Flows on Homogeneous Spaces and Diophantine Approximation on Manifolds

The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and p -adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock–Margulis (strong extremality of nondegenerate submanifolds of ℝ^n ) are generalized to the S -arithmetic setting.

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/\Gamma$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.

We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but satisfies some expansion properties for the adjoint action. Using these dynamical results, we study Diophantine properties of typical points on some self-similar fractals in $${\mathbb {R}}^d$$ . As examples, we show that for any self-similar fractal $$\mathcal {K}\subseteq {\mathbb {R}}^d$$ satisfying the open set condition (for instance any translate or dilate of Cantor’s middle thirds set or of a Koch snowflake), almost every point with respect to the natural measure on $$\mathcal {K}$$ is not badly approximable. Furthermore, almost every point on the fractal is of generic type, which means (in the one-dimensional case) that its continued fraction expansion contains all finite words with the frequencies predicted by the Gauss measure. We prove analogous results for matrix approximation, and for the case of fractals defined by Mobius transformations.

Szekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

The main topic of discussion in this survey is the geometric and metric properties of geodesic and horocycle flows on surfaces of constant curvature and their multidimensional analogues, the so-called G-induced actions (that is, restrictions to a suitable subgroup F of the natural action of G on a homogeneous space of it). One usually takes as F one-parameter or cyclic subgroups of G and the horospherical subgroups corresponding to them (the details are given in Sect. 1). However, in the course of the exposition of our main topic we sometimes mention results relating to the case when F is an n-parameter subgroup of G or a lattice. This class of dynamical systems consists precisely of group actions admitting an extension to a transitive action of a Lie group. We shall not be considering in our survey (or, at any rate, not systematically) the following themes which are close to our main subject matter: Topological dynamics and metric properties of general group actions (in particular, the work of Zimmer on semisimple group actions (Zimmer 1984); Group automorphisms; Affine transformations of homogeneous spaces; G-induced flows on spaces with an infinite invariant measure. The papers (Hopf 1939), (Sullivan 1981) and the recent book (Nicholls 1989) are devoted to the connections between G-induced flows (and also geodesic flows) and the theory of discrete groups of motions of a hyperbolic space; The actions of subgroups on the double cosets related to geodesic (for n > 2) and horocycle (for n > 3) flows on an n-dimensional manifold of constant negative curvature (see (Gel’fand and Fomin 1952) and, in particular, (Mautner 1957)). Flows of this type in somewhat more general form differing from geodesic and horocycle flows have not been investigated.

Contents Introduction §0. Agreements, definitions and preliminaries §1. Surfaces of constant negative curvature §2. Measure rigidity of the horocycle flow §3. Geometric generalizations of Ratner's theorem on measure rigidity of the horocycle flow §4. Quotients and joinings of the horocycle flow §5. Rigidity, quotients and joinings of unipotent flows §6. Dynamics of the horocycle flow §7. Classification of ergodic measures for unipotent flows §8. Uniform distribution of unipotent trajectories §9. Various problems of convergence in the space of measures §10. Structure of orbits, minimal sets, and ergodic measures of homogeneous flows §11. Multiple mixing and measure rigidity of homogeneous flows §12. Ergodic measures and orbit closures for actions of arbitrary subgroups §13. Unipotent flows on homogeneous spaces over local fields §14. Applications to number theory §15. Some open problemsBibliography

We provide a self-contained, accessible introduction to Ratner’s Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early 1980s by Dani and Smillie [DS84] and later reappeared as an illustrative special case [Rat92] of Ratner’s work [Rat91-Rat94] on the equidistribution of unipotent flows in homogeneous spaces. We also prove an interesting probabilistic result due to Breuillard: on the modular surface an arbitrary uncentered random walk on the horocycle through almost any point will fail to equidistribute, even though the horocycles are themselves equidistributed [Bre05]. In many aspects of this exposition we are indebted to Bekka and Mayer’s more ambitious survey [BM00], Ergodic Theory and Topological Dynamics for Group Actions on Homogeneous Spaces.

This thesis uses dynamical systems to study two types of objects: number expansions (Chapters 1 and 2) and translation surfaces (Chapter 3). Our first chapter builds a broad, unifying theory for a large class of continued fraction algorithms producing what we call contracted Farey expansions. These algorithms are based on three ideas: (i) contraction of generalised continued fractions, (ii) induced transformations, and (iii) the natural extension of the Farey tent map. Within this theory, we find several well-studied algorithms; a new subfamily of superoptimal continued fractions with arbitrarily good convergence and approximation properties; and a unifying framework to prove several old and new results in Diophantine approximation. Chapter 2 introduces a new, one-parameter family of functions called skewed symmetric golden maps. Using tools from ergodic theory, we study the relative frequencies of digits typically occurring in number expansions produced by these maps. The central tool for our analysis is a mysterious phenomenon of our functions called matching, which has been recently observed and exploited to understand several other families of functions generating number expansions. Our final chapter deals with translation surfaces, i.e., surfaces obtained by gluing pairs of parallel, equal-length, and oppositely oriented edges of planar polygons. The Veech group of a translation surface is the group of Jacobians of its orientation-preserving affine automorphisms. We develop a novel algorithm to construct translation surfaces with prescribed lattice Veech groups in given strata. Our ideas are also used to obtain obstructions for the realisability of certain Veech groups in certain strata. In particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is the modular group.

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Indefinite quadratic forms and unipotent flows on homogeneous spaces

Smooth embeddings of SL(2) and PGL(2)

In a paper from 2010, Budarina, Dickinson and Levesley studied the rational approximation properties of curves parametrised by polynomials with integral coefficients in Euclidean space of arbitrary dimension. Assuming the dimension is at least three and excluding the case of linear dependence of the polynomials together with P(X) ≡ 1 over the rational number field, we establish proper generalisations of their main result.

Diophantine Approximation and Continued Fraction Expansions of Algebraic Power Series in Positive Characteristic

Диофантовы экспоненты являются одними из самых простых количественных характеристик, отвечающих за аппроксимационные свойства линейных подпространств евклидова пространства. Данный обзор посвящeн описанию современного состояния раздела теории диофантовых приближений, изучающего диофантовы экспоненты и соотношения, которым они удовлетворяют. Мы обсуждаем классические диофантовы экспоненты, возникающие в задаче приближения нуля набором значений нескольких линейных форм в целых точках, их аналоги в теории диофантовых приближений с весами, мультипликативные диофантовы экспоненты, а также диофантовы экспоненты решeток. Особое внимание уделяется принципу переноса. Библиография: 99 названий.

Metaxalone is oxazolidin-2-one derivative, biopharmaceutics classification system class II drug, available in the market as a central muscle relaxant drug, used to alleviate discomforts associated with severe, painful musculoskeletal conditions. The major drawbacks associated with formulation of solid oral dosage forms of metaxalone are its poor aqueous solubility, poor flow property and low compressibility. The aim of present study was to develop different crystalline forms of metaxalone in order to overcome these drawbacks and consequently enhance its dissolution and bioavailability. For this purpose two different approaches were adopted namely cocrystallization and melt sonocrystallization. Cocrystals of metaxalonesaccharin and metaxalone-lactic acid were prepared, with saccharin and lactic acid as coformers, by using solvent evaporation method. Melt sonocrystallization technique was applied for preparation of melt sonocrystallized form of metaxalone. The percentage yield, melting point, particle size, flow property, crystallinity index, solubility and dissolution profiles of developed crystals were evaluated and compared with original metaxalone. Further the developed forms of metaxalone were characterized by differential scanning calorimetry, fourier transform infrared spectroscopy, X-ray diffraction and scanning electron microscopy. The sharp peak of differential scanning calorimetry thermograms and fourier transform infrared spectroscopy spectras indicate towards the formation of crystalline forms with retention of characterstic functionality of original metaxalone. The particle size and crystallinity index of the developed metaxalone crystals were significantly reduced in comparison to the original form. In addition, there was significant enhancement in the flow and compressibility of developed crystals, as indicated by the value of angle of repose, Carr’s index and Hausner ratio. The solubility of prepared crystals was much higher as compared to metaxalone and progressively increased in the following order: metaxalonesaccharin< metaxalone-lactic acid<melt sonocrystallized form of metaxalone. The dissolution studies revealed that 87.08, 92.79 and 99.79 % of the drug was released within 30 min from metaxalone-saccharin, metaxalone-lactic acid and melt sonocrystallized form of metaxalone crystals, respectively compared to 67.63 % release from the original metaxalone drug. Further maximum solubility and faster dissolution was observed with melt sonocrystallized form of metaxalone as compared to metaxalone-lactic acid and metaxalone-saccharin. On the basis of this it can be concluded that solvent evaporation method and melt sonocrystallization techniques are promising techniques that may afford crystals with improved flow property, improved solubility and dissolution.