On generalization of Sierpiński gasket in Lobachevskii plane (Q2404816)

The author defines iterated function systems (IFSs) on the two-dimensional Beltrami-Klein model \(\Lambda\) of hyperbolic geometry. He establishes that \((\Lambda, \rho)\) is a complete metric space where \(\rho(x,y) := \text{arcosh} \frac{1 - \langle x,y\rangle}{\sqrt{1-x^2}\sqrt{1-y^2}}\) is the standard metric on \(\Lambda\). An example of a contractive mapping on \(\Lambda\) is given by \[ \lambda(x) := \frac{x}{|x|}\text{tanh} (t\, \text{artanh} |x|), \qquad t \in (0,1). \] Using this contractive mapping together with some group elements from the translation group of \(\Lambda\), the analogue of the Sierpiński gasket in \(\Lambda\) is constructed. Moreover, a Mandelbrot set associated with a new family of attractors of such IFSs on \(\Lambda\) is investigated.