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The topological classification of the inner mappings on the fully invariantregular components of the wandering set with a special attracting boundary upto the topological conjugacy is defined in terms of distinguishing graph. Twoinner mappings of the same degree are topologically equivalent on their fullyinvariant regular components of a certain class if and only if theirdistinguishing graphs are equivalent.

On the topological classification of dynamics of inner mappings on the regular components of the wandering set with a special attracting boundary