Critical locus for Hénon maps in an HOV region

In 1D, the dynamics and topology of polynomial Julia sets are to a large extent determined by the orbits of the critical points. Complex Hénon maps are polynomial automorphisms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {C}^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , hence they do not have critical points in the usual sense, but each has a non-empty critical locus (i.e. the set of tangencies between the foliations of forward, respectively backward escaping sets). We give a description of the critical locus for Hénon maps in a Hubbard-Oberste-Vorth like region from the complex horseshoe locus, the first description in a non-perturbative setting. We prove that the critical locus is a Riemann surface of infinite genus which is a collection of truncated spheres connected by handles. This model was conjectured by John Hubbard. The critical locus lies in the complement of the Julia set and it naturally completes the picture of any standard horseshoe construction. We also discuss the monodromy of the critical locus and its connection with the Bedford-Smillie stable and unstable critical loci from the Julia sets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Superscript plus"> <mml:semantics> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">J^{+}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Superscript minus"> <mml:semantics> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">J^{-}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .