In this paper we introduce an exponential map of the algebra of germs of vector fields into the group of germs of diffeomorphisms at zero. It is shown that this mapping is not a bijection. A brief review of the key results of the analytic iteration problem is given.
Growth of groups and diffeomorphisms of the interval
Distortion for diffeomorphisms of surfaces with boundary
Large deviation for return times in open sets for axiom A diffeomorphisms
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