On the notions of singular domination and (multi-)singular hyperbolicity
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. ARTICLES .
September 2020 Vol. 63 No. 9: 1721–1744 https://doi.org/10.1007/s11425-019-1764-x
On the notions of singular domination and (multi-)singular hyperbolicity To the Memory of Professor Shantao Liao
Sylvain Crovisier1,∗ , Adriana da Luz2 , Dawei Yang3 & Jinhua Zhang4 1Laboratoire 2Beijing
de Math´ ematiques d’Orsay, CNRS, Universit´ e Paris-Sud, Orsay 91405, France; International Center for Mathematical Research, Peking University, Beijing 100871, China; 3School of Mathematical Sciences, Soochow University, Suzhou 215006, China; 4School of Mathematical Sciences, Beihang University, Beijing 100191, China
Email: [email protected], [email protected], [email protected], [email protected] Received July 29, 2019; accepted August 14, 2020; published online August 24, 2020
Abstract
The properties of uniform hyperbolicity and dominated splitting have been introduced to study the
stability of the dynamics of diffeomorphisms. One meets difficulties when trying to extend these definitions to vector fields and Shantao Liao has shown that it is more relevant to consider the linear Poincar´ e flow rather than the tangent flow in order to study the properties of the derivative. In this paper, we define the notion of singular domination, an analog of the dominated splitting for the linear Poincar´ e flow which is robust under perturbations. Based on this, we give a new definition of multi-singular hyperbolicity which is equivalent to the one recently introduced by Bonatti and da Luz (2017). The novelty of our definition is that it does not involve the blow-up of the singular set and the rescaling cocycle of the linear flows. Keywords MSC(2010)
multi-singular hyperbolicity, singular domination, star vector field, linear Poincar´ e flow 37C10, 37D05, 37D25, 37D30
Citation: Crovisier S, da Luz A, Yang D W, et al. On the notions of singular domination and (multi-)singular hyperbolicity. Sci China Math, 2020, 63: 1721–1744, https://doi.org/10.1007/s11425-019-1764-x
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Introduction
The dynamical properties of a system (a flow or diffeomorphism) which are robust, i.e., which persist under C 1 small perturbations, are often associated with invariant structures in the tangent bundle. A famous example of such a property is the C 1 structural stability which has been characterized [16, 20, 24] by the uniform hyperbolicity of the tangent dynamics. Among other robust properties that have been studied one can mention Liao’s star property [19] (the robust hyperbolicity of all the periodic orbits) and the robust transitivity (the robust existence of a dense orbit). Various forms of hyperbolicity have been introduced to investigate them, such as the dominated splittings that we discuss below. These concepts are in general easier to understand in the case of diffeomorphisms (see, for example, [28] for a survey). The transposition to flows leads to difficulties due to the presence of singularities: the * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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