Toda lattices and positive-entropy integrable systems

This paper studies completely integrable hamiltonian systems on T*Σ where Σ is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}^{n+1}$\end{document} bundle over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{T}^n$\end{document} with an ℝ-split, free abelian monodromy group. For each periodic Toda lattice there is an integrable hamiltonian system on T*Σ with positive topological entropy. Bolsinov and Taimanov’s example of an integrable geodesic flow with positive topological entropy fits into this general construction with the A(1)1 Toda lattice. Topological entropy is used to show that the flows associated to non-dual Toda lattices are typically topologically non-conjugate via an energy-preserving homeomorphism. The remaining cases are approached via the homology spectrum. An energy-preserving conjugacy implies the congruence of two rational quadratic forms over the unit group of a number field F. When F/ℚ is normal a classification of flows is obtained. In degree 3, this results from a well-known result of Gelfond; in higher degrees, the result is conditional on the conjecture that a rationally independent set of logarithms of algebraic numbers is algebraically independent over ℚ.