The \(p\)-adic classification of fractals, and chaos (Q1595735)

The author proposes a \(p\)-adic interpretation of the iterative process of constructing the Sierpinski gasket. The limit procedure of determining the fractal dimension is considered with respect to the \(p\)-adic metric on rational numbers. Then different subsequences of iterations lead, for different \(p\), to different \(p\)-adic fractal dimensions. The second problem under consideration is the behaviour of the dynamical systems \(x\mapsto rx(1-x)\) and \(x\mapsto x^2\). Differences between the real and \(p\)-adic cases are analysed.