Abstract
We slate and prove a theorem on a recurrence relation satisfied by a sequence of functions (f λ(z))λ⩽2 such that f λ converges to a quadratic irrational with an order of convergence equal to k. We define recursively a sequence of functions (F λ(z))λ⩽1 that shadows orbits generated by successive approximations and that are identical to (f k (z))k ⩽1 quadratic irrationals. By defining F 2 as Newton's Map we show that F 3, is Halley's Map. We prove that members of the sequence (F λ(z))λ⩽2 have shadowing properties with order of convergence equal to k
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