This is a program to prove the hyperbolicity of real Henon maps
(x, y) |--> (a - x^2 + by, x): R^2 -> R^2.

INSTALL
==================================================
1. Install MATLAB
   http://www.mathworks.com/

2. Install GAIO (Global Analysis of Invariant Objects)
   http://math-www.uni-paderborn.de/~agdellnitz/gaio/

3. Install CAPD (Computer Assisted Proofs in Dynamics)
   http://capd.wsb-nlu.edu.pl/
   (If you use compiled binary files and do not want to compile 
   C and C++ source codes by yourself, you can skip this step.)

4. Check if MATLAB, GAIO (and CAPD) are properly installed.

5. Expand realhenon.tar.gz, using gunzip and tar. 
   This will create a directory named 'realhenon' which
   contain the following files.


FILES AND DIRECTORIES
==================================================
Source Codes:
       realhenon.cpp           C++ source code for the map definition
       hyp_realhenon.c         C source code for main routine
       single_param.m          MATLAB script
       showhyp.m               MATLAB script
       showsta.m               MATLAB script
       start_comp.m            MATLAB script

Binary Object Files:
       hyp_realhenon.mexmac    Mach-O object file for MacOSX (PowerPC CPU)
       hyp_realhenon.mexglx    ELF object file for LINUX (i486 CPU)

Directories:
       data/                   directory for temporary data and the results
       snapshots/              directory for snapshots taken every one hour

Others:
       README                  This file
       makefile                makefile (GNU make)
       mexopts.sh              mex option file

HOW TO USE (SINGLE PARAMETER VALUE)
==================================================
1. Edit 'parameter setting' lines in single_param.m
   For example, if you want to prove the hyperbolicity for 
   a = 5.4 and b = 1, put
       param_inf = [5.4; 1] 
       param_sup = [5.4; 1]

   Note that since the expression '5.4' as a double precision number
   is not mathematically equivalent to the real number '5.4'. 
   Therefore if you want to be rigorous, the setting should be like
       param_inf = [5.39999999; 1] 
       param_sup = [5.40000001; 1]

2. Launch MATLAB and type
   single_param
   then the computation starts.

3. If the program proves the hyperbolicity of given parameter, 
   it will exit displaying 'Hyperbolic!'.


HOW TO USE (2-PARAMETER FAMILY)
==================================================
   Launch MATLAB and type 'start_comp' then the computation starts.

   The set of hyperbolic parameter values will be saved in
   'data/hyperbolic.tree' and the remaining parameter values in 
   'data/remaining.tree'. These are GAIO tree files.

   To show the hyperbolic parameter values, launch another MATLAB and
   type 'showhyp'. You can see the number of remaining cubes in the
   parameter tree by typing 'showsta'.

   The status of the program will be saved once every hour to the directory 
   'snapshots'. The file 'snapshots/hyperbolic_xxxxxxxx.tree' is a copy of
   'data/hyperbolic.tree' when xxxxxxxx seconds past form the beginning.
   
   If the computation was interrupted for any reason, you can resume the
   computation using the saved data in 'snapshots' and function resume_comp. 
   For example, if there is a file 'data/hyperbolic_00003600.tree', you can
   resume the computation by running 'resume_comp(3600)'.


COMPILATION
==================================================
If you want to make the binary file (i.e. hyp_realhenon.mexXXX) by yourself,
simply type 'make' at the 'realhenon' directory.

Be sure that CPAD and GAIO are installed properly in the directories 
given in 'makefile' and 'mexopts.sh' is properly configured.


==================================================
Zin ARAI 
arai@math.kyoto-u.ac.jp     
http://www.math.kyoto-u.ac.jp/~arai/