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Register a new userSuperdensity of non-integrable billiards and diophantine approximation
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Consider a finite polysquare or square tiled region, a connected, but not necessarily simply-connected, polygonal region tiled with aligned unit squares. Using ideas from diophantine approximation, we prove that a half-infinite billiard orbit in such a region is superdense, a best possible form of time-quantitative density, if and only if the initial slope of the orbit is a badly approximable number. As the traditional approach to questions of density and uniformity via ergodic theory depends on results such as Birkhoffs ergodic theorem which are essentially time-qualitative in nature and do not appear to lead naturally to time-quantitative statements, we appeal to a non-ergodic approach that is based largely on number theory and combinatorics. In particular, we use the famous 3-distance theorem in diophantine approximation combined with an iterative process. This paper improves on an earlier result of the authors and Yang where it is shown that badly approximable numbers that satisfy a quite severe technical restriction on the digits of their continued fractions lead to superdensity. Here we overcome this technical impediment.
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We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ frac{log{n}}{n}log{left|x-frac{P_n}{Q_n}right|}rightarrow -frac{pi^2}{3} quad text{in measure.} $$ for a.e. $x$. In the course of the proof, we reprove known inspiring results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infinite almost everywhere.
We show that on any non-integrable finite polysquare translation surface, superdensity, an optimal form of time-quantitative density, leads to an optimal form of time-quantitative uniformity that we call super-micro-uniformity.
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.
Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-archimedean quantitative approximation allows us to show: - a quantified version of Silvermans and Ingrams recent comparison between the critical height and any ample height on the moduli space $mathcal{M}_d(bar{mathbb{Q}})$, - two improvements of McMullens finiteness of the multiplier maps: reduction to multipliers of cycles of exact given period and an effective bound from below on the period, - a characterization of non-affine isotrivial rational maps defined over the function field $mathbb{C}(X)$ of a normal projective variety $X$ in terms of the growth of the degree of the multipliers.
Because of its ineffectiveness, the usual arithmetic Hilbert-Samuel formula is not applicable in the context of Diophantine Approximation. In order to overcome this difficulty, the present paper presents explicit estimates for arithmetic Hilbert Functions of closed subvarieties in projective space.
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