Homeomorphism of Hereditarily Locally Connected Continua

Let $$f:X\rightarrow X$$ be a hereditarily locally connected continuum homeomorphism and denote respectively by P(f), AP(f) $$\Omega (f)$$ , the sets of periodic points, almost points non-wandering f. We show that any $$\omega $$ -limit set (resp. $$\alpha set) is minimal. Moreover, we (f)=AP(f)$$ . also prove if $$P(f)=\emptyset then there exists unique minimal set. On other hand, $$P(f)\ne \emptyset infinite has adding machine structure absence Li-Yorke pairs. Consequently, partially solve positive entropy conjecture which remains open even in case continuum.