Ultrametric dynamics
The ultrametric \(d\) on a nonempty set \(X\) is a map \(a:X\times X\to \Gamma\) with values in an ordered set \((T_0\leq)\) with the smallest element \(0\), satisfying (1) \(d(x,y)= 0\quad\text{iff }x= y,\) (2) \(d(x,y)= d(y,x),\) (3) \(d(x,y)\leq\gamma\) and \(d(y,z)\leq\gamma\) imply \(d(x,z)\leq\gamma\) for \(x,y,z\in X\), \(\gamma\in\Gamma\). For any \(\gamma\in\Gamma\) and \(x\in X\) the ball \(B_\gamma(x)\) is defined as the set \(\{y\in X\mid d(y,x)\leq \gamma\}\). If there is a point \(z\) with \(d(x,z)= \gamma\), the ball \(B_\gamma(x)\) is called principal. The space is said to be spherically complete if every totally ordered set of balls has nonempty intersection. When this condition holds only for principal balls then the space is called principally complete. The authors study the dynamics on ultrametric spaces generated by a selfmap \(\varphi: X\to X\) or maps \(\psi_1,\psi_2: X\to X'\), satisfying certain conditions. For the selfmap \(\varphi: X\to X\) on \((X,d)\) the following conditions appear in their main result \[ x\neq y \Rightarrow d(\varphi(x), \varphi(y))< d(x,y),\tag{1} \] \[ \varphi(x)\neq x\Rightarrow d(\varphi^2(x), \varphi(x))< d(\varphi(x), x).\tag{2} \] They show that if the space \((X,d)\) is principally complete and \(\varphi\) satisfies condition (1), then \(\varphi\) has a unique fixed point. If it satisfies condition (2), then \(\varphi\) has a fixed point.
scientific article; zbMATH DE number 6122120