We investigate second order lossless digital filters with two's complement overflow. We numerically approximate the fractal set D of points that iterate arbitrarily close to the discontinuity. For the case of eigenvalues of the associated linear map of the form e iθ with θ/π / ∈ Q we present evidence that D has positive two dimensional Lebesgue measure. For θ/π ∈ Q we confirm that D has Lebesgue measure zero. As a by-product we get estimates of the exterior dimension of D. These results imply that if such filters are realized using finite-precision arithmetic then they will have a sizeable fraction of orbits that are periodic with high period overflows.

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The influence of magnetic breakdown on the usual effective-Hamiltonian theory for Bloch electrons in a magnetic field is discussed using a simple two-dimensional rectangular model. The theory is based upon an expansion of the wave function in Wannier functions, but these have to be replaced by generalized Wannier functions (suggested by Blount) to handle breakdown. It is shown how to construct a linear network essentially equivalent to the network derived previously by the author for a nearly-free-electron model. The de Haas-van Alphen effect is discussed by expressing the energy density of states in terms of a time-independent Green's function on the network. It is shown how to construct a wave function lying on a twodimensional network as used by Pippard.* J. M. Luttinger, Phys. Rev. 84, 814 (1951). 3 E. Brown, Phys. Rev. 133, A1038 (1964.