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The characteristic exponents of the falling ball model

We study the characteristic exponents of the Hamiltonian system of $n$ ($\ge 2$) point masses $m_1,\dots,m_n$ freely falling in the vertical half line $\{q|\, q\ge 0\}$ under constant gravitation and colliding with each other and the solid floor $q=0$ 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 that $m_1\ge\dots\ge m_n$ (i. e. the masses do not increase as we go up) and $m_1\ne m_2$.