Occasionally attracting compact sets and compact-supercyclicity

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Let X be a real or complex Banach space and Tₜ:X→ X is a power bounded operator (or a C₀-semigroup). If there exists a "occasionally" attracting compact subset K (for each x in unit ball ₙ ρ(Tⁿ x, K)=0 then there exists attracting finite-dimensional subspace L (for each x in X ₙ ρ(Tⁿ x, L)=0. Also we define the compact-supercyclicity. Each infinity-dimentional X has no compact-supercyclic isometries. If T is a supercyclic and power bounded that Tⁿx vanishes for each x.