Shadowing, transitivity and a variation of omega-chaos

We study a special type of shadowing (DSP) of chain transitive continuous self-maps of compact Hausdorff spaces. We prove some basic properties of DSP. As application of DSP, we obtain sufficient conditions for a statistical variant of $\omega$-chaos and prove the topological genericity of it. We also consider topological distribution of irregular points under the assumption of DSP.