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

Complete invariants and parametrization of expansive Lorenz maps

“…which means the first l words of k − and the first r words of k + , then we have (1) e − = w ∞ − and e + = w ∞ + , i.e., e − and e + are periodic; (2) The following decompositions holds:…”

“…The periodic renormalization is interesting because β-transformation can only be renormalized periodically (see [3][6]) . This kind of renormalization was studied by Alsedà and Falcò [1], Malkin [11]. It was called phase locking renormalization in [1] because it appears naturally in Lorenz map whose rotational interval degenerates to a rational point.…”

“…This kind of renormalization was studied by Alsedà and Falcò [1], Malkin [11]. It was called phase locking renormalization in [1] because it appears naturally in Lorenz map whose rotational interval degenerates to a rational point.…”

Renormalization and $\alpha$-limit set for expanding Lorenz maps

“…This kind of renormalization was studied by Alsedà and Falcò [3], Malkin [19]. It was called phase locking renormalization by Alsedà and Falcò in [3] because it appears naturally in Lorenz map whose rotational interval degenerates to a rational point. As we shall see in Theorem B and Theorem C, the periodic renormalization corresponds to completely invariant closed set with isolated points, while non-periodic renormalization corresponds to Cantor set.…”

Renormalization and conjugacy of piecewise linear Lorenz maps

“…We say that f a,b,c expand on average if the average slope ac + b(1 − c) is greater than 1. It is easy to see that the average slope is greater than 1 if and only if f a,b,c (0) < f a,b,c (1). We are concerned with the renormalization and conjugacy of piecewise linear Lorenz map that expand on average.…”