An Interdisciplinary Journal

2019, Vol.22, No.2, pp.164 - 176

Additive Dimension Theory for Birkhoff Curves
Alexander V. Osipov, Ivan A. Kovalew, and Dmitry W. Serow

The additive dimension for a common boundary of the Wada basins bases (and Wada ocean) accessible points has been defined. One is constituted to be value being inverse to fractional density for the sequence (basis) zero Schnirelmann density and one characterizes only metric property of the boundary (Birkhoff curve). The additive dimension is similar to Hausdorff-Besicovitch dimension. All Wada basin and Wada ocean are quite metrically characterized to be only additive dimension of accessible points. It follows that additive dimension is invariant with respect to a plane diffeomorphism.

Key words: dissipative dynamic system, accessible point, Birkhoff curve, indecomposable continuum (atom), Wada basins, Cantor set, top of umbrella, Schnirelmann density, fractional density, additive basis, Hausdorff-Besicovitch dimension

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