Spectral Dynamics for the Infinite Dihedral Group and the Lamplighter Group

For a tuple A = (A(0), A(1), ... , A(n)) of elements in a Banach algebra B, its projective (joint) spectrum p(A) is the collection of z is an element of P-n such that A(z) = z(0)A(0) + z(1)A(1) + <middle dot><middle dot> <middle dot>+ z(n)A(n) is not invertible. If B is the group C & lowast;-algebra for a discrete group G generated by A(0), A(1), ... , A(n) with a representation rho, then p (A) is an invariant of (weak) equivalence for rho. In [8], B. Goldberg and R. Yang proved that the Julia set J(F) of the induced rational map F for the infinite dihedral group D infinity is the union of the projective spectrum with the extended indeterminacy set. But the extended indeterminacy set EF is complicated. To obtain a better relationship between the projective spectrum and the Julia set, by replacing A(pi)(z) = z(0) + z(1 )pi(a) + z(2)pi(t) with the extended pencil A(pi) (z) = z(0) + z(1)pi(a) + z(2)pi(t) + z(3)pi(at), where rr is the Koopman representation, and using the method of operator recursions, we show that p(A pi) = J(F). Further, we study the spectral dynamics for the Lamplighter group L, and prove that J(Q) = EQ, where Q is the rational map associated with L.