Abstract
Iteration Theory has Functional Equations and Dynamics as close neighbours. Considering a function as a time-one map of a discrete autonomous semi-dynamical system, the difficult problems of fractional-time state and continuous time embedding arise. Right composition by the function considered defines a bounded operator on a suitable Banach space of functions. Thus a connection is established between Iteration Theory and Functional Analysis. Then semigroups of right-composition operators can be investigated, moreover a generalization of fractional and continuous time iterates can be considered: phantom iterates. In another direction, functional-differential equations due to J. Aczel and E. Jabotinsky can be used to study «weak dynamics»: to consider systems having some but not all properties of dynamical systems.
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