On the finiteness of attractors for piecewise maps of the interval

Abstract

We consider piecewise C 2 non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise C 2 non-flat map of the interval displays only a finite number of non-periodic attractors.

Microorganism billiards in closed plane curves

“…We are interested to know the regime in which the Schwarzian derivative of this map is negative. This simplifies the dynamics dramatically, for negativity of the Schwarzian is preserved under forward iterations of the map and much is known in the literature [38,42] about hyperbolicity and the structure of attractors for such maps. The reader can easily check that the Schwarzian…”

“…(2) organizes the (piecewise) global sign of the Schwarzian derivative, Fig. 3 shows the maps F (q), F (q), and Much about the attractor structure for maps with piecewise-negative Schwarzian derivatives has been described in a recent work [42]. In this regime we expect strong attractors in microorganism elliptical billiards without orientation reversals, with all values of q belonging to at least one basin of attraction, and attractors primarily of periodic or change-of-interval type.…”

“…For piecewise negative Schwarzian maps, the maximum number of attractors [42] can be as large as 2 2 card(Sing(F )) , which for a generic elliptical microorganism billiard can be very large. Interestingly, our simulations never revealed more than one attractor, with a basin of attraction including the entire boundary.…”

“…These billiards have a caustic which is itself somewhat elliptical in appearance, but rotated with respect to the table boundary. Note that since these tables preserve orientation, their dynamics are equivalently defined by the one-dimensional map and therefore the inverse, which has everywhere negative Schwarzian, also satisfies the conditions given [42] to have a finite set of attractors with global basin between them of periodic or change-ofinterval type. Yet here we again find only single attractor in the reverse dynamics; all the rotator-like orbits with π/θ o / ∈ Z have periodic orbits as limit sets of the inverse map.…”

Topological and statistical attractors for interval maps

“…In the papers above about contracting Lorenz maps, the existence of only one discontinuity and the preserving orientation property of the maps are crucial in the proofs. Nevertheless, in [BPP19] Brandão, Palis and Pinheiro show that the existence of wandering interval is not an obstruction to prove the finiteness of non-periodic metric attractors for general maps even with several discontinuities. This fact is also true for topological attractors (Proposition 4.9) and so, combining this information about wandering intervals with the Ergodic Formalism, we were able to prove the finiteness of non-periodic attractor for discontinuous maps that are piecewise C 2 (see Theorem A in the next section).…”