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Search Results: 1 - 10 of 2471 matches for " Tomas Persson "
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Absolutely continuous invariant measures for some piecewise hyperbolic affine maps
Tomas Persson
Mathematics , 2005,
Abstract: A class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.
Random iterated function systems with smooth invariant densities
Tomas Persson
Mathematics , 2009,
Abstract: We consider some random iterated function systems on the interval and show that the invariant measure has density in $\mathcal{C}^\infty$. To prove this we use some techniques for contractions in cone metrics, applied to the transfer operator.
Another Proof of a Lemma by L. Shepp
Tomas Persson
Mathematics , 2014,
Abstract: We give a new proof of a lemma by L. Shepp, that was used in connection to random coverings of a circle.
On the Hausdorff Dimension of Piecewise Hyperbolic Attractors
Tomas Persson
Mathematics , 2009,
Abstract: We study non-invertible piecewise hyperbolic maps in the plane. The Hausdorff dimension of the attractor is calculated in terms of the Lyapunov exponents, provided that the map satisfies a transversality condition. Explicit examples of maps for which this condition holds are given.
On a problem by R. Salem concerning Minkowski's question mark function
Tomas Persson
Mathematics , 2015,
Abstract: Minkowski's question mark function is strictly increasing on $[0, 1]$ and hence defines a Stieltjes measure on $[0, 1]$. A problem originating from Salem in 1943, is to determine whether the Fourier series of this measure decay to zero or not. The purpose of this note is to mention that a recent result by Jordan and Sahlsten implies that the Fourier transform decays to zero with a polynomial speed.
Typical points and families of expanding interval mappings
Tomas Persson
Mathematics , 2015,
Abstract: We study parametrised families of piecewise expanding interval mappings $T_a \colon [0,1] \to [0,1]$ with absolutely continuous invariant measures $\mu_a$ and give sufficient conditions for a point $X(a)$ to be typical with respect to $(T_a, \mu_a)$ for almost all parameters $a$. This is similar to a result by D. Schnellmann, but with different assumptions.
A Note on Random Coverings of Tori
Tomas Persson
Mathematics , 2013, DOI: 10.1112/blms/bdu087
Abstract: This note provides a generalisation of a recent result by J\"arvenp\"a\"a, J\"arvenp\"a\"a, Koivusalo, Li, and Suomala, (to appear), on the dimension of limsup-sets of random coverings of tori. The result in this note is stronger in the sense that it provides also a large intersection property of the limsup-sets, the assumptions are weaker, and it implies the result of J\"arvenp\"a\"a, J\"arvenp\"a\"a, Koivusalo, Li, and Suomala as a special case. The proof is based on a recent result by Persson and Reeve from 2013.
Smooth Livsic regularity for piecewise expanding maps
Matthew Nicol,Tomas Persson
Mathematics , 2010,
Abstract: We consider the regularity of measurable solutions $\chi$ to the cohomological equation \[ \phi = \chi \circ T -\chi, \] where $(T,X,\mu)$ is a dynamical system and $\phi \colon X\rightarrow \R$ is a $C^k$ valued cocycle in the setting in which $T \colon X\rightarrow X$ is a piecewise $C^k$ Gibbs--Markov map, an affine $\beta$-transformation of the unit interval or more generally a piecewise $C^{k}$ uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions $\chi$ possess $C^k$ versions. In particular we show that if $(T,X,\mu)$ is a $\beta$-transformation then $\chi$ has a $C^k$ version, thus improving a result of Pollicott et al.~\cite{Pollicott-Yuri}.
The absolute continuity of the invariant measure of random iterated function systems with overlaps
Balazs Barany,Tomas Persson
Mathematics , 2009, DOI: 10.4064/fm210-1-2
Abstract: We consider iterated function systems on the interval with random perturbation. Let $Y_\epsilon$ be uniformly distributed in $[1- \epsilon, 1 + \epsilon]$ and let $f_i \in C^{1+\alpha}$ be contractions with fixpoints $a_i$. We consider the iterated function system $\{Y_\epsilon f_i + a_i (1 - Y_\epsilon) \}_{i=1}^n$, were each of the maps are chosen with probability $p_i$. It is shown that the invariant density is in $L^2$ and the $L^2$-norm does not grow faster than $1/\sqrt{\epsilon}$, as $\epsilon$ vanishes.
Bernoulli Convolutions and 1D Dynamics
Tom Kempton,Tomas Persson
Mathematics , 2015, DOI: 10.1088/0951-7715/28/11/3921
Abstract: We describe a family $\phi_{\lambda}$ of dynamical systems on the unit interval which preserve Bernoulli convolutions. We show that if there are parameter ranges for which these systems are piecewise convex, then the corresponding Bernoulli convolution will be absolutely continuous with bounded density. We study the systems $\phi_{\lambda}$ and give some numerical evidence to suggest values of $\lambda$ for which $\phi_{\lambda}$ may be piecewise convex.
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