The project uses the results of Zin Arai's hyperbolic plateau paper
to determine lower
bounds for topological entropy for whole regions of the Hénon
parameter space. The image below shows these regions in black, which
are the hyperbolic plateaus from Arai's paper.
Below is the main result of this project: rigorous lower bounds for
the topological entropy for Hénon on 43 of the plateaus. The
image visualizes these lower bound in three dimensions.
A specific plateau of interest is the one containing the point (5.4,
-1), which is the top left plateau in the above plot (number 11).
These parameter values were studied by M.J. Davis, R.S. MacKay, and
A. Sannami in their paper Markov shifts in the Hénon
family, where the authors conjectured that there existed a
conjugacy to a certain subshift of finite type. Below is an index
pair for these parameters which yields a subshift with the same
entropy as the one given by MacKay, et. al.. In fact, the subshifts
are strongly shift equivalent; the colored regions in the figure
correspond to the symbols used by MacKay.
Below are nice index pairs for plateau 20 and plateau 12.