12/24/2019, 06:26 AM
I asked this question in 1993 in the following manner,
Consider two fixed points \( \alpha_1, \alpha_2 \) of the complex exponential map \( a^z \) where \( a^{\alpha_1}=\alpha_1 \) and \( a^{\alpha_2}=\alpha_2 \) and their Lyapunov multipliers \( \lambda_1, \lambda_2 \).
Can the forward orbit of \( a^z \) traverse a region of space dominated by \( \alpha_1,\lambda_1 \) to the region dominated by \( \alpha_2,\lambda_2 \)?
My numerical research indicated that the answer can be affirmative. The main problem I faced what the chaotic region between \( \alpha_1, \alpha_2 \). By adjusting \( a\rightarrow 1 \) the chaotic region becomes arbitrarily thin and several hundred iterations can map \( \alpha_1,\lambda_1 \) to \( \alpha_2,\lambda_2 \).
Consider two fixed points \( \alpha_1, \alpha_2 \) of the complex exponential map \( a^z \) where \( a^{\alpha_1}=\alpha_1 \) and \( a^{\alpha_2}=\alpha_2 \) and their Lyapunov multipliers \( \lambda_1, \lambda_2 \).
Can the forward orbit of \( a^z \) traverse a region of space dominated by \( \alpha_1,\lambda_1 \) to the region dominated by \( \alpha_2,\lambda_2 \)?
My numerical research indicated that the answer can be affirmative. The main problem I faced what the chaotic region between \( \alpha_1, \alpha_2 \). By adjusting \( a\rightarrow 1 \) the chaotic region becomes arbitrarily thin and several hundred iterations can map \( \alpha_1,\lambda_1 \) to \( \alpha_2,\lambda_2 \).
Daniel
