Probabilistic Universality in Two-Dimensional Dynamics

Abstract

In this paper we continue to explore infinitely renormalizable Hénon maps with small Jacobian. It was shown in a previous paper by the authors, joint with A. de Carvalho, (J Stat Phys 121(5/6):611–669, 2005) that contrary to the one-dimensional intuition, the Cantor attractor of such a map is non-rigid and the conjugacy with the one-dimensional Cantor attractor is at most 1/2-Hölder. Another formulation of this phenomenon is that the scaling structure of the Hénon Cantor attractor differs from its one-dimensional counterpart. However, in this paper we prove that the unique invariant measure on the attractor assigns a weight to these bad spots which tends to zero on microscopic scales. This phenomenon is called Probabilistic Universality. It implies, in particular, that the Hausdorff dimension of the invariant measure on the attractor is universal. In this way, universality and rigidity phenomena of one-dimensional dynamics assume a probabilistic nature in the two-dimensional world.

Appendix: Open Problems

Let us finish with some questions related to the previous discussion.

Problem I: The collections ${{\mathcal {P}}}_n$, see (8.7), of good pieces that we have constructed are determined by the average Jacobian of the map. Observe that ${{\mathcal {S}}}_n(\theta ^n)$ might be slightly larger than ${{\mathcal {P}}}_n$. It was suggested by Feigenbaum’s experiment, mentioned in the introduction, that the statistics of the remaining bad pieces, might be governed by some universality law. This problem is also related to one of the open problems in [7] on the regularity of the conjugation $h:{{\mathcal {O}}}_F\rightarrow {{\mathcal {O}}}_G$ when $b_F=b_G$.

Problem II: Do wandering domains exist? This question was already formulated in [16]. It is included again because its solution might be obtained by using the techniques developed in this paper.

List of Symbols

$b_F$:

Average Jacobian

$B^n_\omega$:

A piece of the nth-renormalization level

${{\mathcal {B}}}^n$:

Collection of pieces in the nth-renormalization level

${{{\mathcal {B}}}^n[k]}$:

Pieces of ${{\mathcal {B}}}^n$ in $E^k$

$B_n(x)$:

The piece in ${{\mathcal {B}}}^n$ containing $x\in {{\mathcal {O}}}_F$

${\mathbf {B}}$:

The piece B viewed from its proper scale

$\text {Dist}(\phi )$:

Distortion

$D_k$:

Derivative of $\psi ^k_v$ at the tip

$\delta _B$:

Thickness of B

$\Delta _B$:

Absolute thickness of B

$E^k$:

Part of a dynamical partition

$f_*$:

Unimodal renormalization fixed point

$G_k$:

Return map related to the partition by $E^k$

$k_i(B)$:

Depth of the ith-predecessor of B

$\kappa _0(n)$:

Minimal depth to safely push-up

$\kappa (n)$:

Upper bound of the brute-force regime

l(k):

Maximal allowable depth

$\eta _\phi$:

Nonlinearity

${{\mathcal {O}}}_F$:

Invariant Cantor set of F

$\psi ^k_{c,v}$:

Coordinate changes related to the renormalization $R(R^kF)$

$\psi ^{n}_\omega$:

Coordinate change

$\Psi ^{n}_k$:

Coordinate change relating $R^{n-k}(R^kF)$ to $R^n$

${{\mathcal {P}}}_n(k; q_0,q_1)$:

Collection of $q_0,q_1$-controlled pieces

${{\mathcal {P}}}_n$:

Pieces obtained by applying the three regimes

$q_0,q_1$:

Boundary one-dimensional regime

$\sigma$:

Scaling factor of the unimodal renormalization fixed point

$\sigma _B$:

Scaling factor of B

${{\mathcal {S}}}^n(\epsilon )$:

Collection of pieces in ${{\mathcal {B}}}^n$ with $\epsilon$ precision

$t_k$:

Tilt of the derivative of $\psi ^k_v$ at the tip

T:

Tangent line field to ${{\mathcal {O}}}_F$

$\tau _F$:

Tip

X:

The differentiable part of ${{\mathcal {O}}}_F$