Sinai-Bowen-Ruelle measure and the structure of strange attractors in the Lauwerier map. (English) Zbl 0987.37019
This paper is devoted to explore the Sinai-Bowen-Ruelle measure and the structue of strange attractors in the Lauwerier map. The authors consider the Lauwerier map for \(a=4\), that is
\[
F_{a,b}(x,y)= \bigl(bx (1-2y)+ y,\;ay(1-y) \bigr),\;a=4. \tag{1}
\]
They show that (1) possesses a nontrivial topologically transitive attractor \(\Lambda\) which is the closure of the unstable set of some hyperbolic fixed point. Periodic points are dense in \(\Lambda\), and all of them are hyperbolic with eigenvalues uniformly bounded away from 1 in norm. Transversality of the corresponding stable and unstable sets is proved. Moreover the SRB measure supported on the attractor is constructed and its properties are studied.
Reviewer: Messoud Efendiev (Berlin)
MSC:
| 37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
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