On constructing complete solution classes of the Cvitanović-Feigenbaum equation (Q1117535)

The Cvitanović-Feigenbaum equation is studied with the (distant) purpose of determining under what auxiliary conditions the solution is unique. To this end several complete solution classes are constructed, with the aim of studying how these depend on the conditions imposed. Those imposed in this paper always include strict unimodality and, for the most part, continuity. Other conditions which figure are: piecewise linearity, even-ness and concavity. Under the latter condition, with and without the preceding one, two sharp lower bounds are derived on Feigenbaum's scaling factor \(\alpha\), for any value of the exponent z (or \(\epsilon)\). The (unique) solutions corresponding to these lower bounds are exhibited. They are piecewise linear. Also shown is the unique solution describing the dynamics on the attractor. It is piecewise linear but neither even nor continuous.