A point that is minimally expansive and shadowable is topologically stable and GH-stable; a point that is mu-uniformly expansive and mu-shadowable for a Borel measure is strong mu-topologically stable.
Positive Entropy Through Pointwise Dynamics
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abstract
We define some pointwise properties of topological dynamical systems and give pointwise conditions for such a system possesses positive topological entropy. We give sufficient conditions to obtain positive topological entropy for maps which are approximated by maps with the shadowing property in a uniform way.
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Stability Theorems in Pointwise Dynamics
A point that is minimally expansive and shadowable is topologically stable and GH-stable; a point that is mu-uniformly expansive and mu-shadowable for a Borel measure is strong mu-topologically stable.