Quasiperiodically forced interval maps with negative Schwarzian derivative

Abstract

In the study of quasiperiodically forced systems invariant graphs have a special significance. In some cases, it was already possible to deduce statements about the invariant graphs of certain classes of systems from properties of the fibre maps. Here, we study quasiperiodically forced interval maps which are monotonically increasing and have negative Schwarzian derivative. First, we derive some basic results which only require monotonicity. Then we give a classification, with respect to the number and to the Lyapunov exponents of invariant graphs, for this class of systems. It turns out that the possibilities for the invariant graphs are exactly analogous to those for the fixed points of the unperturbed fibre maps.

Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors

“…Part a) of the following proposition can be found in [5]. In a slightly different setting, both parts were proved in the earlier paper [15]. The reader can easily adapt the proof of part b) to the present setting.…”

“…The following proposition is a consequence of negative Schwarzian derivative. Proposition 1.4 ( [15,5]). Let F ∈ F s and ν ∈ P e (S).…”

“…where we used Theorem 1 (in its strengthened form (15)) for the equality. In a similar way one derives the lower bound…”

Chaotically driven sigmoidal maps

“…Let T : T → T be an irrational circle rotation. Such quasiperiodically forced systems were studied by Jäger in [6,7]. He gives the following result [6,Theorem 4.2 and Corollary 4.3] under the aditional assumption that (θ, y) → f θ (y) is continuous: There are three possible cases:…”

How chaotic are strange non-chaotic attractors?

“…r. 14 First of all, for the base dynamics given by f there exist exactly two ergodic invariant measures µ s and µ u which are associated to the invariant graphs by 12) where m denotes the Lebesgue measure on T1. These two measures µ i can be naturally identified withΛ-invariant measuresμ i by embedding them into the invariant 0-torus S := T2 × {0} in the canonical way.…”