The classification of topologically expansive lorenz maps

Abstract

We show that topologically expansive Lorenz maps can be described up to topological conjugacy by their kneading invariants. We also give a simple condition on pairs of symbol sequences which is satisfied if and only if that pair of sequences is the kneading invariant for some topologically expansive Lorenz map. A simple extension of the theorems to the case of expansive maps of the interval with multiple discontinuities is described.

On the topological dynamics and phase-locking renormalization of Lorenz-like maps

“…For this class of maps, a characterization of the set of all kneading pairs was obtained ([12], Theorem 1 ) and the dynamics of each map in the class was completely described by characterizing the set of all itineraries of the map in terms of its kneading pair ([12], Theorem 2*). The first main result of this paper is an extension of [12], Theorem 1, to the Lorenz-like maps (see Theorem A in Subsection 2.1). This extension is obtained as a consequence of the proof of the main result of [1].…”

“…models of the Lorenz equations (see [13], [10], [11], [21] and [22]). A Lorenz map is a map f from the unit interval of the real line into itself which has the following three properties (see Figure 1 (a) ) : 1 ) f is differentiable and monotonic for c, c E (0, 1) ; 2) limxjc .f ~x~ = 1, 1 3) there exists E > 0 such that Also, a map verifying 1), 2) and being topologically expansive (see [12] for a definition) is called a topologically expansive Lorenz map. The topological dynamics of such maps as well as more general classes of maps (still verifying 1) and 2)) have been studied extensively in the literature (see [8], [9], ~12~, [15] and [18]).…”

“…The study of the topological dynamics is often much easier when done by using the symbolic representation of the maps as kneading invariants instead of the maps themselves. In [12] this strategy was used to provide a topological classification of the topologically expansive Lorenz maps. For this class of maps, a characterization of the set of all kneading pairs was obtained ([12], Theorem 1 ) and the dynamics of each map in the class was completely described by characterizing the set of all itineraries of the map in terms of its kneading pair ([12], Theorem 2*).…”

“…To prove Theorem A we will use the coding introduced by Alseda and Manosas in [5] which is an extension of the Kneading Theory for continuous bimodal degree one circle maps introduced by the same authors in [4]. This coding, in the case of the Lorenz maps, is the natural one used by Hubbard and Sparrow in [12]. On the other hand, in [5], Theorem 2* of [12] was extended to the class of Lorenz-like maps (see Proposition 4) and the rotation interval of such maps was characterized in terms of the kneading pair (see Corollary 5).…”

“…This coding, in the case of the Lorenz maps, is the natural one used by Hubbard and Sparrow in [12]. On the other hand, in [5], Theorem 2* of [12] was extended to the class of Lorenz-like maps (see Proposition 4) and the rotation interval of such maps was characterized in terms of the kneading pair (see Corollary 5).…”