Shadowing and Inverse Shadowing for C^1 Endomorphisms

In this paper, we consider the shadowing and the inverse shadowing properties for C^1 endomorphisms. We show that near a hyperbolic set a C^1 endomorphism has the shadowing property, and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods. Moreover, each of these shadowing properties is also ＂uniform＂ with respect to C^1 perturbation.     

Lauwerier吸引子结构和SBR测度

Abstract. In this paper,the Lauwerier map     

Metrics of Hyperbolic Type on Bounded Fractals

On the bounded Sierpinski gasket F we use the set of essential fixed points V ₀ as a boundary and consider the fractal Brownian motion on F killed in V ₀. The corresponding Dirichlet–Laplacian is described in terms of a kind of hyperbolic distance, a metric which explodes near the boundary. We consider Harnack inequalities, Green’s function estimates and (random) products of matrices defining the local energy of harmonic functions. Supported by the DFG research group ‘Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik.’     

An isomorphism theorem for Teichmüller curves

It is proved that for any Fuchsian group Γ such that ℍ/Γ is a hyperbolic Riemann surface, the Teichmüller curve V(Γ) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Γ) onto V(Γ) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmüller curves is deduced, which generalizes a classical result that the Teichmüller curve V(Γ) depends only on the type of Γ and not on the orders of the elliptic elements of Γ when ℍ/Γ is a compact hyperbolic Riemann surface.     

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

Lower bounds to helly numbers of line transversals to disjoint congruent balls

A line ℓ is a transversal to a family F of convex objects in ℝ ^d if it intersects every member of F. In this paper we show that for every integer d ⩾ 3 there exists a family of 2d−1 pairwise disjoint unit balls in ℝ ^d with the property that every subfamily of size 2d − 2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957. Crucial to the proof is the notion of a pinned transversal, which means an isolated point in the space of transversals. Here we investigate minimal pinning configurations and construct a family F of 2d−1 disjoint unit balls in ℝ ^d with the following properties: (i) The space of transversals to F is a single point and (ii) the space of transversals to any proper subfamily of F is a connected set with non-empty interior.     

When Will Every Maximal <Emphasis Type="BoldItalic">F</Emphasis>-free Subposet Contain a Maximal Element?

Let F be a partially ordered set (poset). A poset P is called F-free if P contains no subposet isomorphic to F. A finite poset F is said to have the maximal element property if every maximal F-free subposet of any finite poset P contains a maximal element of P. It is shown that a poset F with at least two elements has the maximal element property if and only if F is an antichain or F ≅ 2 + 2.     

Complementing a finite subgroup of a hyperbolic group by a free factor

We give necessary and sufficient conditions for a finite subgroup H of a hyperbolic group G to contain a free subgroup F of rank two in G such that F and H generate a free product F ∗ H. A verification algorithm for these conditions is pointed out.     

Dichotomies of the set of test measures of a Haar-null set

We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF _σ subset ofX which is neither prevalent nor Haar-null. Research supported by a grant of EPEAEK program “Pythagoras”.     

The Forest Hiding Problem

Let Ω be a disk of radius R in the plane. A set F of unit disks contained in Ω forms a maximal packing if the unit disks are pairwise interior-disjoint and the set is maximal, i.e., it is not possible to add another disk to F while maintaining the packing property. A point p is hidden within the “forest” defined by F if any ray with apex p intersects some disk of F, that is, a person standing at p can hide without being seen from outside the forest. We show that if the radius R of Ω is large enough, one can find a hidden point for any maximal packing of unit disks in Ω. This proves a conjecture of Joseph Mitchell. We also present an O(n ^5/2log n)-time algorithm that, given a forest with n (not necessarily congruent) disks, computes the boundary illumination map of all disks in the forest.     

Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles

Let F′,F be any two closed orientable surfaces of genus g′ > g≥ 1, and f:F→ F be any pseudo-Anosov map. Then we can “extend” f to be a pseudo- Anosov map f′:F′→ F′ so that there is a fiber preserving degree one map M(F′,f′)→ M(F,f) between the hyperbolic surface bundles. Moreover the extension f′ can be chosen so that the surface bundles M(F′,f′) and M(F,f) have the same first Betti numbers. Y. Ni is partially supported by a Centennial fellowship of the Graduate School at Princeton University. S.C. Wang is partially supported by MSTC     

Pants Decompositions of Random Surfaces

Our goal is to show, in two different contexts, that “random” surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g ^7/6−ε . Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil–Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.     

On certain hyperbolic sets

Two invariant sets F of certain diffeomorphisms S that were described by A. Fathi, S. Crovisier, and T. Fisher as examples of hyperbolic sets with the property (unexpected at that time) that, in some neighborhood of such an F, there is no locally maximal set containing F are considered. It is proved that this property, although referring to the behavior of the orbits of S near F, is ultimately determined in the examples mentioned above by a combination of a certain explicitly stated intrinsic property of the action of S on F with the hyperbolicity of F. (This means that if a hyperbolic set F′ for a diffeomorphism S′ is equivariantly homeomorphic to a Fathi-Crovisier or a Fisher set, then F′ has a neighborhood in which S′ has no locally maximal set containing F′.)     

Triangle subgroups of Kleinian groups

We exhibit an interesting new phenomenon concerning certain triangle subgroups Δ of Kleinian groups Γ. Namely the hyperbolic plane Π stabilized by Δ has a precisely invariant tubular neighbourhood. Thus the corresponding 2-orbifoldF ²=∏/Γ _∏ is always embedded in the hyperbolic 3-orbifoldM ³=ℍ³/Γ. We deduce that any two such triangle groups can algebraically intersect only in a finite cyclic subgroup. We give sharp estimates for the radius of these tubular neighbourhoods and present applications concerning the estimation of co-volumes of Kleinian groups containing these triangle subgroups. for J. A. Kalman on the occasion of his 65th birthday Research supported in part by grants from the Australian Research Council, the New Zealand Foundation for Research Science and Technology and the U.K. Scientific and Engineering Research Council.     

The conjugating map for commutative groups of circle diffeomorphisms

For a single aperiodic, orientation preserving diffeomorphism on the circle, all known local results on the differentiability of the conjugating map are also known to be global results. We show that this does not hold for commutative groups of diffeomorphisms. Given a set of rotation numbers, we construct commuting diffeomorphisms inC ^2-ε for all ε>0 with these rotation numbers that are not conjugate to rotations. On the other hand, we prove that for a commutative subgroupF ⊂C ^1+β, 0<β<1, containing diffeomorphisms that are perturbations of rotations, a conjugating maph exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.     

Hölder Regularity of μ-Similar Functions

Given a positive measure μ, d contractions on [0,1] and a function g on ℝ, we are interested in function series F that we call “μ-similar functions” associated with μ, g and the contractions. These series F are defined as infinite sums of rescaled and translated copies of the function g, the dilation and translations depending on the choice of the contractions. The class of μ-similar functions F intersects the classes of self-similar and quasi-self-similar functions, but the heterogeneity we introduce in the location of the copies of g make the class much larger. We study the convergence and the global and local regularity properties of the μ-similar functions. We also try to relate the multifractal properties of μ to those of F and to develop a multifractal formalism (based on oscillation methods) associated with F.     

Approximation of analytic by Borel sets and definable countable chain conditions

LetI be a σ-ideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ ₁ ¹ countable chain condition if there is aΣ ₁ ¹ equivalence relation with uncountably many equivalence classes none of which is inI. Assuming definable determinacy, we show that if the family of Borel sets fromI is definable in the codes of Borel sets, then eachΣ ₁ ¹ set is equal to a Borel set modulo a set fromI iffI fulfils theΣ ₁ ¹ countable chain condition. Further we characterize the σ-idealsI generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property forΣ ₁ ¹ sets mentioned above. It turns out that they are exactly of the formMGR(F)={A : ∀F ∈ F A ∩F is meager inF} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager sets is the unique σ-ideal onR, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition. Research partially supported by NSF grant DMS-9317509.     

On local solutions of degenerate differential inclusions

We study the existence and properties of local solution sets for differential inclusions of the form (Ax)′ ∈ F(t, x), where A is a closed linear surjective operator with nontrivial null space and F is a compact set-valued mapping.     

An improved general extra-gradient method with refined step size for nonlinear monotone variational inequalities

Extra-gradient method and its modified versions are direct methods for variational inequalities VI(Ω, F) that only need to use the value of function F in the iterative processes. This property makes the type of extra-gradient methods very practical for some variational inequalities arising from the real-world, in which the function F usually does not have any explicit expression and only its value can be observed and/or evaluated for given variable. Generally, such observation and/or evaluation may be obtained via some costly experiments. Based on this view of point, reducing the times of observing the value of function F in those methods is meaningful in practice. In this paper, a new strategy for computing step size is proposed in general extra-gradient method. With the new step size strategy, the general extra-gradient method needs to cost a relatively minor amount of computation to obtain a new step size, and can achieve the purpose of saving the amount of computing the value of F in solving VI(Ω, F). Further, the convergence analysis of the new algorithm and the properties related to the step size strategy are also discussed in this paper. Numerical experiments are given and show that the amount of computing the value of function F in solving VI(Ω, F) can be saved about 12–25% by the new general extra-gradient method.