Typical properties of central exponents of diffeomorphisms (Q1081007)

We consider central exponents as functions, not on the space of linear systems indicated above, but on the space of Cauchy problems for nonlinear differential equations. A point of this space is a pair: a nonlinear system of differential equations (some set or other of such systems is provided with a topology in a certain natural sense) is the first element of the pair, and a point of the phase space is the second element of the pair; a nonlinear Cauchy problem can be linearized, and the central exponent of the system of variational equations obtained is considered as a function of the nonlinear Cauchy problem. On such spaces, not only the highest Lyapunov exponent, but also the upper central exponent are not upper semicontinuous everywhere. It turns out however that in such spaces upper semicontinuity of central exponents is typical; we recall that a property is typical if a set of points which is the everywhere dense intersection of a countable collection of open sets has it. The central exponents (with indices \(>1)\) are of at least two kinds. In Differ. Uravn. 19, 1503-1510 (1983; Zbl 0564.34047) the fact that upper semicontinuity is typical of one of them is proved. In the present paper we consider the other kind of central exponents and prove that the upper semicontinuity of the central exponents of this kind is typical.