dynamic systems, general topology, numbers theory

1. Osipov, A.V., Kovalew, I.A. & Serow, D.W., “Additive Dimension Theory for Birkhoff Curves”, 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., Nonlinear Phenomena in Complex Systems, 22:2 (2019), 164-176
2. Osipov, A.V. & Serow, D.W., “Fractional Densities for the Wada Basins”, Fractional density for basis zero Schnirelmann density has been defined. Definition of the fractional density is similar to the Hausdorff-Besicovitch dimension. The existence of the basis zero Schnirelmann density for every Wada basin (Wada ocean) earlier has been proved. This means every Wada basin/ocean are quite topologically characterized to be fractional density. Therefore all fractional densities are invariant with respect to a plane homeomorphism., Nonlinear Phenomena in Complex Systems, 21:4 (2018), 389-394
3. Makarova, M.V., Kovalew, I.A. & Serow, D.W., “Antisymmetric Wada Basins Prime Example: Unstable Antisaddles Case”, Prime example of a dynamic system with two antisymmetric invariant Wada basins has been constructed and one has been illustrated in connotation of the thinking in PostScript. The dynamic system has either saddle and two unstable antisaddle fixed points or inverse saddle and two unstable antisaddle two-periodic points. Onset of Wada basins from the cycles has been considered, Nonlinear Phenomena in Complex Systems, 21:2 (2018), 188-193
4. Osipov, A.V. & Serow, D.W., “Rotation Number Additive Theory for Birkhoff Curves”, Rotation number elementary theory for Birkhoff curves has been constructed. Geometrical (dynamical) and numerical properties for Birkhoff curves being more than two regions common boundary has been studied. Topological number invariants with respect to a dissipative dynamic system on the plane possessing the Birkhoff curve property have been discussed. Simple allocation algorithm of natural numbers has been applied, so that its Schnirelmann density is equal to the rotation number for a region. If the region boundary is a Birkhoff curve then the sequence contains an additive basis zero Schnirelmann density. The basis contains an arbitrary long arithmetic progression. Rotation numbers for regions are defined to be different additive bases zero Schnirelmann density., Nonlinear Phenomena in Complex Systems, 20:4 (2017), 382-393
5. Serow, D.W., “Dissipative Dynamical System with Lakes of Vada”, The circumstantial evidence of the homoclinic point existence for dissipative diffeomorphism on the plane with a single inverse saddle fixed point is adduced, that is to say if closure of the unstable manifold of the single inverse saddle fixed point cuts the plane then there exists homoclinic point. Simple example of the dissipative plane diffeomorphism, such that there exists single fixed point and an inverse saddle is observed. Closure of unstable manifold of the fixed point is continuum containing in the annual ring and it separates the plane on the some regions. The interior of the invariant set containing in the annual ring with respect to the diffeomorphism is nonempty. Moreover the measure of it is positive, 9, no. 4, 2006, 394-398

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