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The characteristic exponents of the falling ball model

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Abstract

We study the characteristic exponents of the Hamiltonian system ofn (>=2) point massesm 1,...,m n freely falling in the vertical half line {q|q>=0} under constant gravitation and colliding with each other and the solid floorq=0 elastically. This model was introduced and first studied by M. Wojtkowski. Hereby we prove his conjecture: All relevant characteristic (Lyapunov) exponents of the above dynamical system are nonzero, provided thatm 1>= ...>=m n (i.e. the masses do not increase as we go up) andm 1≠m 2.

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References

  1. Burago, D., Ferleger, S., Kononenko, A.: Uniform estimates on the number of collisions in semi-dispersing billiards. Manuscript 1995.

  2. Chernov, N.I.: Local ergodicity of hyperbolic systems with singularities. Funct. Anal. and its Appl.27, no. 1, 51–54 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Galperin, G.A.: Elastic collisions of particles on a line. Russ. Math. Surv.33, no. 1, 219–220 (1978)

    Article  MathSciNet  Google Scholar 

  4. Galperin, G.: On systems of locally interacting and repelling particles moving in space. Trudy MMO43, 142–196 (1981)

    MATH  MathSciNet  Google Scholar 

  5. Krámli, A., Simányi, N., Szász, D.: TheK-Property of Three Billiard Balls. Ann. of Math.133, 37–72 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Krámli, A., Simányi, N., Szász, D.: TheK-Property of Four Billiard Balls. Commun. Math. Phys.144, 107–148 (1992)

    Article  MATH  ADS  Google Scholar 

  7. Liverani, C., Wojtkowski, M.: Ergodicity in Hamiltonian Systems. Dynamics Reported4 (New Series), 130–202 (1995)

    MATH  MathSciNet  Google Scholar 

  8. Oseledets, V.I.: The multiplicative ergodic theorem. The Lyapunov characteristic numbers of a dynamical system. Trans. Mosc. Math. Soc.19, 197–231 (1968)

    MATH  Google Scholar 

  9. Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. Math. IHES50, 27–58 (1979)

    MATH  MathSciNet  Google Scholar 

  10. Simányi, N.: TheK-property ofN billiard balls I. Invent. Math.108, 521–548 (1992); II., ibidem110, 151–172 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Sinai, Ya.G., Chernov, N.I.: Ergodic properties of certain systems of 2-D discs and 3-D balls. Russ. Math. Surv. (3)42, 181–207 (1987)

    Article  MathSciNet  Google Scholar 

  12. Simányi, N., Szász, D.: TheK-property of Hamiltonian systems with restricted hard ball interactions. Mathematical Research Letters2, No. 6, 751–770 (1995).

    MATH  MathSciNet  Google Scholar 

  13. Simányi, N., Szász, D.: The Boltzmann-Sinai ergodic hypothesis for hard ball systems. Manuscript 1996

  14. Vaserstein, L.N.: On Systems of Particles with Finite Range and/or Repulsive Interactions. Commun. Math. Phys.69, 31–56 (1979)

    Article  ADS  Google Scholar 

  15. Wojtkowski, M.: Invariant families of cones and Lyapunov exponents. Ergod. Th. Dyn. Syst.5, 145–161 (1985)

    MATH  MathSciNet  Google Scholar 

  16. Wojtkowski, M.: A System of One Dimensional Balls with Gravity. Commun. Math. Phys.126, 507–533 (1990)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Wojtkowski, M.: The System of One Dimensional Balls in an External Field. II. Commun. Math. Phys.127, 425–432 (1990)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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Author notes

  1. Nándor Simányi

    Present address: Mathematical Institute of the Hungarian Academy of Sciences, P.O.B. 127, H-1364, Budapest, Hungary

Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, 16802, University Park, PA, USA

    Nándor Simányi

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  1. Nándor Simányi
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Communicated by Ya. G. Sinai

Research partially supported by the Hungarian National Foundation for Scientific Research, No. 16425.

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Simányi, N. The characteristic exponents of the falling ball model. Commun.Math. Phys. 182, 457–468 (1996). https://doi.org/10.1007/BF02517897

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  • DOI: https://doi.org/10.1007/BF02517897

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