    • (1999) Comm. Math. Phys. , vol.208 , pp. 65-90
    • Gutkin, B.1    Gutkin, E.2    Smilansky, U.3
  • 34
    • 84992704625 scopus 로고    scopus 로고
    • Billiards in Finsler and Minkowski geometries
    • E. Gutkin, S. Tahachnikov. Billiards in Finsler and Minkowski geometries. J. Geom. & Phys. V. 40. 2001/2. P. 277-301.
    • (2001) J. Geom. & Phys. , vol.40 , pp. 277-301
    • Gutkin, E.1    Tahachnikov, S.2
  • 36
    • 0343408151 scopus 로고    scopus 로고
    • On periodic billiard trajectories in obtuse triangles
    • L. Halheisen, N. Hungerhiihler. On periodic billiard trajectories in obtuse triangles. SIAM Review. V. 42. 2000. P. 657-670.
    • (2000) SIAM Review , vol.42 , pp. 657-670
    • Halheisen, L.1    Hungerhiihler, N.2
  • 38
    • 19744378330 scopus 로고    scopus 로고
    • B. Hasselhlatt, A.Katok (editors). Elsevier, Amsterdam
    • B. Hasselhlatt, A.Katok (editors). Handbook of Dynamical Systems. V. 1A. Elsevier, Amsterdam. 2002.
    • (2002) Handbook of Dynamical Systems , vol.1 A
  • 39
    • 84972533671 scopus 로고
    • Convex curves whose points are vertices of billiard triangles
    • N. Innami. Convex curves whose points are vertices of billiard triangles. Kodai Math. J. V. 11. 1988. P. 17-24.
    • (1988) Kodai Math. J. , vol.11 , pp. 17-24
    • Innami, N.1
  • 40
    • 0002661764 scopus 로고
    • The second term of the spectral asymptotic expansion for the Laplace-Beltrami operator on manifolds with a boundary
    • V. Ivrii. The second term of the spectral asymptotic expansion for the Laplace-Beltrami operator on manifolds with a boundary. Func. Anal. & Appl. V. 14. 1980. P. 98-106.
    • (1980) Func. Anal. & Appl. , vol.14 , pp. 98-106
    • Ivrii, V.1
  • 41
    • 51649161951 scopus 로고
    • Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
    • A. Katok. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes Etudes Sci. Publ. Math. V. 51. 1980. P. 137-173.
    • (1980) Inst. Hautes Etudes Sci. Publ. Math. , vol.51 , pp. 137-173
    • Katok, A.1
  • 42
    • 0001212617 scopus 로고
    • The growth rate for the number of singular and periodic orbits for a polygonal billiard
    • A. Katok. The growth rate for the number of singular and periodic orbits for a polygonal billiard. Comm. Math. Phys. V. 111. 1987. P. 151-160.
    • (1987) Comm. Math. Phys. , vol.111 , pp. 151-160
    • Katok, A.1
  • 43
    • 55449104740 scopus 로고    scopus 로고
    • ((In Russian)) Publications of the Moscow Center for Continuing Mathematical Education , in press
    • A. Katok. The billiard table as a mathematician's playground. ((In Russian)) Publications of the Moscow Center for Continuing Mathematical Education. 2002, in press.
    • (2002) The Billiard Table As a Mathematician's Playground
    • Katok, A.1
  • 44
    • 0000612131 scopus 로고
    • Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems
    • A. Katok (With the collaboration of Keith Burns). Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems. Erg. Theor. & Dyn. Sys. V. 14. 1994. P. 757-785.
    • (1994) Erg. Theor. & Dyn. Sys. , vol.14 , pp. 757-785
    • Katok, A.1
  • 47
    • 33645064917 scopus 로고
    • Topological transitivity of billiards in polygons
    • A. Katok, A. Zemlyakov. Topological transitivity of billiards in polygons. Math. Notes. V. 18. 1975. P. 760-764.
    • (1975) Math. Notes , vol.18 , pp. 760-764
    • Katok, A.1    Zemlyakov, A.2
  • 48
    • 0007404656 scopus 로고    scopus 로고
    • Billiards on rational-angled triangles
    • R. Kenyan, J. Smillie. Billiards on rational-angled triangles. Comment. Math. Helv. V. 75. 2000. P. 65-108.
    • (2000) Comment. Math. Helv. , vol.75 , pp. 65-108
    • Kenyan, R.1    Smillie, J.2
  • 49
    • 0000923571 scopus 로고
    • Ergodicity of billiard flows and quadratic differentials
    • S. Kerckhoff, H. Masur, J. Smillie. Ergodicity of billiard flows and quadratic differentials. Ann. Math. V. 124. 1986. P. 293-311.
    • (1986) Ann. Math. , vol.124 , pp. 293-311
    • Kerckhoff, S.1    Masur, H.2    Smillie, J.3
  • 51
    • 84927499190 scopus 로고
    • The existemce of caustics for the billiard problem in a convex domain
    • V. Lazutkin. The existemce of caustics for the billiard problem in a convex domain. Math. USSR Izv. V. 7. 1973. P. 185-214.
    • (1973) Math. USSR Izv. , vol.7 , pp. 185-214
    • Lazutkin, V.1
  • 52
    • 0001697014 scopus 로고
    • Lower bounds for the number of saddle connection and closed trajectories of a quadratic differential
    • Springer-Verlag, New York
    • H. Masur. Lower bounds for the number of saddle connection and closed trajectories of a quadratic differential. In "Holomorphic Functions and Moduli". V. 2. Springer-Verlag, New York. 1988.
    • (1988) Holomorphic Functions and Moduli , vol.2
    • Masur, H.1
  • 53
    • 84971847841 scopus 로고
    • The growth rate of trajectories of a quadratic differential
    • H. Masur. The growth rate of trajectories of a quadratic differential. Erg. Theor. & Dyn. Sys. V. 10. 1990. P. 151-176.
    • (1990) Erg. Theor. & Dyn. Sys. , vol.10 , pp. 151-176
    • Masur, H.1
  • 54
    • 0000534712 scopus 로고
    • Hausdorff dimension of sets of nonergodic measured foliations
    • H. Masur, J. Smillie. Hausdorff dimension of sets of nonergodic measured foliations. Ann. Math. V. 1.34. 1991. P. 455-543.
    • (1991) Ann. Math. , vol.134 , pp. 455-543
    • Masur, H.1    Smillie, J.2
  • 55
    • 77952875853 scopus 로고    scopus 로고
    • Rational billiards and flat structures
    • Elsevier, Amsterdam
    • H. Masur, S. Tahachnikov. Rational billiards and flat structures, pp. 1015-1089 in Handbook of Dynamical Systems. Elsevier, Amsterdam. 2002.
    • (2002) Handbook of Dynamical Systems , pp. 1015-1089
    • Masur, H.1    Tahachnikov, S.2
  • 56
    • 0001271071 scopus 로고
    • Glancing billiards
    • J. Mather. Glancing billiards. Erg. Theor. & Dyn. Sys. V. 2. 1982. P. 397-403.
    • (1982) Erg. Theor. & Dyn. Sys. , vol.2 , pp. 397-403
    • Mather, J.1
  • 58
    • 84974264708 scopus 로고
    • On the number of periodic reflecting rays in generic domains
    • V. Petkov, L. Stojanov. On the number of periodic reflecting rays in generic domains. Erg. Theor. & Dyn. Sys. V. 8. 1988. P. 81-91.
    • (1988) Erg. Theor. & Dyn. Sys. , vol.8 , pp. 81-91
    • Petkov, V.1    Stojanov, L.2
  • 59
    • 0000525568 scopus 로고
    • The billiard ball problem on a table with a convex boundary - An illustrative dynamical problem
    • H. Poritsky. The billiard ball problem on a table with a convex boundary - An illustrative dynamical problem. Ann. Math. V. 51. 1950. P. 446-470.
    • (1950) Ann. Math. , vol.51 , pp. 446-470
    • Poritsky, H.1
  • 60
    • 0035651918 scopus 로고    scopus 로고
    • On triangular billiards
    • J.-Ch. Puchta. On triangular billiards. Comment. Mathem. Helv. V. 76. 2001. P. 501-505.
    • (2001) Comment. Mathem. Helv. , vol.76 , pp. 501-505
    • Puchta, J.-C.1
  • 61
    • 84972495855 scopus 로고
    • Periodic points of the billiard ball map in a convex domain
    • M. Rychlik. Periodic points of the billiard ball map in a convex domain. J. Diff. Geom. V. 30. 1989. P. 191-205.
    • (1989) J. Diff. Geom. , vol.30 , pp. 191-205
    • Rychlik, M.1
  • 63
    • 0000616278 scopus 로고
    • The K-property of N billiard balls. I
    • N. Simanyi. The K-property of N billiard balls. I. Inv. Math. V. 108. 1992. P. 521-548.
    • (1992) Inv. Math. , vol.108 , pp. 521-548
    • Simanyi, N.1
  • 64
    • 0001738670 scopus 로고
    • On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics
    • Ya. Sinai. On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics. Dokl. Akad. Nauk SSSR. V. 153. 1963. P. 1261-1264.
    • (1963) Dokl. Akad. Nauk SSSR , vol.153 , pp. 1261-1264
    • Sinai, Ya.1
  • 65
    • 0001200214 scopus 로고
    • In Russian. English transl
    • In Russian. English transl: Soviet Math. Dokl. V. 4. 1963. P. 1818-1822.
    • (1963) Soviet Math. Dokl. , vol.4 , pp. 1818-1822
  • 66
    • 0001379329 scopus 로고
    • Dynamical systems with elastic reflections: Ergodic properties of dispersing billiards
    • Ya. Sinai. Dynamical systems with elastic reflections: Ergodic properties of dispersing billiards. Uspekhi Mat. Nauk. V. 25. 1970. P. 141-192.
    • (1970) Uspekhi Mat. Nauk. , vol.25 , pp. 141-192
    • Sinai, Ya.1
  • 67
    • 55449124646 scopus 로고    scopus 로고
    • The dynamics of billiard flows in rational polygons
    • (edited by Ya. Sinai). Springer-Verlag, New York
    • J. Smillie. The dynamics of billiard flows in rational polygons, pp. 360-382 in Encyclopedia of Mathematical Sciences vol. 100 (edited by Ya. Sinai). Springer-Verlag, New York. 2000.
    • (2000) Encyclopedia of Mathematical Sciences , vol.100 , pp. 360-382
    • Smillie, J.1
  • 68
    • 26144442638 scopus 로고
    • Geodatische linien auf Polyederflächen
    • P. Stäckel. Geodatische linien auf Polyederflächen. Rend. Circ. Mat. Palermo. V. 22. 1906. P. 141-151.
    • (1906) Rend. Circ. Mat. Palermo , vol.22 , pp. 141-151
    • Stäckel, P.1
  • 69
    • 0001764008 scopus 로고
    • An estimate from above on the number of periodic orbits for semi-dispersed billiards
    • L. Stojanov. An estimate from above on the number of periodic orbits for semi-dispersed billiards. Comm. Math. Phys. V. 124. 1989. P. 217-227.
    • (1989) Comm. Math. Phys. , vol.124 , pp. 217-227
    • Stojanov, L.1
  • 70
    • 84972492806 scopus 로고
    • Note on the periodic points of the billiard
    • L. Stojanov. Note on the periodic points of the billiard. J. Diff. Geom. V. 34. 1991. 835-837.
    • (1991) J. Diff. Geom. , vol.34 , pp. 835-837
    • Stojanov, L.1
  • 71
    • 0000232377 scopus 로고
    • On the K-property of some planar hyperbolic billiards
    • D. Szasz. On the K-property of some planar hyperbolic billiards. Comm. Math. Phys. V. 145. 1992. P. 595-604.
    • (1992) Comm. Math. Phys. , vol.145 , pp. 595-604
    • Szasz, D.1
  • 72
    • 0013238735 scopus 로고    scopus 로고
    • D. Szasz (editor). Encyclopedia of Mathematical Sciences . Springer-Verlag, Berlin
    • D. Szasz (editor). Hard Ball Systems and the Lorentz Gas. Encyclopedia of Mathematical Sciences vol. 101. Springer-Verlag, Berlin. 2000.
    • (2000) Hard Ball Systems and the Lorentz Gas , vol.101
  • 73
    • 0003684740 scopus 로고
    • Soc. Math. de France, Paris
    • S. Tahachnikov. Billiards. Soc. Math. de France, Paris. 1995.
    • (1995) Billiards
    • Tahachnikov, S.1
  • 74
    • 34249973241 scopus 로고
    • Teichmuller curves in modular space, Eisenstein series, and an application to triangulat billiards
    • W. Veech. Teichmuller curves in modular space, Eisenstein series, and an application to triangulat billiards. Inv. Math. V. 97. 1989. P. 553-583.
    • (1989) Inv. Math. , vol.97 , pp. 553-583
    • Veech, W.1
  • 75
    • 51249173967 scopus 로고
    • Modular spaces of quadratic differentials
    • W. Veech. Modular spaces of quadratic differentials. J. Analyse Math. V. 55. 1990. P. 117-171.
    • (1990) J. Analyse Math. , vol.55 , pp. 117-171
    • Veech, W.1
  • 76
    • 5244300118 scopus 로고
    • The billiard in a regular polygon
    • W. Veech. The billiard in a regular polygon. GAFA. V. 2. 1992. P. 341-379.
    • (1992) GAFA , vol.2 , pp. 341-379
    • Veech, W.1
  • 77
    • 33748920096 scopus 로고
    • On the measure of the set of periodic points of a billiard
    • Ya. Vorohets. On the measure of the set of periodic points of a billiard. Mat. Notes. V. 55. 1994. P. 455-460.
    • (1994) Mat. Notes , vol.55 , pp. 455-460
    • Vorohets, Ya.1
  • 78
    • 34548282655 scopus 로고    scopus 로고
    • Ergodicity of billiards in polygons
    • Ya. Vorohets. Ergodicity of billiards in polygons. Mat. Sb. V. 188. 1997. P. 65-112.
    • (1997) Mat. Sb. , vol.188 , pp. 65-112
    • Vorohets, Ya.1
  • 79
    • 0001117305 scopus 로고
    • Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)
    • H. Weyl. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. V. 71. 1912. P. 441-479.
    • (1912) Math. Ann. , vol.71 , pp. 441-479
    • Weyl, H.1
  • 80
    • 0000176923 scopus 로고
    • Invariant families of cones and Lyapunov exponents
    • M. Wojtkowski. Invariant families of cones and Lyapunov exponents. Erg. Theor. & Dyn. Sys. V. 5. 1985. P. 145-161.
    • (1985) Erg. Theor. & Dyn. Sys. , vol.5 , pp. 145-161
    • Wojtkowski, M.1
  • 81
    • 0000928980 scopus 로고
    • Principles for the design of billiards with nonvanishing Lyapunov exponents
    • M. Wojtkowski. Principles for the design of billiards with nonvanishing Lyapunov exponents. Comm. Math. Phys. V. 105. 1986. P. 391-414.
    • (1986) Comm. Math. Phys. , vol.105 , pp. 391-414
    • Wojtkowski, M.1
  • 82
    • 84974509231 scopus 로고
    • Measure theoretic entropy of the system of hard spheres
    • M. Wojtkowski. Measure theoretic entropy of the system of hard spheres. Erg. Theor. & Dyn. Sys. V. 8. 1988. P. 133-153.
    • (1988) Erg. Theor. & Dyn. Sys. , vol.8 , pp. 133-153
    • Wojtkowski, M.1
  • 83
    • 84972500800 scopus 로고
    • Two applications of Jacobi felds to the billiard ball problem
    • M. Wojtkowski. Two applications of Jacobi felds to the billiard ball problem. J. Diff. Geom. V. 40. 1994. P. 155-164.
    • (1994) J. Diff. Geom. , vol.40 , pp. 155-164
    • Wojtkowski, M.1
  • 84
    • 0032348050 scopus 로고    scopus 로고
    • Statistical properties of dynamical systems with some hyperbolicity
    • L. S. Young. Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. V. 147. 1998. P. 585-650.
    • (1998) Ann. Math. , vol.147 , pp. 585-650
    • Young, L.S.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.