Deformations of real rational dynamics in tropical geometry
The author studies the relation between discrete dynamical systems \(z_N=f(z_{N-n},\ldots,z_{N-1})\) where \(f\) is a rational function of the form \[ f_t(\bar{z}) = \sum_{k=1}^mt^{\alpha_k}\bar{z}^{\bar{j}_k},\qquad \alpha_k\in\mathbb{R}, \quad \bar{z}^{\bar{j}_k}=\prod_{i=1}^nz_i^{j_k^i},\quad \bar{j}_k\in\mathbb{Z}^n \] and (max,+)-functions \[ \varphi(\bar{x})=\max(\alpha_1+\bar{j}_1\bar{x},\ldots, \alpha_m+\bar{j}_m\bar{x}),\quad \bar{j}_k\bar{x}=\sum_{i=1}^nj_k^i x_i \] for elementary rational functions \(f\) where the \(f_t\) are independent of \(t\). The author uses the relations between (max,+)-functions and tropical geometry to get results on the iterates of elementary rational functions. The first result on the dynamics states that all orbits of \(\varphi\) are are periodic with the same period \(M\) if the orbits of \(f_t\) are quasi-recursive, i.e.~the inequality \(\frac{1}{C}\leq \frac{z_{N+M}}{z_N}\leq C\) holds true for some constant \(C>0\) independent of \(t\) and the initial values \((z_0,\ldots,z_{N-1})\). While there is a one-to-one correspondence between the family of rational functions \(f_t\) and (max,+)-function \(\varphi\) the same mapping may be represented in d ifferent ways as a (max,+)-function. This allows to define tropical equivalence between rational functions. Using comparison methods an analytic characterization of tropical equivalence is given. For the specific elementary case of \(f(z,w)=\frac{1+w}{z}\) small perturbations are studied and it is shown that some dynamical properties remain invariant under tropical deformations of \(f\).
scientific article; zbMATH DE number 5652311
scientific article