a b s t r a c tLet X denote a locally compact metric space and u : X ? X be a continuous map. In the 1970s Zadeh presented an extension principle helping us to fuzzify the dynamical system (X, u), i.e., to obtain a map U for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets Syst. 61 (1994) 277-283] to generalize Zadeh's original extension principle.In this paper we study basic properties of so-called g-fuzzifications, such as their continuity properties. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on X induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semiconjugacy) between fuzzified dynamical systems.Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, endograph and levelwise topologies.

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Abstract. Motivated by a recent result of Ciesielski and Jasiński we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conjecture of Edrei from 1952, first disproved by Williams in 1954.

We study the properties of Φ-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of Φ-irregular set in terms of entropy on chain recurrent classes and prove that Φ-irregular sets of full entropy are typical. We also consider Φ-level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measures.