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 \e...

The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional...

A homeomorphism f : X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x 6= y, then there is an integer n ∈ Z such that d(fn(x), fn(y)) > c. A homeomorphism f : X → X is continuum-wise expansive if there is c > 0 such that if A is a nondegenerate subcontinuum of X, then there is an integer n ∈ Z such that diam fn(A) > c. Clearly, every expansive homeom...

A centerpiece of Dynamical Systems is comparison by an equivalence relationship called topological conjugacy. Current state of the field is that, generally, there is no easy way to determine if two systems are conjugate or to explicitly find the conjugacy between systems that are known to be equivalent. We present a new and highly generalizable method to produce conjugacy functions based on a f...