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Large deviation asymptotics for Anosov flows. (English) Zbl 0864.58038
Large deviation probabilities is a subject intensively studied by those scientists who are interested in ergodic phenomena. The main purpose of this work is to derive precisely asymptotic formulae for large deviation probabilities for suspensions of subshifts of finite type.
Based on the large deviation property, the author shows that the entropy is analytic and he obtains a strong asymptotic formula under certain conditions, in the case of a transient Anosov flow.
Using some well-known results and methods from the work of Bowen, as well as Ruelle operator and “zeta function” techniques from dynamics, the author analyses the analytic domain of this function. Moreover, he deduces the asymptotic formulae by applying an appropriate Tauberian theorem.
The obtained results are used to study the fluctuations in the volume of Bowen balls and to determine a large deviation expression for a homological nature involving Schwartzmann’s winding cycle.
The final section, precise and accurate as the entire work, contains a more general multidimensional large deviation for suspended flows.
Reviewer: I.Grosu (Iaşi)

MSC:
37A99 Ergodic theory
37D99 Dynamical systems with hyperbolic behavior
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
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