Let f be an orientation preserving homeomorphism of the disc D which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2π/3 or 4π/3, or J has a periodic point of least period n for each n ∈ N∗.

In this paper theorems are proved which provide for lifting and projecting expansive homeomorphisms through pseudocovering mappings so that the lift or projection is also an expansive homeomorphism. Using these techniques it is shown that the compact orientable surface of genus 2 admits an expansive homeomorphism.

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...