arXiv subjects

Lucia D. Simonelli

Publications and source records attributed to Lucia D. Simonelli.

4 recordsLinked to original sources

arXiv2019

Parabolic Flows Renormalized by Partially Hyperbolic Maps

We consider parabolic flows on 3-dimensional manifolds which are renormalized by circle extensions of Anosov diffeormorphisms. This class of flows includes nilflows on the Heisenberg nilmanifold which are renormalized by partially hyperbolic automorphisms. The transfer operators associated to the renormalization maps, acting on anisotropic Sobolev spaces, are known to have good spectral properties (this relies on ideas which have some resemblance to representation theory but also apply to non-algebraic systems). The spectral information is used to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow.

math.DS↗

arXiv2018

Local Limit Theorems for Poisson's Binomial in the Case of Infinite Expectation

Let $ V_{n} = X_{1,n} + X_{2,n} + \cdots + X_{n,n}$ where $X_{i,n}$ are Bernoulli random variables which take the value $1$ with probability $b(i;n)$. Let $λ_{n} = \sum\limits_{i=1}^{n} b(i;n) $, $λ= \lim\limits_{n \to \infty} λ_n,$ and $m_n = \max\limits_{1 \leq i \leq n} b(i;n)$. We derive asymptotic results for $P(V_{n}=k)$ that hold without assuming that $λ< +\infty$ or $m_n \to 0$. Also, we do not assume $k$ to be fixed, but instead, our results hold uniformly for all $k$ which satisfy particular growth conditions with respect to $n$. These results extend known Poisson local limit theorems to the case when $λ= +\infty$. While our results apply to triangular arrays, without the assumption that $m_n \to 0$ they continue to hold for sums of Bernoulli random variables. In this setting, our growth conditions cover a range of values for $k$ not centered at $λ_n$, thus complementing known local limit theorems based on approximation by the normal distribution. In addition, we show that our local limit theorems apply to a scheme of dependent random variables introduced in the work of Sevast'yanov.

math.PR↗

arXiv2018

Countable Markov Partitions Suitable for Thermodynamic Formalism

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 crossections 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.

math.DS↗

arXiv2017

Absolutely Continuous Spectrum for Parabolic Flows/Maps

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.

math.DS↗