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DCDS

We introduce a family of Banach spaces of measures, each containing the set of measures with density of bounded variation. These spaces are suitable for the study of weighted transfer operators of piecewise-smooth maps of the interval where the weighting used in the transfer operator is not better than piecewise Hölder continuous and the partition on which the map is continuous may possess a countable number of elements.
For such weighted transfer operators we give upper bounds for both the spectral radius and for the essential spectral radius.

JMD

By introducing appropriate Banach spaces one can study the spectral properties of the generator of the semigroup defined by an Anosov flow. Consequently, it is possible to easily obtain sharp results on the Ruelle resonances and the differentiability of the SRB measure.

JMD

We provide abstract conditions which imply the existence of a robustly
invariant neighborhood of a global section of a fiber bundle flow. We
then apply such a result to the bundle flow generated by an Anosov
flow when the fiber is the space of jets (which are described by local
manifolds). As a consequence we obtain sets of manifolds
(e.g., approximations of stable manifolds) that are left invariant

*for all*negative times by the flow and its small perturbations. Finally, we show that the latter result can be used to easily fix a mistake recently uncovered in the paper*Smooth Anosov flows: correlation spectra and stability*[2] by the present authors.## Year of publication

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