Arithmetic properties of topological invariants of systems with nonstructurally stable homoclinic trajectories (Q1097563)

This interesting paper deals with the C r-diffeomorphism \({\mathcal T}\) (r\(\geq 3)\), defined on a two-dimensional manifold, possessing a saddle fixed point A with the eigenvalues \(\lambda\), \(\gamma\), such that \(0<| \lambda | <1<| \gamma |\), \(\sigma =| \lambda \gamma | \leq 1\). The existence of the structurally unstable homoclinic trajectory \(\Gamma\) of the saddle fixed point is assumed. A special coordinate system in a neighbourhood of A exists, such that the diffeomorphism \({\mathcal T}\) can be represented in the following form: \(\bar x=\lambda x+f(x,y)x\) 2y, \(\bar y=\gamma y+g(x,y)xy\) 2, \(A=(0,0)\). Let us choose the pair of the points \({\mathcal M}\) \(+,{\mathcal M}\)-\(\in \Gamma\), \({\mathcal M}\) \(+=(x\) \(+,0)\in W\) \(s_{loc}\), \({\mathcal M}\) \(-=(0,y\) \(-)\in W\) \(u_{loc}\), such that \({\mathcal T}\) mM \(-=M\) \(+\) for some \(m\in N\). Let us put \(\vartheta =-\ln | \lambda | /\ln | \gamma |\), \(\tau =(\ln | \gamma |)^{-1}\times \ln | {\mathfrak c}x\) \(+/y\)- \(|\), where \(c\) is a quantity depending on \({\mathcal T}\). The authors show that the quantities \(\vartheta\) and \(\tau\) are topological invariants of the two-dimensional diffeomorphisms, i.e. if \({\mathcal T}_ 1\) is topologically conjugated with \({\mathcal T}_ 2\), then \(\vartheta_ 1=\vartheta_ 2\).