On compact invariant hypersurfaces of discrete dynamical systems (Q731583)

The author proves an analog of the Bendixson-Dulac criterion for discrete dynamical systems. The discrete dynamical system (D), generated by a diffeomorphism \(f: G\to G\), \(G\) is a domain in \(\mathbb R^n\), \(n\geq 2\), is considered. The problem is posed on an upper bound for the maximum number of compact invariant piecewise smooth hypersurfaces for this systems. The main results are contained in two theorems. Th. 1. Suppose that the domain \(U\subset G\) has the homotopy group \(\pi_{n-1}(U)\) of rank \(d(\pi_{n-1}(U))=r\) and there exists a continuous function \(\mu:U\to R\) such that the function \(g(x)\equiv \mu(f(x))\det\text{J}(f(x))-\mu(x)\) is of constant sign on \(U\), where \(\text{J}(f(x))\) is the Jacobi matrix of the diffeomorphism \(f\). Then the system (D) has at most \(r\) compact invariant hypersurfaces in \(U\). Th. 2. If the function \(g(x)=0\) for \(\forall x\in U\), then the system (D) can have at most \(r\) isolated regular compact invariant hypersurfaces in \(U\).