Parameter Dependence in Weakly Hyperbolic Dynamical Systems

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Abstract: DMS-0245359 PI: D DOLGOPYAT, U MarylandSmooth ergodic theory studies the long-time asymptotics of those autonomous systemswhich can be described by a finite number of parameters. Both analytic andnumerical results suggest that most of systems fall in one of the followingclasses: elliptic (quasiperiodic) and hyperbolic (exhibiting sensitive dependence on initial conditions). Systems close to elliptic systemsare described by the famous Kolmogorov--Arnold--Moser theory. By contrastsmall perturbations of hyperbolic systems have been analyzed onlyin a few special cases. This proposal aims at the general perturbationtheory for hyperbolic systems. This includes both the description of theperturbed dynamics up to the certain specific times as well as computing equilibrium characteristics of the perturbedsystem. Because there is sensitive dependence on initial conditions, it is hopeless to deal with perturbations of individualorbits. Rather, using the fact that hyperbolic systems usually havestrong stochastic properties, I will study parameter dependence of frequenciesof visits to a given domain in the phase space. My previous work on statistical properties of hyperbolic systems is essential to this project.During the recent years it has been understood that many systemswe encounter have chaotic behavior. On the other hand thereare several models of chaotic systems which can be exactly solved, which canbe analyzed mathematically. However it can be asked how much onecould trust predictions made on the basis of these models since the specific features which allow us to precisely solve these models might make them in some sense exceptional. In particular since chaotic systems arevery sensitive to initial conditions it is conceivable that a slightchange in the parameters of the system could drastically change theirbehavior. The goal of the current project is to describe the conditionswhen this drastic change does not happen. The main idea is that mostchaotic systems are so complicated that changing parameters does notlead to an appearance of a new pattern of behavior but rather to changing slightly probabilities of already existing patterns.