Program cascade version 0.0.4

Dmitry V. Khmelev

Initial revision 2003-08-20; Last revision 2003-08-27

1  Download

2  File readme

cascade --- locating almost parabolic maps in quadratic family


SUPPORTED ENVIRONMENT

GNU/Linux/XFree/GNUMP

http://www.gnu.org/
http://www.XFree86.org/
http://www.swox.com/gmp/


COMPILATION

I assume that you have X graphical system development kit installed.

Setup GNU Multiple Precision library
(http://www.swox.com/gmp/). Verify that your compiler supports STL
(Standard Template Library) --- e.g. that the gcc version >=3.2
(http://gnuwww.epfl.ch/software/gcc/gcc.html).

Enter make (or gmake) in the directory where sources reside.


BRIEF INSTRUCTION

Given $k$ values of parabolic cascades $p_1$, ..., $p_k$ the program
locates the value of real parameter $c$ such that the mapping $z^2+c$
is $k$ times renormalizable and the order of renormalizations is
$3p_1+2$, ..., $3p_k+2$ (multiplier 3 appears because we consider
pertubation of parabolic periodic point of period 3).

Example of usage:

  cascade 10 10 -mm.cmd -jj.cmd

Output of the command above:

------------ cut ----------
M-set configuration file name: m.cmd
J-set configuration file name: j.cmd
Length of kneading sequence: 1023
2 parabolic cascades of the following orders were used: 10 10
l=-1.75
m=-0.75
r=0.25
Common length is 1 out of 1023
l=-1.75
m=-1.25
r=-0.75
...
Common length is 95 out of 1023
l=-1.74824762437492609024
m=-1.74824762390926480293
r=-1.74824762344360351562
Common length is 1023 out of 1023
Resulting value is -1.74824762390926480293
------------ cut ----------

In this case the value of parameter c=-1.74824762390926480293

Notice that there were also generated m.cmd and j.cmd ---
configuration files for programs 'xbmp' and 'pvmjulia', which can draw
the Julia set and corresponding blowup of Mandelbrot set.

License conditions are described in file LICENSE.txt


ADDITINOAL PROGRAMS

Together with the sources of the program, one can also find the
sources for program 'par3' (previous version of cascade with direct
comparision of kneading sequences) and 'splitseq' for splitting
kneading sequences into blocks. 

par3 README:
This is old variant of 'cascade' program. It can be useful because of
comparision function, where the kneading sequences here are compared 
explicitly.

splitseq README:
This program splits input string into blocks, as described in the book
«One-dimensional dynamics» by W. de Melo and S. van Strien, see page 
109, section 3b (Hofbauer's Tower Construction), Chapter II.


ACKNOWLEDGEMENTS
The author is grateful to Misha Yampolsky for valuable comments.

3  Usage and options summary

user@computer$ ./cascade -h
Usage: cascade [OPTIONS] N1 N2 N3 ...
  N1, N2, ... are expected to be non-negative integers, characterizing
parabolic cascade length (at least one cascade has to be specified).

Options are:
  -s  print kneading sequence to stdout
  -m <filename> output xbmpl command file for drawing M-set to <filename>
  -j <filename> output xbmpl command file for drawing J-set to <filename>
  -p <num=64> set precision to <num> 
  -l <num=-1.75> lower bound for target parameter c
  -r <num=0.25> upper bound for target parameter c
  -q quiet, no messages to stderr
  -d <num=3> Debug ouput level 0..3
  -h  print this help
  -v  print version information

Features:
 1) It is your responsibility to give correct precision value
If precision is not enough, the results are unpredictable.
 2) Ctrl-C can be used to display the progress in lengthy subroutines.
To stop the program push it into background regime by Ctrl-Z and use kill -9.
You can also send a signal kill -2 to the program to check current status.

4  Project revision history

2003-08-20

2003-08-27

5  License

Available at http://www.math.toronto.edu/dkhmelev/PROGS/dynamics/

Author:

Dmitry V. Khmelev dkhmelev((at))math.toronto.edu [change ((at)) to @ in order to get proper address - antispam]

University of Toronto, Department of Mathematics, 100 St George Street, M5S 3G3 ON, Canada

LICENSING TERMS

This program is granted free of charge for research purposes. However you must obtain a license from the author to use it for commercial purposes.

Scientific results produced using the software provided shall acknowledge the use of cascade. The proper references are:

D. Khmelev, Programs in Dynamics http://www.math.toronto.edu/dkhmelev/PROGS/dynamics/

Moreover shall the author of cascade be informed about the publication.

The software must not be modified and distributed without prior permission of the author.

By using cascade you agree to the licensing terms.

NO WARRANTY

BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM ÄS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.