Let \(F(x,y)=(f(x)-\epsilon(x,y),x)\) be an infinite renormalizable Hénon-like map defined on the unit square \(B^0=[0,1]^2\), where \(f\) is a unimodal map with non-degenerate critical point and the perturbation \(\epsilon\) is small enough. Consequently, there exists a nest of \(2^n\)-periodic (under \(F\)) quadrilateral boxes \(B^0 \supset B^1 \supset B^2\supset \cdots\) shrinking to the Cantor attractor \(\mathcal{O}_F=\bigcap_{n=0}^{\infty}\bigcup_{i=0}^{2^n-1}B_i^n\), where the orbit \(\mathcal{B}^n=\{B_i^n=F^i(B^n):0\leq i\leq 2^{n-1}\}\) represents the \(n\)-th renormalization level for \(F\). Moreover, it is known that there exists a unique invariant measure \(\mu\) supported on the attractor. On the other hand, \(\mathcal{O}_F\) is topologically conjugate to the attractor \(\mathcal{O}_{f_{*}}\) of the one-dimensional period-doubling renormalization fixed point \(f_*\) via a suitable conjugation \(h\).
In [A. De Carvalho et al., J. Stat. Phys. 121, No. 5–6, 611–669 (2005; Zbl 1098.37039)], it was proved that the universality and rigidity phenomena, discovered by M.J. Feigenbaum in [J. Stat. Phys. 19, 25–52 (1978; Zbl 0509.58037)], are not valid in the setting of higher-dimensional systems, taking the Hénon family as counterexample. Nevertheless, in the present paper the authors show that the small scale universality is valid for dissipative Hénon maps from a probabilistic point of view.
Indeed, the two main results of the paper are:
(i) Theorem 1. The geometry of the Cantor attractor of a strongly dissipative infinitely renormalizable Hénon-like map \(F\) is probabilistically universal, that is, for the Cantor attractor \(\mathcal{O}_F\) there exists \(\theta<1\) such that \(\mu(S_n(\theta^n))=\mu(\{B\in \mathcal{B}^n: B\, \text{has}\, \epsilon\text{-precision}\})\geq 1- \theta^n,\) where the \(\epsilon\)-precision is defined as follows: if \(B\in \mathcal{B}^n\), and \(B_1, B_2\in \mathcal{B}^{n+1}\) with \(B_1,B_2\subset B\), and if \(I\), \(I_1\), and \(I_2\) are the corresponding pieces of \(f_*\) by conjugation, we say that \(B\) has \(\epsilon\)-precision if after applying an affine map \(A\) we have that the (Hausdorff) distances \(d_H(I,A(B)), d_H(I_1,A(B_1)), d_H(I_2,A(B_2))\) are less than or equal to \(\epsilon\cdot\mathrm{diam}(I)\);
(ii) Theorem 2. The Cantor attractor of a dissipative infinitely renormalizable \(F\) is probabilistically rigid, that is, there exists \(\beta>0\) and a sequence \(X_1\subset X_2\subset \dots\subset \mathcal{O}_{F}\) such that the map \(h:X_N\rightarrow h(X_N)\subset \mathcal{O}_{f_{*}}\) is \((1+\beta)\)-differentiable and \(\mu(X_N)\rightarrow 1\).
As a consequence of Theorem 2, the authors also prove that the Hausdorff dimension is universal in the sense that \(HD_{\mu(\mathcal{O})_F}=HD_{\mu_*(\mathcal{O}_{f_*})}\), where \(\mu*\) is the invariant measure associated to \(\mathcal{O}_{f_*}\).
Although the reader is appropriately guided by the authors, we must mention that the development is extremely technical, with a very deep and fascinating richness from the dynamical point of view.