SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms
Let \(M\) be a compact Riemannian manifold with dimension \(n\) and \(I\) be the unit interval \([0,1]\). The paper under review the author considers the following assumptions on a \(C^2\) diffeomorphism \(F:M\times I\to M\times I\) with \(F(x,t)=(h(x),f_x(t))\) for \((x,t)\in M\times I\): {\parindent=6mm\begin{itemize}\item{} (\(H_1\)) \(f_x(0)=0\) and \(f_x(1)=1\) for every \(x\in M\); \item{} (\(H_2\)) \(h:M\to M\) is an Anosov diffeomorphism preserving the Lebesgue measure \(\mu\); suppose \(h\) has at least two fixed points \(l\) and \(r\); \item{} (\(H_3\)) \(D_t f_l(0)<1\), \(D_t f_r(1)<1\), \(D_t f_l(1)>1\), \(D_t f_r(0)>1\); and \(f_l (t)<t\), \(f_r(t)>t\) for \(t\in(0,1)\); \item{} (\(H_4\)) \(\int_{M\times \{0\}} \log|D_t f_x(0)|d\mu_0<0\) and \(\int_{M\times \{1\}} \log|D_t f_x(1)|d\mu_1<0\); \item{} (\(H_5\)) \(|D_t f_x|<\lambda_{\min}\), where \(\lambda_{min}>1\) is the minimal expanding rate of the Anosov system \(h\) in the unstable directions; where \(D_t f_x(0)=\frac{d}{dt}f_x(t)|_{t=0}\), \(D_t f_x(1)=\frac{d}{dt}f_x(t)|_{t=1}\), and \(\mu_t\) denotes the Lebesgue measure on \(M\times \{t\}\). \end{itemize}} Using Pesin's theory combined with the general theory of differentiable dynamical systems, the author proves (as consequences of two more general results) that under the above assumptions, the system \(f\) admits exactly two physical measures (SRB measures), \(\mu_0\) and \(\mu_1\). Moreover, the basins of the SRB attractors \((M\times \{0\}, \mu_0)\) and \((M\times \{1\}, \mu_1)\) are topologically dense in the total space \(M\times I\). An illustrative example is also given.
scientific article; zbMATH DE number 6157266