For continuous self-maps of compact metric spaces, we initiate a preliminary study of stronger forms of sensitivity formulated in terms of 'large' subsets of N. Mainly we consider 'syndetic sensitivity' and 'cofinite sensitivity'. We establish the following: (i) any syndetically transitive, non-minimal map is syndetically sensitive (this improves the result that sensitivity is redundant in Devaney's definition of chaos), (ii) any sensitive map of [0, 1] is cofinitely sensitive, (iii) any sensitive subshift of finite type is cofinitely sensitive, (iv) any syndetically transitive, infinite subshift is syndetically sensitive, (v) no Sturmian subshift is cofinitely sensitive, (vi) we construct a transitive, sensitive map which is not syndetically sensitive.

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Let f : X → X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ N, f × f 2 × • • • × f m : X m → X m is transitive, (ii) for each m ∈ N, there exists x ∈ X such that the diagonal m-tuple (x, x,. .. , x) has a dense orbit in X m under the action of f × f 2 × • • • × f m. We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii).

We consider non-wandering dynamical systems having the shadowing property, mainly in the presence of sensitivity or transitivity, and investigate how closely such systems resemble the shift dynamical system in the richness of various types of minimal subsystems. In our excavation, we do discover regularly recurrent points, sensitive almost 1-1 extensions of odometers, minimal systems with positive topological entropy, etc. We also show that transitive semi-distal systems with shadowing are in fact minimal equicontinuous systems (hence with zero entropy) and, in contrast to systems with shadowing, the entropy points do not have to be densely distributed in transitive systems.

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We investigate some aspects of the iterative dynamics of a single continuous homomorphism T : X → X of a Hausdorff topological (semi)group X. We show that if X is a Hausdorff topological group and T : X → X is a continuous homomorphism such that either T is syndetically transitive, or T is non-wandering with a dense set of points having orbits converging to the identity element, then T is topologically weak mixing. We also show that for some familiar topological (semi)groups X, there is an (invertible) element a ∈ X such that T : X → X given by T (x) = axa −1 is topologically mixing. As a corollary we get a zero-one law for generic dynamics on certain spaces such as the Cantor space, the Hilbert cube and R k . (2000). 54H20. Mathematics Subject Classification