Dynamical Scale Transform in Tropical Geometry

Tsuyoshi Kato

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789814635387
• Topic: Matemáticas
• Subtopic: Geometría algebraica

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This 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.

--> Contents:

• Iterative Dynamics:
  • Tropical Transform
  • Dynamical Hierarchy
  • Rational Perturbation of Dynamics in Two Variables
  • Pentagram Map and Tropical Geometry
• Iterative 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
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  Readership: Graduate students and researchers in geometry.
  --> Key Features:
  • This book is self-contained and many of the topics in this book are unique, which means they have never appeared elsewhere

Chapter 1

Introduction

The theme of this book is to describe geometry and analysis of dynamical systems from the viewpoint of scale transform. Dynamical systems can create quite complicated structures, and our aim is to understand them by extracting some simple rules from their framework.

(max, +)-functions have two different aspects, as automata in computer science and as Lipschitz functions in global analysis. The former viewpoint produced several concrete mathematical subjects, while the latter allows us to perform some uniform estimates.

Global analysis given by rational functions plays one of the central subjects in dynamical systems. Let f₁, f₂, . . . be a family of rational functions, and consider their orbits:

We will just call them rational orbits. In general such orbits behave in quite complicated manners, and direct analysis of them often causes difficulty in understanding their structure.

In some of the rational dynamics, the structure of their orbits contains a ‘hidden’ framework in a different hierarchy of dynamics. It can be seen by eliminating ‘fluctuation’ in the orbits, which allows us to develop a systematic study of the classes of dynamics. Let us describe our strategy by the following processes:

(A): Extract simple rules from complex systems.

(B): Compare two complex systems which are reduced to the same rules.

(C): Induce characteristic properties of the simple rules and analyze how such properties are reflected in the original complex systems.

(D): Conversely induce some characteristic properties of the complex systems and analyze how such properties are reflected in the simple rules.

(E): Compare different mathematical objects which arise from very different sources but share some characteristics in their simple rules, and discover structural similarity between them.

It will require using a very strong scaling limit to perform (A) above. Actually scale transform can change mathematical structures. Tropical geometry is a kind of dynamical scale transform. The domains change from the positive real line to the real line, and the arithmetics change as:

between these semi-rings. The left hand side is the standard one over the real line, and the arithmetic on the right hand side appears in computer science. From the dynamical viewpoint, the scale transform makes all rational orbits degenerate to 0, and the rule or the constraint which governs the dynamics is replaced drastically to a very different one by a change in their arithmetics.

Let φ(y, x) be a function of two variables over the real line, and consider the orbits inductively defined by:

with initial values (x₀, x₁) ∈ R² . This involves iteratively applying φ many times, since x₂ = φ(x₀, x₁), x₃ = φ(x₁, x₂) = φ(x₁, φ(x₀, x₁)), x₄ = φ(x₂, x₃) = φ(φ(x₀, x₁), φ(x₁, φ(x₀, x₁))), and so on.

Let us consider examples. Let φ(y, x) = max(0, y) − x be a (max, +)- function. It is not so difficult to check that this dynamics is recursive, in the sense that the orbits are periodic with periods 5:

with respect to any initial value. The rule in tropical geometry associates the rational function:

Let us consider the iteration dynamics given by:

It is straightforward to calculate the orbits:

So this is also recursive of period 5.

One may imagine that recursiveness could be generally preserved under the scale transform. In fact it is not the case. Let us choose the function:

and consider the iteration dynamics x_n = ψ(x_n−2, x_n−1). It is recursive of period 9. The associated rational dynamics is given by:

It turns out that it does not give recursive dynamics, which can be seen by computing iterations of 9 times with specific initial values.

We will understand such phenomena as quasi-recursivity so that its framework consists of recursive dynamics but some fluctuation appears in the rational dynamics, which prevents the rigorous recursivity and is eliminated under the scale transform. It turns out that quasi-recursive dynamics by rational functions exactly correspond to the recursive dynamics by (max, +)-dynamics.

Let us list some of concrete processes we describe in this book. We apply tropical geometry techniques to the theory of pentagram maps which arose from classical projective geometry, the theory of automata groups which consist of a class of infinite groups and KdV equations in integrable systems.

(A): Reduce real rational dynamics to (max, +)-dynamics.

(B): (1) Uniform estimates on the rates of rational orbits.

(2) Rough analytic relation on the set of PDE.

(C): (1) Quasi-recursive rational dynamics is equivalent to the recursive (max, +)-dynamics.

(2) Existence of infinite quasi-recursive rational dynamics from Burnside automaton in group theory.

(3) Perturbation and stationary points; contraction corresponds to boundedness.

(4) Uniform bounds on diameter rates of the invariant tori in the integrable system of the pentagram map.

(D): Duality on automata induced from projective duality over rational functions.

(E): Spectral similarity between KdV and the lamplighter group.

Let us list the subjects we treat in this book along the scale transform, where * are described in a general process which leads to an induction of a class of PDE called hyperbolic Mealy systems:

PDE	Rational dynamics	Automaton
KdV	discrete KdV	BBSk
*	*	lamplig...