Paul D. Humke

The space of -limit sets of a continuous map of the interval

We first give a geometric characterization of ω-limit sets. We then use this characterization to prove that the family of ω-limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems.

THE BOREL STRUCTURE OF ITERATES OF CONTINUOUS-FUNCTIONS

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f ∈C[0,1] mapping [0,1] into itself. If f ∈C, f k will denote the kth iterate of f and we put C k = { f k : f ∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I , we let C( f ) denote the set of points at which f is locally constant, i.e. We let N denote the set of positive integers and N N denote the Baire space of sequences of positive integers.

Characterizations of turbulent one-dimensional mappings via -limit sets

The structure of «-limit sets for nonturbulent functions is studied, and various characterizations for turbulent and chaotic functions are obtained. In particular, it is proved that a continuous function mapping a compact interval into itself is turbulent if and only if there exists an «-limit set which is a unilaterally convergent sequence

Measurable Darboux functions

We investigate how certain Darboux-like properties of real functions (including connectivity, almost continuity, and peripheral continuity) are related to each other within certain measurability classes (including the classes of Lebesgue measureable, Borel, and Baire-1 functions).

Symmetric and ordinary differentiation

In 1927, A. Khintchine proved that a measurable symmetrically differentiable function f mapping the real line R into itself is differentiable in the ordinary sense at each point of R except possibly for a set of Lebesgue measure zero. Here it is shown that this exceptional set is also of the first Baire category; even more, it is shown to be a a-porous set of E. P. Dolienko.

CONTRASTING SYMMETRIC POROSITY AND POROSITY

Abstract It is known that the following two fundamental properties of porosity fail for symmetric porosity: 1) Every nowhere dense set A contains a residual subset of points x at which A has porosity 1. 2) If A is a porous set and 0 < p < 1, then A can be written as a countable union of sets, each of which has porosity at least p at each of its points. Here we explore the somewhat surprising extent to which these properties fail to carry over to the symmetric setting and investigate what symmetric analogs do hold.

Revisiting a Century-Old Characterization of Baire Class One, Darboux Functions

dans theorem was that if a primitive subgroup G of the symmetric group Sn contains a 3-cycle, then G must be either An or Sn. In fact, there are many situations in which one can show that a primitive subgroup of Sn must be either An or Sn. One sufficient condition, for example, is that G contains a p-cycle, where p is a prime and p k such that a and pn generate either Sn or An. Of course, this follows from our result too: simply take n > 2k 1, where n is coprirne to g (a). (And indeed, Piccards proof is not really very different from this.) We thank the referee of our paper who alerted us to the existence of Piccards work.

Universally Polygonally Approximable Functions

Abstract It has recently been established that any Baire class one function ƒ : [0, 1] → ℝ can be represented as the pointwise limit of a sequence of polygonal functions whose vertices lie on the graph of ƒ. Here we investigate the subclass of Baire class one functions having the additional property that for every dense subset D of [0, 1], the first coordinates of the vertices of the polygonal functions can be chosen from D.

Analogues of the Denjoy-Young-Saks theorem

In this paper, an analogue of the Denjoy-Young-Saks theorem concerning the almost everywhere classification of the Dini dérivâtes of an arbitrary real function is established in both the case where the exceptional set is of first category and the case where it is o-porous. Examples are given to indicate the sharpness of these results.

Collections of Darboux-like, Baire one functions of two variables

Abstract The collection of Baire class one, Darboux functions from [0, 1] to ℝ is a rich class of functions that has been intensely investigated and characterized over the years. Which Baire class one, real-valued functions defined on [0, 1] × [0, 1] form the most “natural” extension of this class to the two-variable setting is debatable, with many suggestions having been advanced. In light of recent interest in Darboux-like properties for derivatives (i.e. gradients) of differentiable functions of two variables, it seems that now is a good time to consider some of the most feasible notions of “Darboux-like” and investigate the relationships between them.