Multifractal Analysis of Ergodic Averages: A Generalization of Eggleston's Theorem

Let V be a finite set, S be an infinite countable commutative semigroup, { _s , s S} be the semigroup of translations in the function space X = V ^S , A = {A _n } be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let B. We introduce in X a scale metric generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X _f,,Adefined by the following formula:

On Some Aspects of the Deligne–Simpson Problem

The Deligne-Simpson problem in the multiplicative version is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes C _j GL(n, ), so that there exist irreducible (p + 1)-tuples of matrices M _j C _j satisfying the equality M ₁ . . . M _p+1 = I.We solve the problem for generic eigenvalues in the case where all the numbers _j,m() of Jordan blocks of a given matrix M _j with a given eigenvalue and of a given size m (taken over all j, , and m) are divisible by d > 1. Generic eigenvalues are defined by explicit algebraic inequalities of the form a 0. For such eigenvalues there exist no reducible (p + 1)-tuples.The matrices M _j are interpreted as monodromy operators of regular linear systems on the Riemann sphere.     

On a Loewner Umbilic-Index Conjecture for Surfaces Immersed in ℝ4

Loewner's umbilic-index conjecture cannot be extended to surfaces immersed in ⁴. More precisely, given n , there exists an analytic surface immersed in ⁴ having an isolated umbilic point of index n/2.     

Singular Perturbation of Linear Systems with a Regular Singularity

We study singularly perturbed inhomogeneous linear systems which have a regular singularity at the origin. Under some extra assumption, this system has a unique formal solution which is a power series in the variable z and the perturbation parameter . Here we investigate its summability properties. In particular, we show that the series, regarded as a power series in with coefficients depending on z, is 1-summable under some eigenvalue condition. Moreover, we show that this condition in general is sharp.     

Morse Properties for the Minimum Time Function on 2-D Manifolds

Given a two-dimensional smooth manifold M and two smooth vector fields X and Y on M, we want to steer a point p M to a point q M in minimum time using only intergral curves of the vector fields X and Y. Fixing p, we define the minimum time function to reach q. We prove that, generically, is a Morse function in topological sense giving a positive answer to a question of V.I. Arnold.     

Normalizable, Integrable, and Linearizable Saddle Points for Complex Quadratic Systems in \mathbb{C}^2

In this paper, we consider complex differential systems in the neighborhood of a singular point with eigenvalues in the ratio 1 : – with . We address the questions of orbital normalizability, normalizability (i.e., convergence of the normalizing transformation), integrability (i.e., orbital linearizability), and linearizability of the system. As for the experimental part of our study, we specialize to quadratic systems and study the values of for which these notions are distinct. For this purpose we give several tools for demonstrating normalizability, integrability, and linearizability.Our main interest is the global organization of the strata of those systems for which the normalizing transformations converge, or for which we have integrable or linearizable saddles as and the other parameters of the system vary. Many of the results are valid in the larger context of polynomial or analytic vector fields. We explain several features of the bifurcation diagram, namely, the existence of a continuous skeleton of integrable (linearizable) systems with sequences of holes filled with orbitally normalizable (normalizable) systems and strata finishing at a particular value of . In particular, we introduce the Ecalle-Voronin invariants of analytic classifcation of a saddle point or the Martinet-Ramis invariants for a saddle-node and illustrate their role as organizing centers of the bifurcation diagram.     

Integration of Complex Differential Equations

Let X be a polynomial vector field in ²; then it defines an algebraic foliation on P(2). If admits a Liouvillian first integral on P(2), then it is transversely affine outside some algebraic invariant curve S P(2). If, moreover, for some irreducible component S₀ S, the singularities q Sing S are generic, then either is given by a closed rational 1-form or it is a rational pull-back from a Bernoulli foliation This result has several applications such as the study of foliations with algebraic limit sets on P(2)(2), the classification polynomial complete vector fields over ², and topological rigidity of foliations on P(2). We also address the problem of moderate integration for germs of complex ordinary differential equations.     

Newton's Problem of Minimal Resistance for Bodies Containing a Half-Space

We consider Newton's problem of minimal resistance for unbounded bodies in Euclidean space ^d, d 2. A homogeneous flow of noninteracting particles of velocity v falls onto an immovable body containing a half-space {x : (x, n) < 0} ^d, (v, n) < 0. No restriction is imposed on the number of (elastic) collisions of the particles with the body. For any Borel set A {v} of finite measure, consider the flow of cross-section A: the part of initial flow that consists of particles passing through A.We construct a sequence of bodies that minimize resistance to the flow of cross-section A, for arbitrary A. This sequence approximates the half-space; any particle collides with any body of the sequence at most twice. The infimum of resistance is always one half of corresponding resistance of the half-space.     

The Connectedness of Some Varieties and the Deligne—Simpson Problem

The Deligne—Simpson problem (DSP) (respectively, the weak DSP) is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes C_j GL(n, ) or c_j gl(n, ) so that there exist irreducible (respectively, with trivial centralizer) (p + 1)-tuples of matrices M_j C_j or A_j c_j satisfying the equality M₁ ... M_p+1 = I or A₁ + ... + A_p+1 = 0. The matrices M_j and A_j are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on the Riemann sphere. For ((p + 1))-tuples of conjugacy classes one of which is with distinct eigenvalues we prove that the variety {(M₁, ..., M_p+1) | M_j C_j, M₁ ... M_p+1 = I} or {(A₁, ..., A_p+1) | A_j c_j, A₁ + ... + A_p+1 = 0| is connected if the DSP is positively solved for the given conjugacy classes and give necessary and sufficient conditions for the positive solvability of the weak DSP.2000 Mathematics Subject Classification. 15A30, 15A24, 20G05.     

Sub-Riemannian Geometry: One-Parameter Deformation of the Martinet Flat Case

Consider a sub-Riemannian geometry (U,D,g), where U is a neighborhood of 0 in ³, D is a Martinet type distribution identified to ker , being the one-form and g is a metric on D which can be written as a(q)dx ² + 2b(q)dxdy + c(q)dy ², whereq = (x,y,z). In a previous article [1] we proved that g can be written in a normal form where b 0, a = 1 + yF(q ), c = 1 + G(q ), where . Moreover we analyzed the flat case a = c = 1. In this article we study the following one-parameter deformation of the flat case: a = l, c = (1 + y)² where µ . We parametrize the set of geodesics using elliptic functions. This allows us to compute the trace of the sphere and the wave front of small radius on the plane y = 0. We show that the sphere of small radius is not sub-analytic. This analysis clarifies the role of one of the functional invariants in the normal form.     

Local Regularity of Optimal Trajectories for Control Problems with General Boundary Conditions

Let f and g be two smooth vector fields on a manifold M. Given a submanifold S of M, we study the local structure of time-optimal trajectories for the single-input control-affine system ̇q = f(q) + ug(q) with the initial condition q(0) S. When the codimension s of S in M is small (s 4) and the system has a small codimension singularity at a point q₀ S, we prove that all time-optimal trajectories contained in a sufficiently small neighborhood of q₀ are finite concatenations of bang and singular arcs. The proof is based on an extension of the index theory to the case of general boundary conditions.2000 Mathematics Subject Classification. 49K15, 49K30.     

Number of Singularities of a Generic Web on the Complex Projective Plane

Given a generic d-web W_d of degree n in ², we associate with it a triple (S_Wd, |S_Wd, F_Wd), where S_Wd is a surface in T*², the projective cotangent bundle of ², |S_Wd is the restriction of the natural projection T*^{2 2} to S_Wd and F_Wd is a foliation on S_Wd given by a special meromorphic 1-form. The main objective of this article is to calculate the total number of singularities and the sum of the indices of Baum–Bott for the foliation F_Wd in terms of d and n. These results are compared with the case d = 1 (foliation in ²). We also calculate the total number of nodes and cusps of the projection |S_Wd in terms of d and n.2000 Mathematics Subject Classification. Primary: 37F75, Secondary: 34M45.     

Transformations Having Homogeneous Spectra

We show that for a generic automorphism T, the Cartesian product T × T has homogeneous spectrum of multiplicity two. New examples of automorphisms with the property * are presented.     

Nilpotency of Zinbiel Algebras

Zinbiel algebras are defined by the identity (a b) c = a(bc+cb). We prove an analog of the Nagata–Higman theorem for Zinbiel algebras. We establish that every finite-dimensional Zinbiel algebra over an algebraically closed field is solvable. Every solvable Zinbiel algebra with solvability length N is a nil-algebra with nil-index 2^N if p = char K = 0 or p > 2^N – 1. Conversely, every Zinbiel nil-algebra with nil-index N is solvable with solvability length N if p = 0 or p > N – 1. Every finite-dimensional Zinbiel algebra over complex numbers is nilpotent, nil, and solvable.2000 Mathematics Subject Classification. 17A32.     

On the Codimension One Motion Planning Problem

In this paper, we improve the results of [5] related to motion planning problems for corank one sub-Riemannian (SR) metrics. First, we give the exact estimate of the metric complexity, in the generic 3-dimensional case. (Only bounds from above and from below were given in [5].) Second, we show that the general expression for the metric complexity (that was proven to hold generically in the C case, or under certain nonvanishing condition (C) in the analytic case) is, in fact, always true under condition (C), on the complement of a subset of codimension infinity, in the set of C motion planning problems. Both results are constructive, i.e., an asymptotic optimal synthesis is exhibited in both cases.2000 Mathematics Subject Classification. 53C17, 49J15, 34H05.     

Groups of Flagged Homotopies and Higher Gauge Theory

Groups _k (X;) of flagged homotopies are introduced of which the usual (abelian for k > 1) homotopy groups _k (X;p) is the limit case for flags contracted to a point p. The calculus of exterior forms with values in an algebra A is developped of which the limit cases are the differential forms calculus (for A = ) and gauge theory (for 1-forms). Moduli space of integrable forms with respect to higher gauge transforms (cohomology with coefficients in A) is introduced with elements giving representations of _k in G = expA.     

Rational Approximations of Transfer Functions of Some Viscoelastic Rods with Applications to Robust Control

We study rational approximations of the transfer function of a uniform or nonuniform viscoelastic rod undergoing torsional vibrations that are excited and measured at the same end. The approximation is to be carried out in a way that is appropriate, with respect to stability and performance, for the construction of suboptimal rational stabilizing compensators for the rod. The function can be expressed as , where g is an infinite product of fractional linear transformations and is a (generally transcendental) function that characterizes a particular viscoelastic material. First, g(²) is approximated by its partial products g _N(²). For relevant values of ², convergence rates for g _N are analyzed in detail. Convergence suitable for our problem requires the introduction of a new irrational convergence factor, which must be approximated separately. In addition, the fractional linear factors in ²(s) that appear in g _N(²(s)) must be replaced by something rational. When the damping is weak it is possible to do this by separating the oscillatory modes from the creep modes and ignoring the latter; in general, this step remains incomplete. Some numerical data illustrating all the stages of the process as well as the final results for various viscoelastic constitutive relations are presented.     

Limit Cycles for Multidimensional Vector Fields. The Elliptic Case

We consider polynomial vector fields of the form and their polynomial perturbations of degree n. We present a sufficient condition that the perturbed system has an invariant surface close to the plane z = 0. We study limit cycles which appear on this surface. The linearized condition for limit cycles, bifurcating from the curves y ² – x ³ + 3x = h, leads to a certain 2- dimensional integral (which generalizes the elliptic integrals). We show that this integral has a representation R ₁(h)I ₁ + + R _e(h)I _e, where R _j (h) are rational functions with degrees of numerators and denominators bounded by O(n). In the case of constant and one-dimensional matrix A(x,y) we estimate the number of zeros of the integral by const n.     

Extremal Synthesis for Generic Planar Systems

For a generic single-input planar control system we analyze the structure of the set of extremals for the time-optimal problem. Generically all extremals are finite concatenations of regular arcs that are bang or correspond to a smooth feedback. Moreover, the support of extremals is a Whitney stratified set. We collect these information in the definition of extremal synthesis. In the cotangent bundle, we give a topological classification of the singularities of the extremal synthesis and study the projections of the support of extremals (regarded as a two-dimensional object, after normalization) from ² × S ¹ to the plane. With respect to the Whitney classical singularities here we deal with a stratified set with edges and corners, and along with cusps and folds, we find other stable singularities.     

Tangent, Quasitangent, and Peritangent Directions to the Trajectories of Differential Inclusions

We introduce several new types of tangent and generalized tangent directions to the trajectories of autonomous differential inclusions and prove some refinements of the existing results on lower estimates and even exact characterizations of these sets in the case where the differential inclusion is either locally Lipschitz or continuously parametrized.