Random Versus Deterministic Exponents in a Rich Family of Diffeomorphisms

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We study, both numerically and theoretically, the relationship between the random Lyapunov exponent of a family of area preserving diffeomorphisms of the 2-sphere and the mean of the Lyapunov exponents of the individual members. The motivation for this study is the hope that a rich enough family of diffeomorphisms will always have members with positive Lyapunov exponents, that is to say, positive entropy. At question is what sort of notion of richness would make such a conclusion valid. One type of richness of a family—invariance under the left action of SO(n+1)—occurs naturally in the context of volume preserving diffeomorphisms of the n-sphere. Based on some positive results for families linear maps obtained by Dedieu and Shub, we investigate the exponents of such a family on the 2-sphere. Again motivated by the linear case, we investigate whether there is in fact a lower bound for the mean of the Lyapunov exponents in terms of the random exponents (with respect to the push-forward of Haar measure on SO(3)) in such a family. The family 𝒻 that we study contains a twist map with stretching parameter ε. In the family𝒻 we find strong numerical evidence for the existence of such a lower bound on mean Lyapunov exponents, when the values of the stretching parameter ε are not too small. Even moderate values of ε like ε≥10 are enough to have an average of the metric entropy larger than that of the random map. For small ε the estimated average entropy seems positive but is definitely much less than the one of the random map. The numerical evidence is in favor of the existence of exponentially small lower and upper bounds (in the present example, with an analytic family). Finally, the effect of a small randomization of fixed size δ of the individual elements of the family 𝒻 is considered. Now the mean of the local random exponents of the family is indeed asymptotic to the random exponent of the entire family as ε tends to infinity.

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