Dynamical scale transform in tropical geometry (Q2833690)

This book describes geometry and analysis of dynamical systems by looking at their scale transforms via \(\log_t\), where \(t > 1\). Of particular interest is the tropical transform, also known as Maslov dequantization [\textit{G. L. Litvinov} and \textit{V. P. Maslov}, in: Idempotency. Based on a workshop, Bristol, UK, October 3--7, 1994. Cambridge: Cambridge University Press. 420--443 (1998; Zbl 0897.68050)], obtained by \(t \to \infty\).NEWLINENEWLINEThe author applies this strategy to three subjects:NEWLINENEWLINEProjective geometry: Chapters 2--5 are dedicated to the theory of iterated dynamics and pentagram maps. In Chapter 2, the author reduces real rational dynamics to tropical dynamics and shows that quasi-recursive rational dynamics is equivalent to recursive \((\text{max}, +)\)-dynamics. Theorem 5.6 proves, from the viewpoint of scale transform, that the continuous limit of the pentagram map induces the Boussinesq equation.NEWLINENEWLINEInfinite groups: Chapters 6--9 mostly treat (stable) state dynamics, automata groups, and Burnside groups. The idea is to study whether geometric and analytic properties of rational dynamics pass to automata groups in tropical geometry via scale transform.NEWLINENEWLINEMathematical physics: Chapters 10--16 provide a classification of partial differential equations using tropical geometry. Chapter 14 introduces some analytic relation among PDEs in two variables which is based on asymptotic estimates of all positive solutions; then it is shown that two PDEs obtained from tropically equivalent rational functions are related. Chapter 15 and 16 are devoted to the basic analysis of hyperbolic Mealy systems, in particular, the existence of solutions involves the interplay of estimates between piecewise linear and differentiable dynamics.NEWLINENEWLINEThroughout the book, the author also shows that tropical geometry links different subject in mathematics. For instance, Korteweg-de Vries equations and lamplighter groups share structural similarities.NEWLINENEWLINEThe book is self-contained and includes basic concepts in each subject such as iterated dynamics, pentagram map, lamplighter group, Burnside problem, Korteweg-de Vries equations and hyperbolic Mealy systems.