Deformations of real rational dynamics in tropical geometry
type: Item
instance of: scholarly article
title: Deformations of real rational dynamics in tropical geometry
zbMATH Open document ID: 1200.37017
author: Tsuyoshi Kato
published in: Geometric and Functional Analysis. GAFA
publication date: 2010-01-04T00:00:00Z
review text: 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 \(f\).
reviewed by: Jörg Härterich
Mathematics Subject Classification ID: 37C35
Mathematics Subject Classification ID: 14T05
Mathematics Subject Classification ID: 37E30
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 5652311
zbMATH Keywords: elementary rational functions
zbMATH Keywords: tropical geometry
zbMATH Keywords: max-plus function
zbMATH Keywords: tropical deformation
zbMATH Keywords: quasi-recursive function
zbMATH Keywords: tropical equivalence
MaRDI profile type: Publication
full work available at URL: s00039-009-0023-5
OpenAlex ID: W2087920289
cites work: “Recurrence Equations”, An Integrable System
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cites work: Q4661963
cites work: Q4226356
cites work: Q4408031
cites work: Q4323873
DOI: 10.1007/S00039-009-0023-5
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description: scientific article; zbMATH DE number 5652311