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Let      Y Y       be the space      { z ∈  C : z 3 ∈  [ 0 , 1 ] } \{ z \in {\mathbf {C}}:{z^3} \in [0,1]\}       with a metric defined by the arc length. Suppose that      f f       is a continuous map from      Y Y       to itself and      P P       is a finite      f f      -invariant subset. In this paper we construct a continuous map      C P {C_P}       from      Y Y       to itself satisfying      C P | P = f | P {C_P}{|_P} = f{|_P}       which achieves the infimum topological entropies of continuous maps from      Y Y       to itself which agree with      f f       on      P P      .

Topological entropy for finite invariant subsets of 𝑌