Thread Rating:
• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Iterating at fixed points of b^x bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 09/08/2007, 11:36 AM (This post was last modified: 09/08/2007, 11:43 AM by bo198214.) Now let us take it a bit further for a general real base $b>1$. Proposition. The only real base $b>1$ for which the complex function $b^z$ has a fixed point $a$ with $|\exp_b'(a)|=1$ is $b=e^{1/e}$ and in this case the only such fixed point is $e$. Simalarly to the previous post we assume a fixed point being given by $a=re^{i\alpha}$ then for parabolicity we have to prove $|\exp_b'(a)|=\ln(b)|\exp_b(a)|=\ln(b)|a|=\ln(b)r=:s$ and we have the equation system: $\ln(b)r\cos(\alpha)=\ln( r)$ (1) and $\ln(b)r\sin(\alpha)=\alpha$ (2) Both equation squared and added yields $(\ln(b)r)^2=\ln ( r)^2+\alpha^2$ and hence $\alpha=\pm\sqrt{(\ln(b)r)^2-\ln( r)^2}$ For the case $s=1$: $r=1/\ln(b)$ and $\ln( r)=-\ln(\ln(b))$ and so $\alpha=\pm\sqrt{1-(\ln(\ln(b)))^2}$. Now we put this into equation (1) with $x=-\ln(\ln(b))$: $\cos(\pm\sqrt{1-x^2})=x$ We substitute $\sqrt{1-x^2}=:y$, $x=\pm\sqrt{1-y^2}$ $\cos(\pm y)=\pm\sqrt{1-y^2}$. But the functions on both sides are well known. The right side is a circle with radius 1 and the left side is always above or below the right side in the region $y=-1..1$. So equality happens exactly at $y=0$. This implies $\ln( r)=x=\pm 1$ however only $\ln( r)=1$ satisfies equation (1). Then $b=\exp( \exp( -x))=e^{1/e}$ is the only real base for which $\exp_b$ has a parabolic fixed point and this fixed point is given by $\alpha=0$ and $\ln( r)=1$ and is hence the real number $e$. « Next Oldest | Next Newest »

 Messages In This Thread Iterating at fixed points of b^x - by bo198214 - 09/08/2007, 10:02 AM The fixed points of e^x - by bo198214 - 09/08/2007, 10:34 AM The fixed points of b^x - by bo198214 - 09/08/2007, 11:36 AM RE: Iterating at fixed points of b^x - by jaydfox - 09/12/2007, 06:23 AM RE: Iterating at fixed points of b^x - by bo198214 - 09/12/2007, 09:54 AM RE: Iterating at fixed points of b^x - by GFR - 10/03/2007, 11:03 PM RE: Iterating at fixed points of b^x - by Gottfried - 10/04/2007, 06:53 AM RE: Iterating at fixed points of b^x - by bo198214 - 10/04/2007, 01:30 PM RE: Iterating at fixed points of b^x - by bo198214 - 10/04/2007, 05:54 PM RE: Iterating at fixed points of b^x - by Gottfried - 10/04/2007, 11:05 PM RE: Iterating at fixed points of b^x - by GFR - 01/31/2008, 03:07 PM

 Possibly Related Threads... Thread Author Replies Views Last Post tetration from alternative fixed point sheldonison 22 29,230 12/24/2019, 06:26 AM Last Post: Daniel Are tetrations fixed points analytic? JmsNxn 2 3,018 12/14/2016, 08:50 PM Last Post: JmsNxn Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,577 03/19/2016, 10:44 AM Last Post: fivexthethird Derivative of exp^[1/2] at the fixed point? sheldonison 10 10,526 01/01/2016, 03:58 PM Last Post: sheldonison [MSE] Fixed point and fractional iteration of a map MphLee 0 2,162 01/08/2015, 03:02 PM Last Post: MphLee iterating x + ln(x) starting from 2 tommy1729 2 2,679 04/29/2013, 11:35 PM Last Post: tommy1729 attracting fixed point lemma sheldonison 4 9,744 06/03/2011, 05:22 PM Last Post: bo198214 cyclic points tommy1729 3 4,470 04/07/2011, 07:57 PM Last Post: JmsNxn iterating non-analytic tommy1729 0 1,817 02/08/2011, 01:25 PM Last Post: tommy1729 Branch points of superlog mike3 0 2,327 02/03/2010, 11:00 PM Last Post: mike3

Users browsing this thread: 1 Guest(s)