08/16/2009, 11:15 PM
Ok, my explanation for the phenomenon is roughly:
The orbit of a point \( z_0 \) from the upper halfplane under \( \log \) is a right winding spiral around the primary fixed point in the upper halfplane.
This spiral gets bigger the farther \( z_0 \) is located from the origin.
The spiral getting bigger means the lower part of the spiral gets closer to the real axis.
Indeed already the second iteration of \( z \) (though it is usually not the minimum of the spiral) converges to the real axis when the imaginary part of \( z_0 \) goes to infinity.
\( z_0 = x_0 + iy_0 \)
\( x_1 = \ln(|z|) \)
\( y_1 = \arctan(y_0/x_0) \)
\( y_2 = \arctan(\arctan(y_0/x_0)/\ln(\sqrt{x_0^2+y_0^2}) \)
for big arguments the logarithm grows faster than the arctan, thatswhy if we increase a big \( y_0 \) the quotient \( \arctan(y_0/x_0)/\ln(|z|) \) decreases and hence \( y_2 \) decreases.
As long as the radius of a number \( z_n \) with \( 0<\Im(z_n)<\pi \) is too big, the imaginary part of the logarithm \( \Im(z_{n+1})=\Im(\log(z_n)) \) (which is the angle of \( z_n \)) will be smaller than the imaginary part of \( z_n \).
But doesnt this mean that the singularities that come close to the real axis somehow accumulate around 0?
At least the minimum of the spiral will be around 0. I think I could even calculate bounds on the real axis around 0 in which the minimum of every such spiral has to live.
The orbit of a point \( z_0 \) from the upper halfplane under \( \log \) is a right winding spiral around the primary fixed point in the upper halfplane.
This spiral gets bigger the farther \( z_0 \) is located from the origin.
The spiral getting bigger means the lower part of the spiral gets closer to the real axis.
Indeed already the second iteration of \( z \) (though it is usually not the minimum of the spiral) converges to the real axis when the imaginary part of \( z_0 \) goes to infinity.
\( z_0 = x_0 + iy_0 \)
\( x_1 = \ln(|z|) \)
\( y_1 = \arctan(y_0/x_0) \)
\( y_2 = \arctan(\arctan(y_0/x_0)/\ln(\sqrt{x_0^2+y_0^2}) \)
for big arguments the logarithm grows faster than the arctan, thatswhy if we increase a big \( y_0 \) the quotient \( \arctan(y_0/x_0)/\ln(|z|) \) decreases and hence \( y_2 \) decreases.
As long as the radius of a number \( z_n \) with \( 0<\Im(z_n)<\pi \) is too big, the imaginary part of the logarithm \( \Im(z_{n+1})=\Im(\log(z_n)) \) (which is the angle of \( z_n \)) will be smaller than the imaginary part of \( z_n \).
But doesnt this mean that the singularities that come close to the real axis somehow accumulate around 0?
At least the minimum of the spiral will be around 0. I think I could even calculate bounds on the real axis around 0 in which the minimum of every such spiral has to live.