On the Scaling Behavior in a Map of a Circle onto Itself

A scaling·theoretical framework for a map of a circle onto itself is proposed, and it gives a way of understanding the appearance of unusual scaling behaviors for a particular parameter value found by Shenker and reveals a simple relation between scaling exponents, v = 2x. Recently Shenker!) has numerically shown the existence of remarkable scaling behaviors in a map of a circle onto itself as follows: T(B)=B+Q- tr sin 2J[B, (1) where Q and K are constants, and the variable B parametrizes the circle, so that B+ 1 must be identified with B. It is no doubt important to investigate the properties of such a map since it can be regarded as a one-dimensional version of the standard mapping which plays an important role in the study of stochastic instabilities in Hamiltonian dynamics. 2)-4) Moreover, the map can be useful to study a certain type of routes to turbulence such that the transition to a turbulent state takes place via quasiperiodic state, i.e., torus. S) Let us now describe Shenker's results concentrating on a part with which we are mainly concerned in this paper. He determined a set of values of Q, Qi(K), for which TQi ( 0)- Pi = 0 holds as Q is varied with K fixed, where Q i and Pi are the (i + 1)th and the ith Fibonacci numbers,*) respectively. Then, it was found that in the limit i~oo, Qi(K) converge to a certain value Q(K) in the following way: r Qi(K)-Qi-l(K) iI.~Qi+l(K)-Qi(K) (O~K <1