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The bifurcation set as a topological invariant for one-dimensional dynamics
Nonlinearity ( IF 1.7 ) Pub Date : 2021-03-09 , DOI: 10.1088/1361-6544/abb78c
Gabriel Fuhrmann 1 , Maik Grger 2 , Alejandro Passeggi 3
Affiliation  

For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of (some of) their endpoints. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological invariant for one-dimensional dynamics. We provide a detailed description of this invariant in the realm of transitive maps and observe that it carries fundamental dynamical information. In particular, for transitive non-minimal piecewise monotone maps, the bifurcation set encodes the topological entropy and strongly depends on the behavior of the critical points.



中文翻译:

分岔集作为一维动力学的拓扑不变量

对于单位间隔或圆上的连续映射,我们将分岔集定义为那些间隔空洞的集合,这些空洞的幸存集对其端点的(某些)任意小的变化敏感。通过假设全局视角并关注该集合的几何和拓扑特性,而不是幸存的单个孔集,我们获得了一维动力学的新拓扑不变量。我们在传递映射领域详细描述了这个不变量,并观察到它携带基本的动态信息。特别是,对于传递性非最小分段单调映射,分岔集编码拓扑熵并强烈依赖于临界点的行为。

更新日期:2021-03-09
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