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Authors
Mauricio Garay
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Arithmetic class are closed subsets of the euclidean space which generalise arithmetical conditions encoutered in dynamical systems, such as diophantine conditions or Bruno type conditions. I prove density estimates for such sets using Dani-Kleinbock-Margulis techniques.
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Inspired by the work of Zagier, we study geometrically the probability measures $m_y$ with support on the closed horocycles of the unit tangent bundle $M=text{PSL}(2,mathbb{R})/text{PSL}(2,mathbb{Z})$ of the modular orbifold $text{PSL}(2,mathbb Z)$. In fact, the canonical projection $mathfrak{p}:Mtomathbb{H}/text{PSL}(2,mathbb Z)$ it is actually a Seifert fibration over the orbifold with two especial circle fibers corresponding to the two conical points of the modular orbifold. Zagier proved that $m_y$ converges to normalized Haar measure $m_o$ of $M$ as $yto0$: for every smooth function $f:Mto mathbb R$ with compact support $m_y(f)=m_0(f)+o(y^frac12)$ as $yto0$. He also shows that $m_y(f)=m_0(f)+o(y^{frac34-epsilon})$ for all $epsilon>0$ and smooth function $f$ with compact support in $M$ if and only if the Riemann hypothesis is true. In this paper we show that the exponent $frac12$ is optimal if $f$ is the characteristic function of certain open sets in $M$. This of course does not imply that the Riemann hypothesis is false. It is required the differentiability of the functions in the theorem.
We establish the (level-1) large deviation principles for three kinds of means associated with the backward continued fraction expansion. We show that: for the harmonic and geometric means, the rate functions vanish exactly at one point; for the arithmetic mean, it is completely degenerate, vanishing at every point in its effective domain. Our method of proof employs the thermodynamic formalism for finite Markov shifts, and a multifractal analysis for the Renyi map generating the backward continued fraction digits. We completely determine the class of unbounded arithmetic functions for which the rate functions vanish at every point in unbounded intervals.
We introduce a new method to establish time-quantitative density in flat dynamical systems. First we give a shorter and different proof of our earlier result that a half-infinite geodesic on an arbitrary finite polysquare surface P is superdense on P if the slope of the geodesic is a badly approximable number. We then adapt our method to study time-quantitative density of half-infinite geodesics on algebraic polyrectangle surfaces.
We study approximation schemes for shift spaces over a finite alphabet using (pseudo)metrics connected to Ornsteins $bar{d}$ metric. This leads to a class of shift spaces we call $bar{d}$-approachable. A shift space $bar{d}$-approachable when its canonical sequence of Markov approximations converges to it also in the $bar{d}$ sense. We give a topological characterisation of chain mixing $bar{d}$-approachable shift spaces. As an application we provide a new criterion for entropy density of ergodic measures. Entropy-density of a shift space means that every invariant measure $mu$ of such a shift space is the weak$^*$ limit of a sequence $mu_n$ of ergodic measures with the corresponding sequence of entropies $h(mu_n)$ converging to $h(mu)$. We prove ergodic measures are entropy-dense for every shift space that can be approximated in the $bar{d}$ pseudometric by a sequence of transitive sofic shifts. This criterion can be applied to many examples that were out of the reach of previously known techniques including hereditary $mathscr{B}$-free shifts and some minimal or proximal systems. The class of symbolic dynamical systems covered by our results includes also shift spaces where entropy density was established previously using the (almost) specification property.
Let $lambdain (1,sqrt{2}]$ be an algebraic integer with Mahler measure $2.$ A classical result of Garsia shows that the Bernoulli convolution $mu_lambda$ is absolutely continuous with respect to the Lebesgue measure with a density function in $L^infty$. In this paper, we show that the density function is continuous.
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