Author(s): Dr. Satya Prakash Gupta

Abstract: This Paper we characterize in this paper, we need the following terminology and concepts. Let N={0,1,2,…,} be the set of natural numbers and let Z={0,±2…,} be the set of integers. For a set A, |A| denotes the cardinality of the set A. If f :X→X is map (=continuous function) of a topological space X, then f0=Id (n≥1) denotes the composition with itself n times. The orbit of a point x∈X under f, denoted by 0+(x, f), is the set {fn├ (x)┤|n∈N}. Also if f: X→X is a homeomorphism, then we put f-n=(f-1)n(n≥1), where f-1 is the inverse of f. The two-sided orbit of a point x∈X under f, denoted by O+(x,f), is the set {fn├ (x)┤|n∈Z}. A point x∈X is called a periodic point of f if there exists a positive number N∈N such that fN(x)=x. A point x∈X is called an almost periodic point of f provided that for any neighborhood U of x in X, there exists N∈N such that {fn+i├ (x)┤|i=0,1,2,.........N}∩U≠ for all N∈N. We denote the set of all almost periodic points of f by AP(f). A subject W of X is invariant of f if W ≠and f(W)⊆W.A subset W of X is a minimal set of f if W is a closed invariant set off and W does not contain any proper closed invariant set off. A map f: X→X is minimal if X is a minimal set off. It is well known that if f: S1→S1 is an irrational rotation of the unit circle SI, then f is minimal.

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