Homeomorphism of Hereditarily Locally Connected Continua

Abstract

Let f : X → X be a hereditarily locally connected continuum homeomorphism, we show that any ω-limit set (resp. α-limit set) is minimal. Moreover, we show that Ω(f ) = AP (f ) = R(f ). We also prove that if P (f ) = ∅, then there exists a unique minimal set. On the other hand, if P (f ) = ∅ then we prove that any infinite minimal set has the adding machine structure and the absence of Li-Yorke pairs. Consequently, we partially solve the positive entropy conjecture which remains open even in the case of hereditarily locally connected continuum.

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