Robust transitivity of singular hyperbolic attractors
- PDF / 489,299 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 30 Downloads / 168 Views
Mathematische Zeitschrift
Robust transitivity of singular hyperbolic attractors Sylvain Crovisier1 · Dawei Yang2 Received: 26 January 2020 / Accepted: 20 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non-isolated singularities inside the non-wandering set. A typical example of a singular hyperbolic set is the Lorenz attractor. However, in contrast to uniform hyperbolicity, singular hyperbolicity does not immediately imply robust topological properties, such as the transitivity. In this paper, we prove that on an open and dense subset of the space of C 1 vector fields of a compact manifold, any singular hyperbolic attractors is robustly transitive.
1 Introduction In 1963 Lorenz [21] studied some polynomial ordinary differential equations in R3 and he found a strange attractor with the help of computers. By trying to understand the chaotic dynamics in Lorenz’ systems, [2,18,19] constructed some geometric abstract models which are called geometrical Lorenz attractors: these are robustly transitive non-hyperbolic chaotic attractors with singularities in three-dimensional manifolds. In order to study attractors containing singularities for general vector fields, Morales– Pacifico–Pujals [26] first gave the notion of singular hyperbolicity in dimension 3. This notion can be adapted to the higher dimensional case, see [10,12,22,31]. In the absence of singularity, the singular hyperbolicity coincides with the usual notion of uniform hyperbolicity; in that case it has many nice dynamical consequences: spectral decomposition, stability, probabilistic description and so on. However there also exist open classes of vector fields exhibiting singular hyperbolic attractors with singularity: the geometrical Lorenz attractors are such examples. In order to have a description of the dynamics of general flows, we thus need to
Sylvain Crovisier was partially supported by the ERC project 692925 NUHGD. Dawei Yang was partially supported by NSFC (11822109, 11671288, 11790274, 11826102).
B
Dawei Yang [email protected]; [email protected] Sylvain Crovisier [email protected]
1
Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Saclay, 91405 Orsay, France
2
School of Mathematical Sciences, Soochow University, Suzhou 215006, People’s Republic of China
123
S. Crovisier, D. Yang
develop a systematic study of singular hyperbolicity in the presence of singularity. This paper contributes to that goal. We do not expect to be able to describe the dynamics of arbitrary vector fields. Instead, we consider the Banach space Xr (M) of all C r vector fields on a compact manifold M without boundary and focus on a subset G which is dense and as large as possible. A successful approach consists of considering subsets that are C 1 -residual (i.e. containing a dense Gδ subset with respect to the C 1 -topology), but this does not immediately allow us to handle smoother
Data Loading...
