Lyapunov exponent and topological entropy plateaus in piecewise linear maps

Abstract

We consider a two-parameter family of piecewise linear maps in which the moduli of the two slopes take different values. We provide numerical evidence of the existence of some parameter regions in which the Lyapunov exponent and the topological entropy remain constant. Analytical proof of this phenomenon is also given for certain cases. Surprisingly however, the systems with that property are not conjugate as we prove by using kneading theory.

Continuity of entropy for Lorenz maps

“…Lorenz maps and their topological entropy have been and still are investigated intensely, see for instance [5,7,9,10,13,14,15,17,18,19,22,23,25,26,38]. The simplest example of a Lorenz map is a (normalised) β-transformation, and the topological entropy of such a transformation is equal to ln(β); this was first shown in [17,33].…”

Matching in a family of piecewise affine maps

“…Let us observe that, since 1 is a fixed point for Q γ for all γ ∈ R, we get that if γ ∈ Q s ; then both positive and negative orbit of γ end up in 1, and derivatives can be made to match. Furthermore, γ / ∈ P M γ , see (4). This implies that the matching persist under a small perturbation in γ, and hence matching is an open and dense condition.…”

Entropy locking

“…Proof. We use formula (5). If the interval on which f is acting has length γ, then the length of each lap of f n is not larger than γ/α n .…”

“…For a piecewise continuous piecewise monotone map f (with the finite number of laps), the usual definition of its topological entropy is (5) h…”

“…In 2013, V. Botella-Soler, J. A. Oteo, J. Ros and P. Glendinning [5] noticed that for certain values of λ and µ both Lyapunov exponent and topological entropy of T λ,µ,b remain constant as b varies in some interval of values close to 0, although the kneading sequence varies. In 2014, H. Bruin, C. Carminati, S. Marmi and A. Profeti [3] explained this phenomenon by observing that it is caused by matching with index zero, where for some k > 0 we have T k b (0 − ) = T k b (0 + ).…”