About

This is a set of programs to rigorously verify the conditions given in "Symbolic dynamics for Hénon maps near the boundary of the horseshoe locus" by Yuki Hironaka and Yutaka Ishii.

Requirement

You need to have the following programs installed properly on your computer.

Download and Compilation

  1. Download hironaka-ishii.tar.gz and extract it. Then a directory named hironaka-ishii will be created.
  2. Change the current directory to hironaka-ishii/src and edit makefile therein according to your environment. You need to set variables INCLUDES so that your C++ compiler can find Boost include files, and set CXX to the name of your C++ compiler (by default, it is g++).
  3. At the directory hironaka-isii/src, type make to compile all programs. Binary files will be installed in hironaka-ishii/bin.

How to Run the Programs

  1. First we need to generate the data files. Change the current directory to hironaka-ishii/run and execute Python scripts data_interpolation_pos.py and data_interpolation_nes.py therein. These scripts read the data of boxes at selected parameter values (stored in "hironaka-ishii/data/positive_distinguished" and "hironaka-ishii/data/negative_distinguished") and generate the data at other parameter values by linear interpolation. The generated data files will be installed in hironaka-ishii/data/positive and hironaka-ishii/data/negative. The size of files generated by this procedure will be approximately 1.2GB.
  2. Run the scripts run_Numerical_Check_*.py at hironaka-ishii/run to verify according "Numerical Check" statements in the paper. The script run_Numerical_Check_pp.py is for verifying the condition on the periodic points of period 7 used in the proof of Proposition A.1. These script files have two parameters process_number and max_process to specify the number of processes that you want to run in parallel. If you want to run multiple processes in parallel, set max_process to the number of processes and run the script for process_number = 0, 1, ..., max_process - 1. To run the program in a single process mode, call these scripts without a parameter. If the script exits normally with no error message, the verification is completed.

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