Real and Complex horseshoes

• The paper [BSh] discusses the parameter locus of complex horseshoe maps and showcases some Conjectures that Hubbard made on the subject. By the definition, the complex horseshoe locus is inside the locus of hyperbolicity. By rigorous computer work, Zin Arai was able to identify certain regions of hyperbolic parameters, which he calls hyperbolic plateaus. The Hubbard conjectures concerned the existence of nontrivial loops inside parameter space and the automorphisms of the shift map that would be induced if such loops existed. Zin Arai has continued with his computer work has proved the actual existence of some such nontrivial loops in the parameter space of complex horseshoes. Further, he has determined their monodromy.
• An important question is to describe the region in real parameter that corresponds to real horseshoes. The first case is the case of the most basic horsesehoes: of degree 2. This was first done for maps of small jacobian. Then in a remarkable work, Zin Arai and Yutaka Ishii extended this to the full case of all degree 2 horseshoes.
• The small jacobian paper uses the "3-box system" of "crossed mappings". The paper [BSSymbolicH] uses the same system of crossed mappings to produce a symbolic characterization of the real horseshoe locus. This system is well suited for maps with small Jacobian and near the Chebyshev point. The paper of Arai and Ishii mentioned above introduces different systems of crossed mappings for dealing with horseshoes whose jacobians can be arbitrarily large.
• One of the basic properties of real horseshoes is that they are "real maps of maximal entropy", by which we mean that the entropy of the real map is the same as the entropy of its complexification. (Or, in other words, the entropy does not increase if we extend the real map to its complexification.) The tool for dealing with this case is "quasi-hyperbolicity": in [BS8] we show that real maps of maximal entropy are always quasi-hyperbolic. In [BSrme] we use quasi-hyperbolicity to show that: Real maps of maximal entropy are (uniformly) hyperbolic unless there is a tangency between the stable and unstable manifolds of a pair of saddle points of periods 1 or 2.