Abstract

All nonhyperbolic automorphisms of the 2-torus are not structurally stable, and it is generally impossible to predict the dynamics of their arbitrarily small perturbations. In this paper, given a representative of each algebraic conjugacy class of nonperiodic nonhyperbolic maps, a one-parameter family of diffeomorphisms is constructed, in which the zero value of the parameter corresponds to the given map and the nonzero values, to Morse–Smale diffeomorphisms. According to results of V. Z. Grines and A. N. Bezdenezhnykh, a Morse–Smale diffeomorphism of a closed orientable surface which induces a nonperiodic action on the fundamental group has nonempty heteroclinic set. It is proved that, in all of the constructed families, the diffeomorphisms corresponding to nonzero parameter values have nonempty orientable heteroclinic sets in which the number of orbits is determined by the automorphism being perturbed.

中文翻译：

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2-环面非双曲代数自同构的摄动

摘要

2-环面的所有非双曲自同构在结构上都不稳定，并且通常不可能预测它们任意小的扰动的动态。在本文中，给定非周期非双曲映射的每个代数共轭类的代表，构造一个单参数微分同胚族，其中参数的零值对应于给定映射，非零值对应于 Morse-Smale 微分同胚。根据 VZ Grines 和 AN Bezdenezhnykh 的结果，可定向封闭曲面的 Morse-Smale 微分同胚对基本群产生非周期作用，具有非空异宿集。事实证明，在所有构建的家庭中，