4 research outputs found
Renormalization and α-limit set for expanding Lorenz maps
We establish a one-to-one correspondence between the renormalizations and proper totally invariant closed sets (i.e., α-limit sets) of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. We describe the minimal renormalization by constructing the minimal totally invariant closed set, so that we can define the renormalization operator. Using consecutive renormalizations, we obtain complete topological characteriza- tion of α-limit sets and nonwandering set decomposition. For piecewise linear Lorenz map with slopes ≥ 1, we show that each renormalization is periodic and every proper α-limit set is countable
Multidimensional Rovella-like attractors
We present a multidimensional flow exhibiting a Rovella-like attractor: a
transitive invariant set with a non-Lorenz-like singularity accumulated by
regular orbits and a multidimensional non-uniformly expanding invariant
direction. Moreover, this attractor has a physical measure with full support
but persists along certain0909.1033 submanifolds of the space of vector fields.
As in the 3-dimensional Rovella-like attractor, this example is not robust. The
construction introduces a class of multidimensional dynamics, whose suspension
provides the Rovella-like attractor, which are partially hyperbolic, and whose
quotient over stable leaves is a multidimensional endomorphism to which
Benedicks-Carleson type arguments are applied to prove non-uniform expansion.Comment: 45 pages, 14 figures; improved introduction with more citations to
other relevant related works. To appear in Journal of Differential Equation