Dynamical Scale Transform in Tropical Geometry

The following sections are included:

• Preface
• Contents

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…

The following sections are included:

• Deformation of semi-ring structure
• Tropical algebra
• Tropical transform
• Uniqueness of the scale transform
• Intermediate selection
• Tropical equivalence
• Examples
• Projective duality
• Curves in ℂP²

The following sections are included:

• Iterated dynamics
• Dynamical hierarchy
• Basic estimates
• Comparison between rational orbits
• Quasi-recursiveness

The following sections are included:

• Stationary dynamics
• Contraction vs boundedness
• Unbounded orbits
• Traces
• Return map for trace
• Perturbation of recursive maps
• A recursive map

The pentagram map consists of a beautiful integrable system, which was originally initiated by R. Schwartz from the viewpoint of the dynamical system on the set of convex polygons in projective geometry. Given an n-gon on the plane, the result of the action called the pentagram map, is also another n-gon given by the convex hull of the intersection points of consecutive shortest diagonals…

Roughly speaking an automaton is a rule defined over finite sets. Even though it is so abstract, it can produce quite interesting objects in the fields of not only algebra but also geometry and analysis. In chapter 7, we shall treat automata groups in relation to geometric group theory, and cell automata of integrable systems in relation to mathematical physics.

The most general form of such abstract rules will be the Turing machine, but in this book we will introduce a more restrictive class of automata which is enough to apply to such objects passing through tropical geometry.

Automata groups are a class in finitely generated groups which contain several instances with quite important and characteristic properties in group theory. It is beyond the scope of this book to give complete references, but let us quickly explain a few important results…

Automaton is defined over finite sets, but once it is expressed by (max; +)- functions, it canonically extends to a map over the real vector space. If one starts from an automaton, there are no canonical extensions, which are actually so exible. Here we verify that there still exist effective extensions to reect dynamical properties of automata to rational dynamics passing through tropical geometry.

Let us recall quasi-recursivity in 3:5. We have considered dynamics in the space time free case. More concretely let φ be a (max; +)-function of n variable, and consider the one dimensional dynamics x_N = φ(x_N−n,…,x_N−1). Let f_t be the tropical correspondence to φ, and consider the corresponding dynamics z_N = f_t(z_N−n,…,z_N−1) with the initial data x₀ = log_t z₀,…,x_n−1 = log_t z_n−1.

We have verified that φ is always recursive over ℝ whenever f_t is the case over (0; ∞) for all t > 1. However the converse is not true in general. What we verified is that φ is recursive if and only if f_t is quasi-recursive.

Below we study recursivity in state dynamics.

The KdV (Korteweg–de Vries) equation is a main subject in mathematical physics, and there have been many researches on the analysis of their solutions. On the other hand the lamplighter group is also an important subject in group theory, and there have been many researches on this group particularly on random walks. Despite their importance, these subjects are rather separated from each other in the mathematical fields. So who could imagine that both the KdV equation and the lamplighter group can share structural similarities! Direct comparison of their structures would cause quite complicated situations, however we might be able to compare their frameworks after scale transform in tropical geometry, which makes their structures much simpler.

The following sections are included:

• Markov operator
• Markov operators for BBS_k=1 and lamplighter group
• Spectral computation for BBS translation (k = 1)
• Numerical computation of spectra for BBS (k ≥ 2)
• Stochastic matrices and ergodicity
• Ergodicity on the boundary of the binary tree

Tropical geometry is a kind of scale transform between dynamical systems, which provides a correspondence between automata and real rational dynamics. It allows us to treat rational dynamics as a class such that equivalent rational dynamics share the same analytic behaviors on the large scale in some sense…

Let us treat partial differential equations by introducing a time parameter. The idea is also to roughly compare solutions to different PDEs as in the previous chapter, however this extension allows us to compare rather different types of partial differential equations, and can cover much broader classes…

Tropical geometry provides us with an important prototype in mathematical physics. Particularly we have seen that the Korteweg—de Vries (KdV) equation admits BBS automaton as its framework. It allows us to study analytic aspects at the same time for three categories of dynamical systems which sit in different hierarchies mutually. The classes of dynamics which can be analyzed by these scaling limits are rather broad beyond integrable systems. In fact it is easy to find discrete dynamical systems which are far from integrable, but which are transformed into integrable cell automata. If we consider a situation when two dynamical systems are transformed into the same integrable cell automaton, where one is integrable and the other is not, then it will be quite natural to try to classify such dynamical systems by some systematic method, which should include wide classes of dynamical systems as above…

Let us recall the state system of the rational dynamics given by elementary rational functions:

where the initial values are given by .

We shall follow the same process as previous chapters to induce partial differential equations which are systems in this case. In general it depends on the equations whether global solutions exist or they are unique. In the case of the Mealy automata we have considered, they are in the class of first order hyperbolic systems, which we call hyperbolic Mealy systems.

We study the basic analysis of hyperbolic Mealy systems including the existence and uniqueness.

We develop a basic analysis of the hyperbolic Mealy systems of partial differential equations in our sense. In particular we verify the existence and uniqueness of their global solutions. Then we apply the previous results to the large scale analysis of them.

The following sections are included:

• Bibliography
• Index