**Title:** The Toda Lattice and Positive-Entropy Integrable Systems

**Speaker:** Professor Leo Butler

**Speaker Info:** Northwestern University

**Brief Description:**

**Special Note**:

**Abstract:**

The Periodic Toda Lattice is a well-studied, integrable system that is intimately linked to the theory of Kac-Moody Lie algebras.We show that if $\Sigma$ is a ${\bf T}^{n+1}$ bundle over ${\bf T}^n$, with a totally real monodromy group, and $\Psi$ is a Kac-Moody Lie algebra, there is an analytic hamiltonian system on $T^* \Sigma$ such that:

-it is smoothly but not real-analytically integrable; -it has positive topological entropy; -on an open, dense set it is semi-conjugate to a periodic Toda Lattice flow.

The case where $n=1$ and $\Psi = A^{(1)}_1$ is shown to be the Bolsinov-Taimanov example [Inv. Math. 2000].

In addition, we show that if $\Psi'$ is "dual" to $\Psi$, then the systems are topologically conjugate.

The construction of the hamiltonians depends on an arbitrary labeling of the elements of $\Psi$. We show that the topological conjugacy of two flows constructed using two labelings depends on an old conjecture of S. Lang concerning the algebraic independence of logarithms of algebraic numbers.

Assuming the Lang conjecture, we show that it is possible to show that some labelings give non-conjugate flows.

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