Paths through inverse limits

In Bani?, ?repnjak, Merhar and Milutinovi? (2010) [2] the authors proved that if a sequence of graphs of surjective upper semi-continuous set-valued functions fn:X→X² converges to the graph of a continuous single-valued function f:X→X, then the sequence of corresponding inverse limits obtained from fn converges to the inverse limit obtained from f. In this paper a more general result is presented in which surjectivity of fn is not required. The result is also generalized to the case of inverse sequences with non-constant sequences of bonding maps. Finally, these new theorems are applied to inverse limits with tent maps. Among other applications, it is shown that the inverse limits appearing in the Ingram conjecture (with a point added) form an arc.     

Quasi-monotone mappings on θn-continua

A continuous function f from a continuum X onto a continuum Y is quasi-monotone if, for every subcontinuum M of Y with nonvoid interior, f^-1(M) has a finite number of components each of which is mapped onto M by f. A θ_n-continuum is one that no subcontinuum separates into more than n components. It is known that if f is quasi-monotone and X is a θ₁-continuum, then Y is a θ₁-continuum or a θ₂-continuum that is irreducible between two points. Examples are given to show that this cannot be generalized to a θ_n-continuum and n + 1 points for any n >1, but it is proved that if f is quasi-monotone and X is a θ_n-continuum, then Y is a θ_n-continuum or a θ_n+1-continuum that is the union of n + 2 continua H,S₁,S₂,…,S_n+1, whe for each i, S_i is the closure of a component of Y H, S_i is irreducible from some point P_i to H, and H is irreducible about its boundary. Some theorems and examples are given concerning the preservation of decomposition elements by a quasi-monotone map defined on a θ_n-continuum that admits a monotone, upper-semicontinuous decomposition onto a finite graph.     

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf _n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?f_n(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf _A:X→R ^ω such that A=f _A ^?1 (A). Splitting spaces will be studied systematically.     

Closed subsets of the domain whose image has the dimension of the range

It is shown that if dim Y < ∞ and if f(X) = Y is a mapping between compact metric spaces such that 1 ? m ? dim f^-1(y)?n for all y ? Y, then there exists a closed set K ? X such that dim K ? n ? m and dim f(K) = dim Y. This answers a question posed by J. Keesling and D. Wilson.     

Embedding Tn-like continua in Euclidean space

Many authors have been concerned with embedding ∏-like continua in Rⁿ where ∏ is some collection of polyhedra or manifolds. A similar concern has been embedding ∏-like continua in Rⁿ up to shape. In this paper we prove two main theorems. Theorem: If n ? 2 and X is Tⁿ-like, then X embeds in R²ⁿ. This result was conjectured by McCord for the case H¹(X) finitely generated and proved by McCord for the case that H¹(X) = 0 using a theorem of Isbell. The second theorem is a shape embedding theorem. Theorem: If X is Tⁿ-like, then X embeds in Rⁿ⁺² up to shape. This theorem is proved by showing that an n-dimensional compact connected abelian topological group embeds in Rⁿ⁺². Any Tⁿ-like continuum is shape equivalent to a k-dimensional compact connected abelian topological group for some 0 ? k ? n.     

Limits of inverse limits

We study the following problem: if a sequence of graphs of upper semi-continuous set valued functions fn converges to the graph of a function f, is it true that the sequence of corresponding inverse limits obtained from fn converges to the inverse limit obtained from f?     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

Closed maps on metric spaces

The following result is proved: Let Y be the image of a metric space X under a closed map f. Then every ?f^-1(y) is Lindelöf if and only if Y has a point-countable k-network.     

Limit laws of entrance times for homeomorphisms of the circle

Given a homeomorphismf of the circle with irrational rotation number and a descending chain of renormalization intervalsj _n off, we consider for each interval the point process obtained by marking the times for the orbit of a point in the circle to enterJ _n. Assuming the point is randomly chosen by the unique invariant probability measure off, we obtain necessary and sufficient conditions which guarantee convergence in law of the corresponding point process and we describe all the limiting processes. These conditions are given in terms of the convergent subsequences of the orbit of the rotation number off under the Gauss transformation and under a certain realization of its natural extension. We also consider the case when the point is randomly chosen according to Lebesgue measure,f being a diffeomorphism which isC ¹-conjugate to a rotation, and we show that the same necessary and sufficient conditions guarantee convergence in this case.     

On weighted lattice paths

A weighted lattice path from (1, 1) to (n, m) is a path consisting of unit vertical, horizontal, and diagonal steps of weight w. Let f(0), f(1), f(2), … be a nondecreasing sequence of positive integers; the path connecting the points of the set $\text{{(n, m) ¦ f(n ? 1) ? m ? f(n), n = 1, 2, …}}$ will be called the roof determined by f. We determine the number of weighted lattice paths from (1, 1) to (n + 1, f(n)) which do not cross the roof determined by f. We also determine the polynomials that must be placed in each cell below the roof such that if a 1 is placed in each cell whose lower left-hand corner is a point of the roof, every k × k square subarray comprised of adjacent rows and columns and containing at least one 1 will have determinant $\text{x(}\text{k}\text{2}\text{)}$.     

F-limit points in dynamical systems defined on the interval

Given a free ultrafilter p on ? we say that x ∈ [0, 1] is the p-limit point of a sequence (x _n ) _n∈? ? [0, 1] (in symbols, x = p -lim _n∈? x _n ) if for every neighbourhood V of x, {n ∈ ?: x _n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f ^p : [0, 1] → [0, 1] is defined by f ^p (x) = p -lim _n∈? f ⁿ (x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f ^p . For a filter F we also define the ω ^F -limit set of f at x. We consider a question about continuity of the multivalued map x → ω _f ^F (x). We point out some connections between the Baire class of f ^p and tame dynamical systems, and give some open problems.     

Extraordinary dimension of maps

We consider the extraordinary dimension dimL introduced recently by Shchepin [E.V. Shchepin, Arithmetic of dimension theory, Russian Math. Surveys 53 (5) (1998) 975-1069]. If L is a CW-complex and X a metrizable space, then dimLX is the smallest number n such that ΣnL is an absolute extensor for X, where ΣnL is the nth suspension of L. We also write dimLf?n, where is a given map, provided dimLf⁻¹(y)?n for every y∈Y. The following result is established: Supposeis a perfect surjection between metrizable spaces, Y a C-space and L a countable CW-complex. Then conditions (1)-(3) below are equivalent:

(1)

dimLf?n;

(2)

There exists a dense andGδsubsetGofC(X,In)with the source limitation topology such thatdimL(f×g)=0for everyg∈G;

(3)

There exists a mapis such thatdimL(f×g)=0;If, in addition, X is compact, then each of the above three conditions is equivalent to the following one;

(4)

There exists anFσsetA⊂Xsuch thatdimLA?n−1and the restriction mapf|(X?A)is of dimensiondimf|(X?A)?0.

     

On resolvable primal spaces

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If f is a selfmap on X, the sets A?X with f (A)?A are the closed sets of an Alexandroff topology called the primal topology 𝒫(f ) associated with f. We investigate resolvability for primal spaces (X, 𝒫(f)). Our main result is that an Alexandroff space is resolvable if and only if it has no isolated points. Moreover, n-resolvability and other related concepts are investigated for primal spaces.     

A small decrease in the degree of a polynomial with a given sign function can exponentially increase its weight and length

A Boolean function f: {?1, +1} ⁿ → {?1, +1} is called the sign function of an integer-valued polynomial p(x) if f(x) = sgn(p(x)) for all x ∈ {?1, +1} ⁿ . In this case, the polynomial p(x) is called a perceptron for the Boolean function f. The weight of a perceptron is the sum of absolute values of the coefficients of p. We prove that, for a given function, a small change in the degree of a perceptron can strongly affect the value of the required weight. More precisely, for each d = 1, 2, ..., n ? 1, we explicitly construct a function f: {?1, +1} ⁿ → {?1, +1} that requires a weight of the form exp{Θ(n)} when it is represented by a degree d perceptron, and that can be represented by a degree d + 1 perceptron with weight equal to only O(n ²). The lower bound exp{Θ(n)} for the degree d also holds for the size of the depth 2 Boolean circuit with a majority function at the top and arbitrary gates of input degree d at the bottom. This gap in the weight values is exponentially larger than those that have been previously found. A similar result is proved for the perceptron length, i.e., for the number of monomials contained in it.     

Very Strange Projective Curves

Let Y ? Pⁿ, n ≥ 3, be an integral non-degenerate very strange projective curve, i.e. assume that the general hyperplane section of Y is not in linearly general position; hence we are in characteristic p. Let π: X → Y be the normalization. Set d? deg(Y), g? Pa(X) and L? π* (Oy(1)). Here we prove that d > 2g?2 and in particular h¹(X,L) = 0 and h⁰(X,L) = d+1?g ≥ (d?1)/2 > n+1.     

On the asymptotic behavior of the bestL ∞-approximation by polynomials

LetE _n (f) denote the sup-norm-distance (with respect to the interval [?1, 1]) betweenf and the set of real polynomials of degree not exceedingn. For functions likee ^x , cosx, etc., the order ofE _n (f) asn→∞ is well known. A typical result is $$2^{n - 1} n!E_{n - 1} (e^x ) = 1 + 1/4n + O(n^{ - 2} ).$$ It is shown in this paper that 2 ^n?1 n!E _n?1(e ^x ) possesses a complete asymptotic expansion. This result is contained in the more general result that for a wide class of entire functions (containing, for example, exp(cx), coscx, and the Bessel functionsJ _k (x)) the quantity $$2^{n - 1} n!E_{n - 1} \left( f \right)/f^{(n)} \left( 0 \right)$$ possesses a complete asymptotic expansion (providedn is always even (resp. always odd) iff is even (resp. odd)).     

Ruzsa’s constant on additive functions

A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20.     

Certain finite dimensional maps and their application to hyperspaces

In [8] Y. Sternfeld and this author gave a positive answer to the following longstanding open problem: Is the hyperspace (=the space of all subcontinua endowed with the Hausdorff metric) of a 2-dimensional continuum infinite dimensional? This result was improved in [9] where it was shown that for every positive integer numbern a 2-dimensional continuum contains a 1-dimensional subcontiuum with hyperspace of dimension ≥n. And it was asked there: Does a 2-dimensional continuum contain a 1-dimensional subcontinuum with infinite dimensional hyperspace? In this note we answer this question in the positive. Our proof applies maps with the following properties. A real valued mapf on a compactumX is called a Bing map if every continuum that is contained in a fiber off is hereditarily indecomposable.f is called ann-dimensional Lelek map if the union of all non-trivial continua which are contained in the fibers off isn-dimensional. It is shown that for dimX=n+1 the maps which are both Bing andn-dimensional Lelek maps form a denseG _σ-subset of the function spaceC(X, I)     

Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.