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By a topological dynamical system (TDS for short) (X,f),we mean a compact metric space X together with a surjective continuous map f:X rarr X.The chaoticity of a TDS is a centrum topic of the research since the introducing the term of chaos in 1975 by Li and York, known as Li-York chaos today, and notions of sensitivity is the kernel in the definition of chaos in the Ruelle-Takens chaos and Schweizer-Smital...
In this paper, we prove that there exists a subsystem of a one-sided symbolic space with two symbols such that the set-valued map on it is topologically ergodic, topologically double ergodic, topologically transitive and topologically weakly mixing.
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