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In this paper, for iterated function systems, we define the classic conceptof the dynamical systems: topological conjugacy of diffeomorphisms. Wegeneralize the Hartman-Grobman theorem for one dimensional iterated functionsystems on R. Also, we introduce the basic concept of structural stability foran iterated function system and so we investigate the necessary condition forstructural stability of an iterated function system on R.

Hartman-Grobman Theorem for IFS