Absolutely continuous spectrum for parabolic flows/maps
Lucia D. Simonelli
Discrete & Continuous Dynamical Systems - A 2018, 38(1): 263-292 doi: 10.3934/dcds.2018013

We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive certain spectral results for skew products over translations and Furstenberg transformations.

keywords: Parabolic dynamical systems spectral theory commutator methods
Countable Markov partitions suitable for thermodynamic formalism
Michael Jakobson Lucia D. Simonelli
Journal of Modern Dynamics 2018, 13(1): 199-219 doi: 10.3934/jmd.2018018

We study hyperbolic attractors of some dynamical systems with apriori given countable Markov partitions. Assuming that contraction is stronger than expansion, we construct new Markov rectangles such that their cross-sections by unstable manifolds are Cantor sets of positive Lebesgue measure. Using new Markov partitions we develop thermodynamical formalism and prove exponential decay of correlations and related properties for certain Hölder functions. The results are based on the methods developed by Sarig [26]-[28].

keywords: Thermodynamic formalism Markov partitions Gurevich pressure hyperbolic attractors SRB measure

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