A spaceXis said to be almost totally disconnected if the set of its degenerate components is dense inX. We prove that an almost totally disconnected compact metric space admits a minimal map if and only if either it is a finite set or it has no isolated point. As a consequence we obtain a characterization of minimal sets on dendrites and local dendrites.

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The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is given.

Abstract. A result by Franzová and Smítal shows that a continuous map of the interval into itself is chaotic if and only if its topological sequence entropy relative to a suitable increasing sequence of nonnegative integers is positive. In the present paper we prove that for any increasing sequence of nonnegative integers there exists a chaotic continuous map with zero topological sequence entropy relative to this sequence.

Abstract. In connection with the Entropy Conjecture it is known that the topological entropy of a continuous graph map is bounded from below by the spectral radius of the induced map on the first homology group. We show that in the case of a piecewise monotone graph map, its topological entropy is equal precisely to the maximum of the mentioned spectral radius and the exponential growth rate of the number of periodic points of negative type. This nontrivially extends a result of Milnor and Thurston on piecewise monotone interval maps. For this purpose we generalize the concept of Milnor-Thurston zeta function incorporating in the Lefschetz zeta function. The methods developed in the paper can be used also in a more general setting.