A homeomorphism of a compact metric space is tight provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every C diffeomorphism of a closed surface factors to a tight homeomorphism of a generalized cactoid (roughly, a surface with nodes) by a semi-conjugacy whose fibers carry zero entropy. Stony Brook IMS Preprint #2002/04 ...

Let T be a tent map with the slope strictly between √ 2 and 2. Suppose that the critical point of T is not recurrent. Let K denote the inverse limit space obtained by using T repeatedly as the bonding map. We prove that every homeomorphism of K to itself is isotopic to some power of the natural shift homeomorphism.

We propose a relaxed notion of shadowing. In particular, for homeomorphism f on metric space, we ask whether every approximate orbit is near some actual nearby system. distinguish the cases where close or arbitrarily to f. prove relations between these notions and ordinary shadowing present various examples. finally discuss an application, $$C^0$$ -closing lemma chain recurrent points topologic...