The twisted cocycle over interval exchange renormalizations has a symmetric Lyapunov spectrum with at least κ+1 zero exponents, and is fully degenerate for rotation-type permutations on their homology torus.
Binary constant-length substitutions and Mahler measures of Borwein polynomials
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Twisted cocycle for interval exchange transformations: Invariant structures and Lyapunov spectrum
The twisted cocycle over interval exchange renormalizations has a symmetric Lyapunov spectrum with at least κ+1 zero exponents, and is fully degenerate for rotation-type permutations on their homology torus.