Renormalization in the Hénon family, II: the heteroclinic web

We study highly dissipative Hénon maps

with zero entropy. They form a region Π in the parameter plane bounded on the left by the curve W of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in Π, but there exist infinitely many different topological types of such maps (even away from W). We also prove that in the infinitely renormalizable case, the average Jacobian b _F on the attracting Cantor set \({\mathcal{O}}_{F}\) is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside \({\mathcal{O}}_{F}\) if and only if there are no heteroclinic tangencies.

\({\mathcal{A}}_{F}\) :

global attractor, Sect. 4

\({\mathcal{A}}^{n}_{F}\) :

n^th-scale attractor, Sect. 6

b _F :

average Jacobian, Sect. 2

B ₀ :

non-escaping points, Sect. 4

\(B_{v^{n}}\) :

renormalization piece around the tip of level n, Sect. 2

\(\hat{\beta}_{n}\) :

fixed point of R ⁿ F, Sect. 3

β _n :

periodic point, Sect. 3

\(\beta'_{n}\) :

periodic point, Sect. 3

\(\mathcal{C}_{F}\) :

non-laminar points in \({\mathcal{A}}_{F}\), Sect. 6

γ _∞ :

curve, Fig. 7

γ _j :

curve, Fig. 7

Γ_∞ :

curve, Fig. 7

Γ _j :

curve, Fig. 7

D _n :

periodic domain containing the tip, Sect. 3

D ^τ :

domain bounded by \(W^{s}_{{\mathrm{loc}}}(\tau)\) and W ^u(β ₀), Sect. 8

E _k,n :

heteroclinic points, Sect. 4

\(E^{s}_{k,n}\) :

heteroclinic points, Sect. 9

\(\Phi^{n}_{0}\) :

coordinate change, Sect. 3

\(\Phi^{k+1}_{k}\) :

coordinate change, Sect. 3

\({\mathcal{H}}^{n}_{\Omega}(\overline{\varepsilon})\) :

n-times renormalizable maps, Sect. 2

\({\mathcal{I}}_{\Omega}(\overline{\varepsilon})\) :

infinitely renormalizable maps, Sect. 2

\({\mathcal{I}}^{n}_{\Omega}(\overline{\epsilon})\) :

n-times renormalizable maps with a periodic attractor of period 2ⁿ, Sect. 2

\(K^{u}_{n}\) :

fundamental domain in W ^u(β _n ), Sect. 4

\(K^{s}_{n}\) :

fundamental domain in W ^s(β _n ), Sect. 9

\(\mathcal{K}_{k,n}(\overline{\varepsilon})\) :

maps with heteroclinic tangencies, Definition 4.5

\(\mathcal{UK}_{k,n}(\overline{\varepsilon})\) :

maps in \(\mathcal{K}_{k',n'}(\overline{\varepsilon})\) with k≤k′<n′≤n, Definition 4.5

κ _F :

topological invariant, Sect. 8

λ _n :

stable eigenvalue of β _n , Sect. 6

μ _n :

unstable eigenvalue of β _n , Sect. 6

\(\hat{M}^{n}_{i}\) :

component stable manifold of \(\hat{\beta}_{n}\), Sect. 3

\(M^{n}_{i}\) :

component stable manifold of β _n , Sect. 3

\({\mathcal{O}}_{F}\) :

critical Cantor set, Sect. 2

\({\mathcal{P}}_{F}\) :

the set of periodic point of F

\(\hat{p}^{n}_{i}\) :

heteroclinic point of R ⁿ F, Sect. 3

\(p^{n}_{i}\) :

heteroclinic point of F, Sect. 3, also Fig. 2

q _i :

heteroclinic point of R ⁿ F, Sect. 4, also Fig. 3

\(q'_{i}\) :

heteroclinic point of R ⁿ F, Sect. 4, also Fig. 3

σ :

scaling factor of the unimodal fixed point, Sect. 2

Trap  _n :

n^th-trapping region, Sect. 4

T _n :

saddle region of β _n , Sect. 4

U _n :

local unstable manifold of β _n in T _n , Sect. 4

S _n :

local stable manifold of β _n in T _n , Sect. 4

τ _F :

tip, Sect. 2

Authors and Affiliations

1. Stony Brook University, Stony Brook, NY, USA

   M. Lyubich & M. Martens

Corresponding author

Correspondence to M. Lyubich.

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