From Osher Doctorow
Robert L. Devaney in "A survey of exponential dynamics," 2001 or 2002, http://math.bu.edu/people/bob/paper/augsburg.pdf as well as in various publications including a book has examined the exponential function analog of polynomial Julia/Mandelbrot sets which has an explosion when (c1 + ic2)exp(z) has c2 = 0 and c1 > 1/e. Here z = x + iy. According to Devaney, the appearance of so-called indecomposable continua in the chaotic Julia set for c1 > 1/e (in my notation) accounts for the explosion. In the explosion, the Julia set jumps from being nowhere dense to becoming the whole plane. The Rare Event Theory analysis is quite simple and proceeds as follows. 1) c exp(z) = (c1 + ic2)(x + iy) = c1 (cos u - sin u) = p1(c1 sin u - 1, c1 cos u) = 1 where c2 = 0 and x = cos(u), y = sin(u) because exp(z) = cos(theta) + isin(theta) for angle theta in the usual complex representation and we set u = theta. The equating to 1 optimizes p1. Setting the middle expression of (1) equal to 1 results in the following. 2) c1 cos(u) - c1 sin(u) = 1 and therefore: 3) c1(cos(u) - sin(u) = 1 But dividing both sides of (3) by c1 not equal to 0 results in: 4) cos(u) - sin(u) = 1/c1 This is impossible for u between 0 and pi/2 radians and for c1 = 1/e since both cosine and sine functions are between 0 and 1 there. If 0 < c1 < 1/e, then 1/c1 > 1/e and again (4) is impossible. The only place where (4) is not contradictory is where c1 > = 1 which is most of the interval between 1/e and infinity except for (1/e, 1). The fact that the chaotic and non-chaotic regions have reversed in the Rare Event Theory representation for u between 0 and pi/2 is probably not that dangerous because an entire quarter of the complex plane, where u is between pi/2 and pi, is excluded since cosine is negative there and sine is positive there and thus (4) would be contradictory. The pathology is in the function itself c exp(z) more than in its domain or range. Osher Doctorow