Gonchenko, S. V.; Shil’nikov, L. P. Arithmetic properties of topological invariants of systems with nonstructurally stable homoclinic trajectories. (English. Russian original) Zbl 0635.58025 Ukr. Math. J. 39, No. 1, 15-21 (1987); translation from Ukr. Mat. Zh. 39, No. 1, 21-28 (1987). 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 \(\vartheta_ 1=\vartheta_ 2\). Reviewer: A.Klíč Cited in 4 ReviewsCited in 15 Documents MSC: 37B99 Topological dynamics 57R50 Differential topological aspects of diffeomorphisms 54H20 Topological dynamics (MSC2010) Keywords:stable manifold; saddle quantity; topological conjugacy; C r- diffeomorphism; saddle fixed point; eigenvalues × Cite Format Result Cite Review PDF Full Text: DOI References: [1] N. K. Gavrilov and L. P. Shil’nikov, ?Three-dimensional dynamical systems similar to a system with nonstructurally stable homocinic curve. I,? Mat. Sb.,88, No. 4, 475-492 (1972); II,90, No. 1, 139-157 (1973). [2] J. Palis, ?A differentiable invariant of topological conjugacies and moduli of stability,? Asterisque, No. 51, 335-346 (1978). · Zbl 0396.58015 [3] S. V. Gonchenko and L. P. Shil’nikov, ?Dynamical systems with nonstructurally stable homoclinic curves,? Dokl. Akad. Nauk SSSR,286, No. 5, 1049-1053 (1986). [4] S. V. Gonchenko, ?Nontrivial hyperbolic subsets of systems with a nonstructurally stable homoclinic curve,? in: Methods of Qualitative Theory of Differential Equations [in Russian], Gorkii State Univ. (1984), pp. 89-102. [5] S. Newhouse, J. Palis, and F. Takens, ?Bifurcations and stability of families of diffeomorphisms,? Math. IHES, No. 57, 5-72 (1983). · Zbl 0518.58031 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.