ON NEGATIVE LIMIT SETS FOR ONE-DIMENSIONAL DYNAMICS FRANCISCO BALIBREA, GABRIELA DVORN ´ IKOV ´ A, MAREK LAMPART, AND PIOTR OPROCHA * Abstract. In this paper we study the structure of negative limit sets of maps on the unit interval. We prove that every α-limit set is an ω-limit set, while the converse is not true in general. Surprisingly, it may happen that the space of all α-limit set of interval map is not closed in the Hausdorff metric (thus some ω-limit sets are never obtained as α-limit sets). Moreover, we prove the set of all recurrent points is closed if and only if the space of all α-limit sets is closed. 1. Introduction Positive limit sets, so-called ω-limit sets, of the maps of the interval were deeply studied by many authors. For example [1] (see also [5]) shows that a nonempty subset M of the unit interval can be an ω-limit set if it is a union of intervals or a nowhere dense set. In this context it is also known that the space of all ω-limit sets is closed in the Hausdorff metric (see [4]) and that each ω-limit set is contained in the maximal one (see [4] or [15]). Recently these results were extended onto other classes of one-dimensional spaces, e.g. circle [12] or topological graphs [10]. While for homeomorphisms negative limit sets (called α-limit sets in the present paper) can be defined exactly in the same way as ω-limit sets, for non invertible maps it is not so obvious. One possibility is to take as an α-limit set the set of all accumulation points of the sequence f -n ({x}). For example this approach is represented in [6]. There is also another possibility. Instead of looking at all possi- ble preimages we can simply pick one negative trajectory and check accumulation points of this sequence. Of course obtained set will be usually smaller than the one obtained in the first approach, however it seems to better mimic the situation for homeomorphism. Therefore we adopt this notation in the paper. It is noteworthy that the idea of tracking a single negative trajectory can be generalized even more like in [8] where special α-limit sets were defined. While α-limit sets seem to be very similar to ω-limit sets, they were not much studied so far. The reason for this can be twofold. First, it is much harder to deal with them, mainly because there are multiple choices for point in a negative trajectory. Secondly, images of open sets are usually not open under iteration of non-invertible map, so some tools like Baire Category Theorem can be harder to apply. As we mentioned before, ω-limit sets are well characterized for interval maps, especially if the entropy of map is zero (e.g. see [5] and the references therein). In 1991 Mathematics Subject Classification. Primary: 37E05, Secondary: 37B20, 37B40. Key words and phrases. interval map, negative trajectory, limit set, solenoid. * Corresponding author. 1 Nonlinear analysis: theory, methods & applications. 2012, vol. 75, issue 6, p. 3262-3267. http://dx.doi.org/10.1016/j.na.2011.12.030 DSpace VŠB-TUO http://hdl.handle.net/10084/90232 20/03/2012