The chaotic set and asymptotic periodic point of descendible mapping;

If is a n-dimensional descendible continuous self-mapping, the necessary and sufficient conditions are given to f which is of n-dimensional self-mapping with its square to be of type 2∞ by using the feature of descendible mapping,as well as itsRf\R(f) being countable.

If f is a n-dimensional descendible continuous self-mapping,the characters of descendible mapping can be used.

Some Dynamical Properties of Descendable mappings.;

Let f∈C 0(I×I,I×I) ,if f is descendable mapping and has no abnormal point, Using character of descendable mapping and Cartesian product and closure operation, one_dimensional self_mapping is extended to two_dimensional self_mapping.

Suppose thatΦ:A→A is an additive map and m,n are positive integers.

Using the additivity of the matrix,it is proved that every additive map preserving the lattices of invariant subspaces is of the form:Φ(A)=αA+φ(A)I(A∈Tn),where α is a nonzero scalar,φ:TnF is an additive map and I∈Tn is an identity.

The form of each additive map φ:M→B(X) is proved that if there exist nonzero real m and n such that (m+n)φ(A2)-mAφ(A)-nφ(A)A ∈FI holds for all A ∈ M, then φ(A)=λA , where λ ∈F.

A note on the linearity of an additive mapping;

We study in this paper the structure of additive mappings on triangular matrix algebras which preserve commutativity.