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Among the orbit patterns that force only eventually fixed trajectories, we completely describe the forcing relation, by answering the question: which orbit patterns force which others? (for details see [7])). In another [2] it is the language of locating periodic points of period 1 and 2. In yet another [6], it is an index set for the set of orbit-patterns. Statement of Main Result.There is a natural bijection between the language {L, R} * and the set of all EF orbit patterns. The forcing relation on the latter set receives a neat description when framed in the terminology of theory of languages. We are able to find four rules of derivation in {L, R} * so that the following theorem becomes true: An orbit pattern α forces another orbit pattern β if and only if the corresponding word of β can be derived from that of α using four rules of derivation (described below).

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In this article, we provide a simple geometric proof of the following fact: The existence of transitive normalized maps induced by linear operators is possible only when the underlying space's real dimension is either 1 or 2 or infinity. A similar result holds for projective transformation as well.

We exhibit a single interval map (called universal map) that admits all those orbit patterns which are available in the first Sharkovsky class. An interval map is said to be in the first Sharkovsky class if every periodic point of it is a fixed point. This provides a way to find universal maps in the class of contractions on interval. We also characterize all such universal maps in the first Sharkovsky class. However, this result is not true in higher Sharkovsky classes, i.e., there is no universal function for n-th Sharkovsky class when n > 1.