For a continuous map $f$ of the interval $I$, a set $W\subset I$ is an {\it $\alpha$-limit set} if $W$ is the set of limit points of a sequence $\{ x_n\} _{n=0}^\infty$ in $I$ such that, for any $n$, $f(x_{n+1})=x_n$. Denote by $\alpha (f)$ the system of $\alpha$-limit sets. We prove the following main results. (i) Any minimal set belongs to $\alpha (f)$. (ii) Any $\alpha$-limit set is an $\omega$-limit set which is either minimal, or is contained in a basic set. (iii) If $f$ has zero topological entropy then $\alpha (f)$ is the system of minimal sets. (iv) Any $\omega$-limit set contained in a basic set is an $\alpha$-limit set. (v) The set $\alpha (f)$ need not be closed in the Hausdorff metric; in contrast to this, it is known that the system of $\omega$-limit sets is compact in the Hausdorff metric. (vi) If $f$ has zero topological entropy then $\alpha (f)$ is closed in the Hausdorff metric if and only if the set ${\rm Rec}(f)$ of recurrent points of $f$ is closed in the standard metric in $I$. (vii) Finally, we show that a compact set $W\subset I$ is an $\alpha$-limit set of a suitable continuous map $f:I\rightarrow I$ if and only if $W$ either is a finite union of intervals, or a nowhere dense set. Consequently, $W$ is an $\alpha$-limit set of a continuous map of $I$ if and only if it is an $\omega$-limit set of a continuous map of $I$.
The author is an undergraduate student of Mathematical Analysis at the
Mathematical Institute of the Silesian University in Opava.
This paper is a part of her Master degree thesis, and will be
published in a journal. Paper will be presented to SVO\v C 2003,
the Mathematical Students Competition, May 27 - 29, 2003, B. Bystrica,
Slovakia.