These are the 8 papers in the [BS] series

• The first of these papers is [BS1], which defines the invariant currents, as well as the invariant measure, and it gives the basic structure of Hénon maps in the hyperbolic case. We think of this invariant measure as "the" invariant measure because it is just given to us "for free" since it is the equilibrium measure of the invariant set K (or, equivalently, as the equilibrium measure of J). You can look at [BT1] for a longer discussion of the "equilibrium measure" of a compact subset of Cⁿ.


• [BS2] gives the convergence to the invariant current. For instance, the normalized pullbacks of a stable disk converge to the stable current. As a consequence, every stable manifold is dense in J⁺.


• The paper [BS3] shows that the invariant measure is mixing and has entropy log(d). Further, the Lyapunov exponents of the invariant measure are nonzero (in fact, their absolute values are at least log(d)>0), and thus the invariant measure is hyperbolic.


• In collaboration with Misha Lyubich, we characterized "the" invariant measure as the unique measure of maximal entropy. This paper introduces the measure-theoretic sense of "laminar currents", which was subsequently re-done much better by Dujardin in three papers: laminar currents, structure of currents, intersection of currents. Among the things that these papers do is to give more general settings in which we can conclude that a current is laminar. Also, they give a more clear understanding of the connection between the (easier?) "analytic" intersection product as defined in [BT1] and the (more subtle?) "geometric" intersection product. The ability to identify these two sorts of intersection product is an important technical tool.
  This paper also shows that "the" invariant measure gives the limiting distribution of the saddle points.


• The paper [BS5] defines critical points both in the escaping locus U⁺ and in the intersection of U⁺ and U^-. It shows that, in the 2nd case, the critical locus is never empty; and in the first case, it defines the "critical measure" and gives an integral formula for the Lyapunov exponent in terms of the Green function G⁺ and the critical measure. The Lyapunov exponent is equal to log(d) if and only if the critical measure is zero.


• [BS6] discusses the connectivity of J. For instance, J is connected if and only if each unstable slice of K⁺ is connected. Equivalently, iff there are no critical points in the unstable slice. Equivalently, J is connected if and only if the Lyapunov exponent is log(d).


• The theory of external rays are developed in [BS7]. If J is connected and hyperbolic, then J⁺ is essentially a complex solenoid. This allows us to define external rays land show that they land continuously in this case. This shows that when J is hyperbolic and connected, it is also a finite quotient of the real solenoid. This is the 2-dimensional counterpart to the 1-dimensional result which says that if a polynomial Julia set is hyperbolic and connected, then it is a finite quotient of the circle.


• [BS8] introduces quasi-expanding and quasi-hyperbolic maps. Since all real Hénon maps of (maximal) entropy log(d) are shown to be quasi-hyperbolic, his paper is the foundation for our work on real maps of maximal entropy, which includes the case of real horseshoes.

Expository paper with a focus on "external rays"

Here are the notes from a course I gave on external rays at Postech in Pohang, Korea. In the case of polynomial maps of C, the external rays are parametrized by points of the circle, And in the quadratic case, the points of the circle are represented as a 1-sided sequence of 0's and 1's. In the quadratic Hénon case, the external rays are parametrized by the real solenoid, and they are represented symbolically as bi-infinite sequences of 0's and 1's. If J is hyperbolic and connected, it is given as a quotient of the real solenoid, and this may be represented as a quotient of the full 2-shift on 0's and 1's, modulo and identification of the pairs of sequences corresponding to external rays which have the same landing points.

Real and complex horseshoes

• Click here for the page on real and complex horseshoes

Semi-Parabolic Implosion

• This is the paper [BSU] on parabolic implosion written with John Smillie and Tetsuo Ueda. It is a revision of the earlier version which we put on the arXiv; we couldn't put this revision on the arXiv because the graphics files are too big. I hope you will read the paper and enjoy the pictures.