Clark Butler

Continuity of Lyapunov exponents for cocycles with invariant holonomies

We prove a conjecture of Viana which states that Lyapunov exponents vary continuously when restricted to

ARROW RIBBON GRAPHS

GL(2,\mathbb{R})

A quasi-tree expansion of the Krushkal polynomial

-valued cocycles over a subshift of finite type which admit invariant holonomies that depend continuously on the cocycle.

Rigidity of equality of Lyapunov exponents for geodesic flows

We introduce an additional arrow structure on ribbon graphs. We extend the dichromatic polynomial to ribbon graphs with this structure. This extended polynomial satisfies the contraction–deletion relations and behaves naturally with respect to the partial duality of ribbon graphs. From a virtual link, we construct an arrow ribbon graph whose extended dichromatic polynomial specializes to the arrow polynomial of the virtual link recently introduced by H. Dye and L. Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.