Connected and not arcwise connected invariant sets for some 2-dimensional dynamical systems. (English) Zbl 1185.37057
Let \(T:\mathbb R^2\to\mathbb R^2\) be a diffeomorphism of class \(C^1\). Let \(DT(P)\) denotes the Jacobi’s matrix of \(T\) for \(P=(x,y)\) and \(|K|\) denotes the area of \(K\subset\mathbb R^2\). The main theorem of the present paper:
Assume that the conditions:
Assume that the conditions:
- (i)
- there exists a compact, simply connected set \(K\) with piecewisely smooth boundary such that \(T(K)\subset K\),
- (ii)
- \(|T^n(K)|\to 0\) for \(n\to\infty\),
- (iii)
- \(K\) contains at least two distinct fixed points, and for one of them say \(P_1\), the eigenvalues of \(DT\left( P_1\right)\) say \(\lambda_1\) and \(\lambda _2\) satisfy the inequality \( \lambda_1<-1<\lambda_2<0 \)
are fulfilled. Then the set \(\Omega :=\bigcap _{n=0}^{\infty} T^n(K)\) is connected and not arcwise connected, invariant, compact with zero Lebesgue measure.
Reviewer: Andrzej Piatkowski (Łódź)
MSC:
37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
37E99 | Low-dimensional dynamical systems |
37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |