An Exceptional Set in the Ergodic Theory of Expanding Maps on Manifolds

Under a general hypothesis an expanding map T of a Riemannian manifold M is known to preserve a measure equivalent to the Liouville measure on that manifold. As a consequence of this and Birkhoff’s pointwise ergodic theorem, the orbits of almost all points on the manifold are asymptotically distributed with regard to this Liouville measure. Let T be Lipschitz of class τ for some τ in (0,1], let Ω(x) denote the forward orbit closure of x and for a positive real number δ and let E(x₀, δ) denote the set of points x in M such that the distance from x₀ to Ω is at least δ. Let dim A denote the Hausdorff dimension of the set A. In this paper we prove a result which implies that there is a constant C(T) > 0 such that dimE(x₀,d) 3 dimM - \fracC(T)|logd| \dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\vert\!\log \delta \vert} if τ = 1 and dimE(x₀,d) 3 dimM - \fracC(T)log|logd|\dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\log \vert \log \delta \vert} if τ < 1. This gives a quantitative converse to the above asymptotic distribution phenomenon. The result we prove is of sufficient generality that a similar result for expanding hyperbolic rational maps of degree not less than two follows as a special case.     

Approximation of elements of a generalized Orlicz sequence space by nonlinear, singular kernels

Let l be a generalized Orlicz sequence space generated by a modular (x) = ∑_{i − 0}^∞ _i(¦t_i¦), X = (t_i), with s-convex functions _i, 0 < s 1, and let K_w,j: R₊ → R₊ for j=0,1,2,…, w ε Wwhere is an abstract set of indices. Assuming certain singularity assumptions on the nonlinear kernel K_w,j and setting T_wx = ((T_wx)_i)_{i = 0}^∞, with (T_wx)_i = ∑_{j = 0}ⁱ K_{w,i − j}(¦t_j¦) for x = (t_j), convergence results: T_wx → x in l are obtained (both modular convergence and norm convergence), with respect to a filter of subsets of the set .     

On the Lipschitz equivalence of self‐affine sets

Recently Lipschitz equivalence of self‐similar sets on has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w. Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.     

Geometric approach to a dilation theorem

Problem: Given operators A_j ? O on Hilbert space $\text{H}$, with ΣA_j = 1, to find commuting projectors E_j on a Hilbert space $\text{H}$ ? $\text{H}$ such that (for all _j) x¹A_jy = x¹E_jy for, x, y ∈ $\text{H}$. This paper gives an explicit construction, quite different from the familiar solution.     

On the definition of Bourgain points

This paper is devoted to study of the so-called Bourgain points (B-points) for functions from L^∞(ℝ). In 1993, Bourgain showed that for a real-valued bounded function f, the set E_f of B-points is everywhere dense and has the maximal Hausdorff dimension, dim _H(E_f ) = 1; in addition, the vertical variation of the harmonic extension of f to the upper half-plane is finite at B-points. An essentially simpler definition of B-points is given compared to that in the original works by Bourgain. A geometric characterization of B-points for Cantor-like sets is obtained. Bibliography: 7 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 219–236.     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART II): In terms of a given Hamiltonian function the 1-form w = dH + ?_j|dπ^j is defined, where {?_j:j = 1,…, n} denotes an invariant basis of the planes of the distribution D_n. The latter is said to be canonical if w = 0 (which is analogous to the definition of Hamiltonian vector fields in symplectic geometry). This condition is equivalent to two sets of canonical equations that are expressed explicitly in term of the derivatives of H with respect to its positional arguments. The distribution D_n is said to be pseudo-Lagrangian if dπ^j(?_j,V_h) = 0; if D_n, is both canonical and pseudo-Lagrangian it is integrable and such that H = const. on each leaf of the resulting foliation. The Cartan form associated with this construction [9] is defined a II = π² ? ? πⁿ. If π is closed, the distribution D_N is integrable, and the exterior system {π^j} admits the representation ψ^j = dS^j in terms of a set of 0-forms S^j on M. If, in addition, the distribution D_N is canonical, these functions satisfy a single first order Hamilton-Jacobi equation, and conversely. Finally, a complete figure is constructed on the basis of the assumptions that (i) the Cartan form be closed, and (ii) that the distribution D_n, be both canonical and integrable. The last of these requirements implies the existence of N functions ψ^A that depend on x^h and N parameters w^B, whose derivatives are given by ?ψ^A (x^h, w^B)/?x^j = B^A _j (x^h, ψ^B (x^h,w^B)). The complete figure then consists of two complementary foliations: the leaves of the first are described by the functions ψ^A and satisfy the standard Euler-Lagrange equations, while the second, that is, the transversal foliation, is represented by the aforementioned solution of the Hamilton-Jacobi equation. The entire configuration then gives rise in a natural manner to a generalized Hilbert independent integral and consequently also to a generalized Weierstrass excess function.     

Atomic decompositions of function spaces with Muckenhoupt weights,and some relation to fractal analysis

This paper deals with atomic decompositions in spaces of type B^s_p,q (?ⁿ , w), F^s_p,q (?ⁿ , w), 0 < p < ∞, 0 < q ≤ ∞, s ∈ ?, where the weight function w belongs to some Muckenhoupt class A_r. In particular, we consider the weight function w^Γ_κ (x) = dist(x, Γ)^κ, where Γ is some d ‐set, 0 < d < n, and κ > –(n – d). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hausdorff measure of uniform self-similar fractals

Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R ^d . Denote by D the Hausdorff dimension, by H ^D (E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and H ^D (E) = diam(E) ^D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover, we present a parametrised UIFS, where ω depends on c and show the inequatily H ^D (E) < diam(E) ^D , if c is small enough.     

Stability of semilinear ill-posed problems with a prescribed energy bound

In the present paper we discuss the stability of semilinear problems of the form A^αu + G^α(u) = ? under assumption of an a priori bound for an energy functional E^α(u) ? E, where α is a parameter in a metric space M. Following [11] the problem A^αu + G^α(u) = ?, E^α(u) ? E is called stable in a Hilbert space H at a point α ? M if for any ??H, E, ? > 0 there exists δ > 0 such that for any functions u^α1, u^α2 satisfying A^αju^αj + G^αj(u^αj) = ?^αj, E^αj(u^αj) ? E, j = 1,2 we have ‖u^α1 ? u^α2_H ? ? provided ρ_M(α_j, α) ? δ, ‖?^αj ? ?‖_H ? δ, j = 1,2. In the present paper we obtain stability conditions for the problem A^αu + G^α(u) = ?, E^α(u) ? E.