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:

(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.