Multidimensional Rovella-Like Attractors

Abstract

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 and persists along certain submanifolds of the space of vector fields. As in the 3-dimensional Rovella-like attractor, this example is not robust. As a sub-product of the construction we obtain a new class of multidimensional non-uniformly expanding endomorphisms without any uniformly expanding direction, which is interesting by itself. Our example is a suspension (with singularities) of this multidimensional endomorphism. non-uniform expansion. V.A. and M.J.P. were partially supported by CNPq-Brazil/FAPERJ-Brazil/Pronex Dynamical Systems.

On the Structure of Lorenz Maps

“…From the end of the 20th century until now the main contributions are from M. Martens and W. de Melo [32] and G. Keller and M. St. Pierre [48]. We also can cite results on the existence of SRB-measures (R. Metzger [33]), combinatorial structure of contracting Lorenz maps (Labarca and Moreira [27,26]), thermodynamical formalism (M.J. Pacífico and M. Todd [43]), construction of contracting Lorenz maps (and flows) in higher dimensions (V. Araujo, A. Castro, M.J. Pacífico, and V. Pinheiro [1]).…”

“…Thus, if (a, a ) is the connected component of ). Now we have two possibilities: a is a saddle-noddle (if x < f (x) for a < x < c, see Figure 9.1(2)) or a attracts by both side (for instance, if Df (a) < 1), see Figure 9.1 (1). In any case we get [a, b] ⊂ β(O + f (a) (if a is sable-noddle, for every a < x < c there is some…”

“…In the famous work of Lorenz [29], studying the solution of the following system of differential equations (1) it appears for the first time what was thought to be an attractor with characteristics that became, a posteriori, the ones that defines a "strange attractor" (Nevertheless, for a long time, no one proved that this attractor in the original Lorenz work exhibits in fact these characteristics. It was later shown by Tucker [51] that it does).…”

“…The dynamics of "Lorenz" plateau maps will decompose in three blocks: (1) the set Λ of points whose orbits avoid the critical region C(f t ) := [c 0 , c 1 ], (2) the set of points thats eventually go into the left side of the critical region and follow this critical value, i.e., Γ 0 := {x ; O + ft (x) ∩ [c 0 , c)} and (3) the points that visit the right side and follow its critical value, that is, Γ 1 := {x ; O + ft (x) ∩ (c, c 1 ]}. By Mañe, Leb(Λ) = 0.…”