Continuous two-valued discrete-time dynamical systems and actions of two-valued groups
Konstantin M. Posadskiy
TL;DR
The paper studies continuous $2$-valued discrete-time dynamical systems on $\mathbb{C}$ and whether they can arise from the action of a $2$-valued group. It defines the class $\mathcal{P}_2(\mathbb{C})$ via $P_z(w)=w^2+2p_1(z)w+p_0(z)$ and analyzes $T(z)=-p_1(z)\pm\sqrt{p_1(z)^2-p_0(z)}$, showing a sharp obstruction: in the non-degenerate case, if $p_0$ has any root of odd multiplicity, $T$ cannot be defined by a $2$-valued group. The work also demonstrates that the condition is not sufficient by constructing counterexamples (including a $T$ of the form $(c\pm\sqrt{\gamma(z)})^2$) and provides a constructive example where a non-strongly-invertible dynamics, $T(z)=(1\pm\sqrt{z})^2$, is indeed definable via a Buchstaber–Novikov $2$-valued group; together, these results delineate when continuous $2$-valued dynamics are group-definable and reveal rich structure at the intersection of multivalued maps and generalized group actions.
Abstract
We study continuous 2-valued dynamical systems with discrete time (dynamics) on $\mathbb{C}$. The main question addressed is whether a 2-valued dynamics can be defined by the action of a 2-valued group. We construct a class of strongly invertible continuous 2-valued dynamics on $\mathbb{C}$ such that none of these dynamics can be given by the action of any 2-valued group. We also construct an example of a continuous 2-valued dynamics on $\mathbb{C}$ that is not strongly invertible but can be defined by the action of a 2-valued group.