09/08/2007, 03:04 PM
Proposition. All non-real fixed points \( a \) of \( b^x \), \( b>1 \) are repelling, i.e. \( |\exp_b'(a)|>1 \).
With the previous considerations:
\( a=re^{i\alpha} \)
\( |\exp_b'(a)|=\ln(b)|\exp_b(a)|=\ln(b)|a|=\ln(b)r=:s \)
(1) \( s\cos(\alpha)=\ln( r) \)
(2) \( s\sin(\alpha)=\alpha \)
\( \sin(\alpha)=\frac{\alpha}{s} \).
If \( \alpha>0 \) (\( a \) non-real) is a solution of the above equation then must \( s>1 \) which is clear from comparing the graphs of both sides.
With the previous considerations:
\( a=re^{i\alpha} \)
\( |\exp_b'(a)|=\ln(b)|\exp_b(a)|=\ln(b)|a|=\ln(b)r=:s \)
(1) \( s\cos(\alpha)=\ln( r) \)
(2) \( s\sin(\alpha)=\alpha \)
\( \sin(\alpha)=\frac{\alpha}{s} \).
If \( \alpha>0 \) (\( a \) non-real) is a solution of the above equation then must \( s>1 \) which is clear from comparing the graphs of both sides.
