Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Fixpoints of the dxp() - has there been discussion about it?

I was just reviewing some properties of the dxp-function, the map: x-> b^x-1 .

For functions without constant terms it is widely common to accept the fractional iterate via the powerseries generated by fractional matrix-powers as "natural" embedding into the continuous flow - and to ignore issues of other fixpoints (besides the fixpoint zero). However, I looked at the powerseries, for instance 2^x-1, developed at the other fixpoint - and not surprisingly, the fractional iterates differ in the range for x, where the range of convergence of the different power series for their schroeder-functions overlap, as well as we've discussed this with x-> b^x (where b=e or b=sqrt(2) etc) for instance in the "bummer" thread.

So at this moment I became curious about the common acceptance of the "natural" embedding of iterations of 2^x-1 or exp(x)-1 or in general of functions having f(0)=0 and f'(0)<>0 in a flow based on their powerseries around zero.

Has someone come across that discussion in the context of the dxp? A discussion about the "wobbling" - the sinusoidal different solutions per unit-height-iteration-interval when centered around the different fixpoints? And an explicite justification for the common choice?
Gottfried Helms, Kassel

Possibly Related Threads...
Thread Author Replies Views Last Post
  b^b^x with base 0<b<e^-e have three real fixpoints Gottfried 1 1,855 11/07/2017, 11:06 AM
Last Post: sheldonison
  2 fixpoints , 1 period --> method of iteration series tommy1729 0 1,180 12/21/2016, 01:27 PM
Last Post: tommy1729
  2 fixpoints related by power ? tommy1729 0 1,192 12/07/2016, 01:29 PM
Last Post: tommy1729
  2 real fixpoints again ....... tommy1729 10 7,777 02/23/2016, 10:17 PM
Last Post: tommy1729
  Fixpoints a,b with f ' (a) = f ' (b) = 1 tommy1729 0 1,403 09/15/2015, 08:29 AM
Last Post: tommy1729
  alternative fixpoints = branches ? tommy1729 0 1,515 10/11/2014, 08:50 AM
Last Post: tommy1729
  Wild conjecture about 2 fixpoints. tommy1729 0 1,605 05/03/2014, 10:56 PM
Last Post: tommy1729
  The mystery of 2 fixpoints. tommy1729 0 1,395 05/03/2014, 10:11 PM
Last Post: tommy1729
  Iteration series: Different fixpoints and iteration series (of an example polynomial) Gottfried 0 2,603 09/04/2011, 05:59 AM
Last Post: Gottfried
  Discussion of TPID 6 JJacquelin 3 5,574 10/24/2010, 07:44 AM
Last Post: bo198214

Users browsing this thread: 1 Guest(s)