Random Set and Homeomorphisms up to Topological Conjugacy

Abstract

The reason for this work is to surrender Homeomorphisms to Topological Conjugacy and thorough irregular set hypothesis content to the investigation of nonlinear arbitrary examination, for example, the investigation of general numerical models, estimate hypothesis, settled point hypothesis, and soundness of utilitarian conditions in arbitrary spaces. The application of topological conjugacy decomposition and random circle homeomorphism showcases the topological space properties that re-iteration of any given topological function to itself must generate Conjugacy irrespective of the number of times a function is spilt into new sub shifts. Additionally, for homeomorphism of random circles there must be finitely many g-minimal connected components indicating that a reversal of the process, as outlined with the topological decomposition must result in non-degeneracy of the sets.  For instance, in classical dynamical systems theory, a common question to ask is whether, for a given pair of self-maps (f, g) of some compact metric space, there is a topological conjugacy from f to g. In this paper, we consider the “analogous” question for a noisy pair of maps {(fα, gα)} α∈∆ where α is drawn randomly from some probability space (∆, B (∆), ν). In our case, a random map (fα)α∈∆ is viewed “dynamically” by considering i.i.d. iterations; topological conjugacy is then understood in the “random dynamical systems” framework, namely as a topology-preserving cohomology between the co cycles generated by random maps (fα)α∈∆ and (gα)α∈∆ over the shift map on (∆Z ,B(∆) ⊗Z , ν⊗Z).  

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