• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Jabotinsky's iterative logarithm andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 06/14/2008, 12:44 AM bo198214 Wrote:The categories (non-mathematical) are different: $\text{ilog}$ maps a function to a function (or better a formal powerseries to a formal powerseries) while the Abel function maps values to values. And before writing $\text{ilog}^{-1}$ you should assure that it is invertible, which stronly seems not to be the case. I realize why this is the case now. It would be like trying to reconstruct g(t) from g'(0). However, since: $\mathcal{A}[f](x) = \int \frac{dx}{\mathcal{J}[f](x)}$ and $f^{\circ t}(x) = \mathcal{A}[f]^{-1}(\mathcal{A}[f](x) + t)$ it could be argued that it should be possible to invert the iterative logarithm, provided the Abel function is invertible. Also, I've been starting to realize more and more, that this is really amazing! Jabotinsky was a master of iteration. In "Analytic Iteration" cited above, he gives this formula (3.10) in original and my notations: $L(F^{\circ s}(z)) = \frac{\partial}{\partial z}F^{\circ s}(z) \cd L(z) = \frac{\partial}{\partial s}F^{\circ s}(z)$ $\mathcal{J}[f](f^{\circ t}(x)) = \frac{\partial}{\partial x}f^{\circ t}(x) \cd \mathcal{J}[f](x) = \frac{\partial}{\partial t}f^{\circ t}(x)$ where the relationship between L and ilog are $L(x) = \mathcal{J}[f](x) = \text{ilog}(f)$. What I find most interesting about this formula is that right after it, Jabotinsky says: Jabotinsky Wrote:This double equation is fundamental in the theory of iteration. It can be used and extended in many ways.what did he mean by this? Obviously, he knew how important this was, but from reading it, it seems that he did not realize its connection to Abel functions. Does Ecalle mention the relationship to Julia functions or Abel functions? Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Jabotinsky's iterative logarithm - by bo198214 - 05/21/2008, 06:00 PM RE: Jabotinsky's iterative logarithm - by Ivars - 05/22/2008, 09:02 AM RE: Jabotinsky's iterative logarithm - by bo198214 - 05/22/2008, 10:48 AM RE: Jabotinsky's iterative logarithm - by Ivars - 05/22/2008, 01:13 PM RE: Jabotinsky's iterative logarithm - by andydude - 05/22/2008, 05:53 PM RE: Jabotinsky's iterative logarithm - by bo198214 - 05/22/2008, 06:48 PM RE: Jabotinsky's iterative logarithm - by andydude - 05/22/2008, 07:43 PM RE: Jabotinsky's iterative logarithm - by andydude - 05/22/2008, 09:22 PM RE: Jabotinsky's iterative logarithm - by Ivars - 05/23/2008, 07:05 AM RE: Jabotinsky's iterative logarithm - by bo198214 - 05/23/2008, 09:31 AM RE: Jabotinsky's iterative logarithm - by bo198214 - 05/23/2008, 09:24 AM RE: Jabotinsky's iterative logarithm - by andydude - 05/23/2008, 07:43 PM RE: Jabotinsky's iterative logarithm - by bo198214 - 05/23/2008, 10:36 PM RE: Jabotinsky's iterative logarithm - by andydude - 05/24/2008, 06:37 AM RE: Jabotinsky's iterative logarithm - by andydude - 06/14/2008, 12:44 AM RE: Jabotinsky's iterative logarithm - by Gottfried - 05/23/2008, 11:06 PM RE: Jabotinsky's iterative logarithm - by bo198214 - 05/24/2008, 05:49 AM RE: Jabotinsky's iterative logarithm - by Gottfried - 05/24/2008, 06:56 AM RE: Jabotinsky's iterative logarithm - by bo198214 - 05/24/2008, 08:44 AM RE: Jabotinsky's iterative logarithm - by Gottfried - 05/24/2008, 09:24 AM RE: Jabotinsky's iterative logarithm - by bo198214 - 05/24/2008, 10:03 AM RE: Jabotinsky's iterative logarithm - by Gottfried - 05/24/2008, 04:03 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Jabotinsky IL and Nixon's program: a first categorical foundation MphLee 10 3,310 05/13/2021, 03:11 PM Last Post: MphLee Is bugs or features for fatou.gp super-logarithm? Ember Edison 10 14,850 08/07/2019, 02:44 AM Last Post: Ember Edison Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 15,674 06/10/2019, 04:29 AM Last Post: Ember Edison True or False Logarithm bo198214 4 14,111 04/25/2012, 09:37 PM Last Post: andydude Base 'Enigma' iterative exponential, tetrational and pentational Cherrina_Pixie 4 14,151 07/02/2011, 07:13 AM Last Post: bo198214 Principal Branch of the Super-logarithm andydude 7 20,481 06/20/2011, 09:32 PM Last Post: tommy1729 Logarithm reciprocal bo198214 10 31,315 08/11/2010, 02:35 AM Last Post: bo198214 Kneser's Super Logarithm bo198214 18 63,446 01/29/2010, 06:43 AM Last Post: mike3 Iterative square root like square root recursion limit bo198214 1 6,168 02/23/2009, 03:01 PM Last Post: Gottfried Unique Holomorphic Super Logarithm bo198214 3 9,613 11/24/2008, 06:23 AM Last Post: Kouznetsov

Users browsing this thread: 1 Guest(s)