Newly reducible polynomial iterates

Reducible and ∂-reducible Handle Additions

Let M be a simple 3-manifold with F a component of ∂M of genus at least two. For a slope α on F , we denote by M(α) the manifold obtained by attaching a 2-handle to M along a regular neighborhood of α on F . Suppose that α and β are two separating slopes on F such that M(α) and M(β) are reducible. Then the distance between α and β is at most 2. As a corollary, if g(F ) = 2, then there is at mos...

Iterates of Volterra

Given k ∈ L(0, 1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator Vk defined on L(0, 1) by (Vku)(t) = ∫ t 0 k(t − s)u(s) ds, and its iterates (V n k )n∈N . We construct some much simpler sequences which, as n → ∞, are asymptotically equal in the operator norm to V n k . This leads to a simple asymptotic formula for ‖V n k ‖ and to a simple...

Iterates of Fractional Order

This problem typifies the general one of iteration. Let g(x) be the &th order iterate of g [i.e. g°(x) = x, g(x) = g(g(x))]. The iteration problem is that of attaching a consistent meaning to this expression for fractional k (in the sense of preserving the additive law of exponents). A n / satisfying (1) is thus g(x). By ideas similar to those discussed herein, we can find the most general g an...

Random Iterates of Monotone Maps

On Linear Dependence of Iterates

A functional equation related to a problem of linear dependence of iterates is considered.