Dynamics and Chaotic of Polynomials on Quasi Banach Spaces

In the present paper, the concepts of a quasi-metric space, quasi-Banach space have been introduced. We prove some facts which are defined on these spaces and define some polynomials on quasi-Banach spaces and studied their dynamics, such as, quasi cyclic and quasi hypercyclic. We show the existence of quasi chaotic in the sense of Devaney (quasi D-chaotic) polynomials on quasi Banach space of q-summable sequences lq , 0 q 1 such polynomials P is defined by P((xi)i)=(p(xi+m))i where p:C→C, p(0) = 0. In general we also prove that P is quasi chaotic in the sense of Auslander and Yorke (quasi AY-chaotic) if and only if 0 belong to the Julia set of p, ∀m∈N. And then we prove that if the above polynomial P on lq , 0 q 1 is quasi. AY-chaotic then so is λP where λ∈R+ with λ≥1 and Pn for each n≥2.