Kato T. Dynamical Scale Transform in Tropical Geometry

World Scientific Publishing Co. Pte. Ltd., 2017. — 255 p. — ISBN10: 9814635367This book provides comprehensive analysis of dynamical systems in tropical geometry, which include the author's significant discoveries and pioneering contributions. Tropical geometry is a kind of dynamical scale transform which connects real rational dynamics with piecewise linear one presented by max and plus algebras. A comparison method is given which estimates orbits corresponding to different rational dynamics by reduction to the piecewise linear dynamics.Both rational and piecewise linear dynamics appear in many important branches of mathematics. Tropical geometry can play a role or function to bridge between different subjects in mathematics. This book contains detailed accounts of basic strategy on how to apply tropical geometry to analysis in various mathematical subjects by presenting several applications which include: a rough classification of partial differential equations from the point of view of global behavior of solutions; construction of the infinite quasi-recursive rational dynamics, based on the automaton of the Burnside group by Aleshin-Grigorchuk; study on nearly periodicity of the pentagram map on the moduli space of the twisted polygons; spectral coincidence between lamplighter group in theory of automata groups and Box and ball systems corresponding to KdV equation in soliton theory.This book is self-contained, and detailed accounts of theory of automata groups, BBS and the pentagram map are also included
Topics
Earth Sciences
Geometry & Topology
Mathematical Physics
Algebraic GeometryIterative dynamics
Tropical transform
Dynamical hierarchy
Rational perturbation of dynamics in two variables
Pentagram map and tropical geometry
Evolutional dynamics
State dynamics
Automata groups
Stable state dynamics
Rational Burnside problem
KdV equation and box-ball systems
Spectral similarity between BBS and lamplighter group
Rough comparison between various differential equations
Evolutional dynamics
Rough analytic relation on the set of partial differential equations
Mealy type dynamics
Hyperbolic system of partial differential equations
Analysis of hyperbolic Mealy systems