Semiconjugacies to angle-doubling

Abstract

Abstract. A simple consequence of a theorem of Franks says that whenever a continuous map, g, is homotopic to angle-doubling on the circle, it is semiconjugate to it. We show that when this semiconjugacy has one disconnected point inverse, then the typical point in the circle has a point inverse with uncountably many connected components. Further, in this case the topological entropy of g is strictly larger than that of angle-doubling, and the semiconjugacy has unbounded variation. An analogous theorem holds for degree-D circle maps with D > 2.

Monotonicity of entropy for real multimodal maps

“…Since the only fixed point of T 2s |J is p, it follows that U = W . This proves assertion (5). To prove (6) note that each attracting periodic orbit of T necessarily intersects ∪ b i=1 Z i,T because T and the unstunted sawtooth map S 0 agree outside this set.…”

“…-W (T ) is the set of points which are eventually mapped into the interior of a block of plateaus, see equation (5). -T is the set of maps with the same W (T ), see equation (6).…”

“…Step 2. First assume that c is a critical point as in case (5) so that c is mapped to the boundary of a component U of B(f ) and f −1 (U) ∪ {c} contains a (real) neighborhood of c, see Figure 6 on the right. Then we can choose the family g t so that g t (c) is in the interior of B of J.…”

“…Pick t 0 > 0 small and letĝ = g t 0 . Next we need to take special care of case (5) in the situation when there exists a critical point…”

Multi-chaos from Quasiperiodicity

“…Step 1: The projection semi-conjugacy [6]. By assumption, M has a left eigenvector v L m corresponding to the eigenvalue m. Since M is integer-valued, v L m can be chosen to have integer entries.…”