Symplectic blenders near whiskered tori and persistence of saddle-center homoclinics

Abstract

A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some center subspace has a higher topological dimension than the set itself. We prove that, for any $C^s$ symplectic diffeomorphism (where $s=2,\dots\infty,ω$), if it has a one-dimensional whiskered torus with a homoclinic orbit, then a symplectic blender can be created by an arbitrarily $C^s$-small perturbation. Using this result, we show that the non-transverse homoclinic intersection between the invariant manifolds of a saddle-center periodic point is persistent, in the sense that the original system lies in the $C^s$-closure of a $C^1$-open set of symplectic diffeomorphisms where those having saddle-center homoclinics are dense. Our results also hold in the corresponding continuous-time settings.

Figures (9)

• Figure 1: The schematic picture for a saddle-center periodic point with a homoclinic orbit.
• Figure 2: A schematic picture for a partially-hyperbolic cubic homoclinic tangency, where the space above $\mathbb{A}$ represents $W^\mathrm{u}_{\mathrm{loc}}(\mathbb{A})$ and the space below $\mathbb{A}$ represents $W^\mathrm{s}_{\mathrm{loc}}(\mathbb{A})$. Here $\ell^{\mathrm{ss}}$ and $\ell^{\mathrm{uu}}$ are the strong-stable and strong-unstable leaves, respectively. The orange piece of $\gamma$ is mapped by ${{S}}$ to a curve tangent to $\gamma$ at $\pi^\mathrm{s}(M)$.
• Figure 3: A schematic picture for the return maps, where the vertical lines $\ell^{\mathrm{ss}}$ and $\ell^{\mathrm{uu}}$ are the strong-stable and strong-unstable leaves, respectively. The two points in $\Pi^-$ and $\Pi^+$ are $M^+$ and $M^-$, respectively, and the image of $W^\mathrm{u}_{\mathrm{loc}}(\gamma)$ by $T_1$ intersects $W^\mathrm{s}_{\mathrm{loc}}(\mathbb{A})$ along the red curve, which is tangent to $W^\mathrm{s}_{\mathrm{loc}}(\gamma)$ at $M^+$ and whose projection by $\pi^\mathrm{s}$ is tangent to $\gamma$ in $\mathbb{A}$ at $\pi^\mathrm{s}(M^+)$.
• Figure 4: The case where $N=1$, and $\mathcal{K}_q$ has two elements. The purple boxes are the two preimages of $\Pi$. The one-dimensional disc $\ell$ intersects at least one of the preimages, and hence its image under $T_k$ for some $k\in \mathcal{K}_q$ again contains a piece (the orange one) intersecting one of the preimages.
• Figure 5: Illustration to Lemma \ref{['lem:incli']} ($(r,\varphi)$ projection). For any small piece $\ell$ of $S(\gamma_1)$ containing the point of a transverse intersection of $S(\gamma_1)$ with $\gamma_2$, there exists $k$ such that the $k$-th iterate of $\ell$ approaches $\gamma_2$ from both sides. The scattering map $S$ is defined for partially-hyperbolic heteroclinic intersections in Remark \ref{['rem:scat_trans']}.

Theorems & Definitions (120)

• Remark 1.1
• Theorem A
• Proposition 1.2: Persistent intersections
• Theorem B
• Remark 1.3
• Theorem C
• Theorem D
• Remark 1.4
• Corollary E
• Conjecture 1.5