DCDS
An alternative approach to generalised BV and the application to expanding interval maps
Oliver Butterley
Discrete & Continuous Dynamical Systems - A 2013, 33(8): 3355-3363 doi: 10.3934/dcds.2013.33.3355
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.
keywords: spectral gap. transfer operator Generalised bounded variation expanding map
JMD
Smooth Anosov flows: Correlation spectra and stability
Oliver Butterley Carlangelo Liverani
Journal of Modern Dynamics 2007, 1(2): 301-322 doi: 10.3934/jmd.2007.1.301
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.
keywords: Transfer operator resonances differentiability SRB.
JMD
Robustly invariant sets in fiber contracting bundle flows
Oliver Butterley Carlangelo Liverani
Journal of Modern Dynamics 2013, 7(2): 255-267 doi: 10.3934/jmd.2013.7.255
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.
keywords: Transfer operators resonances differentiability of SRB. Bundle flows

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