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 the inconsistency depending on fixpoint-selection bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 03/07/2008, 08:30 PM (This post was last modified: 03/09/2008, 08:51 PM by bo198214.) Gottfried Wrote:Now I'd like to understand, what is the source of the difference of selection e=1 vs d=1. The most obvious difference was in the given case Hm, I had another look at your transformations (which indeed puzzled me). I came up with this: Your first transformation: $x' = 4 x -1$, $x'' = \frac{1}{4} x + \frac{1}{4}$, $g_0(x)=f(x'')'$ gives the same result as the regular iteration at the upper fixed point, because: $x'' = \tau\circ \mu(x)$ where $\tau(x)=x+\frac{1}{4}$ and $\mu(x)=\frac{1}{4}x$ so $g_0(x)=\mu^{-1}\circ \tau^{-1}\circ f\circ\tau\circ\mu=\mu^{-1}\circ h_0 \circ \mu$ and $h_0$ is the function with upper fixed point moved to 0. But now it is easily derivable that the regular iteration $(\mu^{-1}\circ h\circ \mu)^{\circ t}=\mu^{-1}\circ h^{\circ t}\circ \mu$, where $h^{\circ t}$ is the regular iteration of $h$. This has to do with that the regular iteration is the only one which posses the limit $\lim_{x\downarrow 0} (f^{\circ t})'(x)=\left(\lim_{x\downarrow 0} f'(x)\right)^t$. So we have $f_0^{\circ t}= \tau\circ \mu\circ g_0^{\circ t}\circ \mu^{-1}\circ \tau^{-1}=\tau\circ h_0^{\circ t}\circ \tau^{-1}$. However your second transformation $x' = -4 x +1$, $x'' = -\frac{1}{4} x + \frac{1}{4}$, $g_1(x)=f(x'')'$ has as translation $\tau(x)=x+\frac{1}{4}$ the same translation as in your previous transformation! Only $\mu$ changes to $\mu(x)=-\frac{1}{4}x$. So this second transformation moves also the *upper* fixed point to 0. And we have seen that we can dismiss any multiplicative conjugation $\mu^{-1}\circ f\circ \mu$ with respect to regular iteration. So it has of course the same result (fractional iteration of $f$) as your first transformation. « Next Oldest | Next Newest »

 Messages In This Thread the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 03:06 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 05:20 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 10:14 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 11:17 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/04/2008, 11:25 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/04/2008, 11:32 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/04/2008, 11:43 AM RE: the inconsistency depending on fixpoint-selection - by GFR - 02/07/2008, 09:02 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 02/07/2008, 09:49 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/03/2008, 11:20 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/06/2008, 04:06 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/06/2008, 04:58 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/06/2008, 05:25 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 03:16 AM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/07/2008, 10:56 AM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 07:17 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/07/2008, 08:30 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/07/2008, 09:38 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 01:22 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 01:52 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/08/2008, 04:21 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 03/09/2008, 09:30 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 02:10 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 04:41 PM RE: the inconsistency depending on fixpoint-selection - by Gottfried - 03/14/2008, 07:19 PM RE: the inconsistency depending on fixpoint-selection - by bo198214 - 04/29/2008, 08:15 AM

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