The ℑ-density topology $T_{\Im }$ on ℝ is a refinement of the natural topology. It is a category analogue of the density topology [9, 10]. This paper is concerned with ℑ-density continuous functions, i.e., the real functions that are continuous when the ℑ-densitytopology is used on the domain and the range. It is shown that the family $C_{\Im }$ of ordinary continuous functions f: [0,1]→ℝ which have at least one point of ℑ-density continuity is a first category subset of C([0,1])= f: [0,1]→ℝ: f is continuous equipped...

In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. 348 (1996), 1357–1372] about characterization of $\omega$-limit sets from the class $𝒞(I,I)$ of continuous maps of the interval to the class $𝒞(𝕊,𝕊)$ of continuous maps of the circle. Among others we give geometric characterization of $\omega$-limit sets and then we prove that the family of $\omega$-limit sets is closed with respect to the Hausdorff metric.

The system of Abel equations α(ft(x)) = α(x) + λ(t), t ∈ T, is studied under the general assumption that $f_t$ are pairwise commuting homeomorphisms of a real interval and have no fixed points (T is an arbitrary non-empty set). A result concerning embeddability of rational iteration groups in continuous groups is proved as a simple consequence of the obtained theorems.

Let $W\left(I\right)$ denote the family of continuous maps $f$ from an interval $I=[a,b]$ into itself such that (1) $f\left(a\right)=f\left(b\right)\in \{a,b\}$; (2) they consist of two monotone pieces; and (3) they have periodic points of periods exactly all powers of $2$. The main aim of this paper is to compute explicitly the topological sequence entropy $h_D\left(f\right)$ of any map $f\in W\left(I\right)$ respect to the sequence $D={\left(2^{m-1}\right)}_{m=1}^{\infty }$.