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We prove that the Hausdorff dimension of a set Fand the Hausdorff dimension of its image ϕ(F)with respect to the measure μ:=λoϕ−1are equal under coding map ϕif μis Gibbs measure and λis Lebesque measure.

Abstract: Generalized Thue-Morse sequences are introduced into vibrational question of a hain of masses linked by springs of constant strength, and increasing property is proven of spectrum Δn of linear operator In about the vibrational model.