Minicourse on Complex Hénon Maps (joint with Misha Lyubich) Lecture 2: Currents and their applications Lecture 3: Currents cont d.; Two words about parabolic implosion Lecture 5.5: Quasi-hyperbolicity Eric Bedford Stony Brook U. For notes, comments and references, go to link MAT 655 on www.math.stonybrook.edu/ ebedford
Theorem (Friedland-Milnor) The polynomial automorphisms of C 2 that are dynamically nontrivial are compositions of complex Hénon maps. f(x, y) = (p(x) by, x) p(x) = x d + is a polynomial of degree d 2, and b 0 constant. The Green function of K + is the rate of escape to infinity: G + 1 (x, y) := lim n d n log+ f n (x, y) G + f = d G + K + := {(x, y) : f n (x, y) } = {G + = 0} U + := C 2 K + J + := K + = U +, F + := C 2 J +
Classically, potential theory gives a correspondence: in R 1 ( d dt) 2: 1 2 t δ 0 functions measures in C 1 Laplacian : log z 2πδ 0 Potential Theory For compact K C Green function G K harmonic measure µ K Picture to show computation. Theorem Green function of the filled Julia set K = K(p): 1 G K (z) = lim n d n log+ p n (z) Theorem (Brolin) For non-exceptional z 0, µ K = lim n 1 d n a p n (z 0) δ a Moral: This replaces Julia set by a measure. Introduces a different sense of convergence. Idea of proof.
Currents I Distribution Theory = (Calculus) Theorem (Schwartz) T D, T 0 T is a measure. Theory of Currents = (Calculus + Linear Algebra) The currents of dimension k are elements of the dual space of test k-forms. Basic example. Let M be an oriented k-manifold. The current of integration [M] is the functional which acts on test k-forms as φ [M], φ := φ The operator d acts on currents of integration: d[m] = [ M]. There is a deep theory of currents (Geometric Measure Theory) developed for the study of minimal surfaces. We will use currents as a language and need only their basic properties. Gives a difference sense of convergence. M
If α m = α m,j dz j, then iα m ᾱ m is a positive form. If t m 0, then α := m t mα m ᾱ m is positive. A current T is positive if T, α 0 for all positive α. Theorem (Lelong) Positive Currents If T = T j,k dz j d z k is a positive current, then each T j,k is a (signed) measure. The analogue of the Laplacian in C n : dd c = 2i = 2i 2 z j z k dz j d z k dd c is a vector-valued operator that carries the information of the restriction of the 2-D Laplacian to any complex line in C n. Theorem (Poincaré-Lelong) If h is a holomorphic function, then 1 2π ddc log h = [h = 0]
Miracles from Complex Analysis If M C 2 is a complex submanifold, then it has a canonical orientation. The current of integration [M] is positive. Convergence is different in the real/complex cases. Theorem (Wirtinger) If β = i 2 dz d z + i 2dw d w = dx dy + du dv, then Area(M) = β M
Invariant current µ + := 1 2π ddc G + G + is pluri-harmonic on the set G + > 0. Thus Supp(µ + ) = J + We may slice µ + on any complex disk D C 2 : µ + D := 1 2π (ddc D )(G + D ) and this slice is just the harmonic measure of the slice D K +. Analogue of Brolin s Theorem: Theorem (B-Smillie) If A = {h = 0} C 2 is any algebraic curve, then for some c > 0 d n [f n A] = d n f n [A] cµ + There is no exceptional A. Convergence measured by slices. Sketch of proof (more or less same as Brolin).
Rigidity of K + Theorem (B-Smillie) If V is a complex subvariety of C 2, then it is not possible that V K +, and if V is algebraic, then V K +. Theorem (Fornæss-Sibony) If T is a positive, closed, current in C 2 which is supported on K +, then T = cµ + for some c > 0. In particular, if V is a complex subvariety of C 2, then we cannot have V K +.
Laminar currents Our currents will be generated by stable/unstable manifolds. These are Riemann surfaces which are (usually) equivalent to C. These do not have singularities, but they generate laminations. A lamination is locally homeomorphic to a product T, where T C is compact, and C is the unit disk. The set T is the transversal, and the sets {t} are the plaques of the lamination. The homeomorphism is required to be holomorphic (conformal) on each plaque. A measure λ on the transversal measures the lamination. We define the uniformly laminar current S λ = λ(t) [{t} ] t T If measures λ 1 and λ 2 are related by holonomy, then S λ1 = S λ2. Laminarity gives us expansion/contraction.
Hyperbolic Hénon maps A Hénon map is hyperbolic if it is hyperbolic on J. Problems/Questions: How can you tell whether a given map is hyperbolic? Are there convenient criteria for hyperbolicity? Are there any nice families of hyperbolic maps? Theorem (B-Smillie) Suppose that f is a hyperbolic, Hénon map which is dissipative. Then 1. There are no wandering Fatou components. 2. Sink orbits O 1,..., O N, and int(k + ) = j Basin(O j) 3. J + = x J W s (x), and J j O j = x J W u (x) 4. J = J 5. The currents µ ± correspond to the Ruelle-Sullivan picture. Moral: Always have Ruelle-Sullivan (even without hyperbolicity).
µ + as generalized stable manifold Theorem Let p be a saddle point, and let D D W s (p) be a complex disk such that D has no mass for µ D. If c := Mass(µ D ) > 0, then lim n d n [f n (D)] = cµ + Idea of proof: Length-Area Principle. Corollary Every stable manifold is a dense subset of J +. Corollary If B is a basin of attraction (point or rotational), then B = J +. All basins have the same boundary. Supports can change under convergence.
Laminarity of µ + For general f, the current µ + has a measure-theoretic laminarity in the following sense: There are positive, uniformly laminar currents T + j that: The total variation measures T + j j T + j = µ + The laminar currents T + j with the properties are carried on disjoint sets. can be made of Pesin boxes. We hope to discuss quasi-hyperbolicity in Lecture #5. In this case, then there are stable/unstable manifolds, but these may not form a lamination.
Invariant measure µ := µ + µ The fact that µ ± have continuous potentials means that we can take the wedge product µ + µ. If we can do this, then we will have f (µ) = (f µ + ) (f µ ) = d µ + d 1 µ = µ Analytic definition: If T is a positive closed current, and dd c u 0, with u continuous, then we define the action on a test function φ to be T dd c u, φ := T, u dd c φ This establishes T dd c u as a positive distribution. By Schwartz, we conclude that T dd c u is a positive measure. Exercise: µ + µ + = µ µ = 0 (which suggests that µ ± is laminar).
Properties of µ Theorem If χ is a continuous function, and if c := χ µ, then Thus µ is mixing. d n f n (χµ + ) µ cµ Idea of proof: go back to the Length-Area principle. Theorem If b 1, then the Lyapunov exponents of µ are χ log b log d < 0 < log d χ + Theorem (B-Lyubich-Smillie) µ is the unique measure of maximal entropy = log d. Further, J = Supp(µ) = Closure{Saddle Points}
Quasi-expansion We let f be a complex Hénon map and recall the set K = {(x, y) C 2 : f n (x, y) is bounded for all n Z} We will work with J J := J + J, which is J = Supp(µ) = Shilov K = Closure({Periodic saddles}) Let p be a periodic saddle point, and let Ep s/u denote the stable/unstable directions at p. We will use the rate of escape (Green) function G + (x, y) := 1 d n log+ f n (x, y) to define a metric on Ep u.
Normalized parametrizations of unstable manifolds The unstable manifold W u (p) is conformally equivalent to C and thus has a complex affine structure. There is a uniformization ψ p : C W u (p) J C 2, ψ p (0) = p We may replace ψ p (ζ) by ψ p (βζ) to obtain the normalization max ζ 1 G+ (ψ p (ζ)) = 1 This normalization determines β uniquely. Let S denote the set of all saddle periodic points, and set Ψ S = {ψ p : p S} to be all the normalized unstable parametrizations.
canonical metric Let 1 denote the unit tangent vector on C. We define a norm # p on E u p by the identity Dψ p (1) # p = 1 (This metric may not be uniformly equivalent to Euclidean metric.) For each p, there is a number λ p C such that The identity yields the conclusion f ψ p (ζ) = ψ f(p) (λ p ζ) G + f = d G + λ p > 1 f to be quasi-expanding if there is 1 < κ < λ p for all p S. Theorem A complex Hénon map is quasi-expanding if and only if Ψ S is a normal family of maps from C to C 2.
another characterization of quasi-expansion Fix r > 0. For p S, we let W u p (r) be the connected component of W u (p) B p (r) containing p. We say that f satisfies proper, bounded area condition if there exists r > 0 and A < such that W u p B p (r) is closed, and Area(W u p ) A Theorem (B-Smillie) f is quasi-expanding if and only if it satisfies the proper, bounded area condition.
Unstable manifolds W u If f is quasi-expanding, we let ˆΨ denote the set of all normal limits ψ x = lim j ψ pj with p j x J. The set Wx u := ψ x (C) is independent of ψ x. Theorem (Lyubich-Peters) W u x is nonsingular. Theorem If f is quasi-expanding, then W u has locally finite folding.
Geometry implies uniformly hyperbolicity If J carries a Riemann surface lamination, then f satisfies the proper, bounded area condition and is thus quasi-expanding. Theorem If J + and J are both laminated, and if the laminations are transversal along J, then f is uniformly hyperbolic.
real maps of maximal entropy: real dynamics = complex dynamics Hénon map f : R 2 R 2 is real, maximal entropy if h top (f R ) = log(d) (= h top (f C )) Theorem (B-Lyubich-Smillie) The following are equivalent for a real Hénon map: h top (f R ) = log d J R 2 S R 2 For all p, q S, W s (p) W u (q) R 2. In all of these cases, we have J = J.
Quasi-hyperbolicity: either uniform hyperbolicity or tangency Theorem (B-Smillie) If f is real, maximal entropy, then f is quasi-hyperbolic Theorem (B-Smillie) If f is quasi-hyperbolic, then f is hyperbolic on J if and only if there is no tangency between W s (p) and W u (q) for some p, q S. Horseshoes References for quasi-expansion and quasi-hyperbolicity: Loreno Guerini and Han Peters, When are J and J equal?, arxiv:1706.00220 Eric Bedford and John Smillie, Hyperbolicity and quasi-hyperbolicity in polynomial diffeomorphisms of C 2, arxiv:1601.06268