For maps with several neutral fixed points, every measure in the simplex of fixed-point measures has a basin of attraction of full Hausdorff dimension 1.
Occupation times of sets of infinite measure for ergodic transformations
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Hausdorff Dimension of Sets of Generic Points for Non-statistical Dynamical Systems
For maps with several neutral fixed points, every measure in the simplex of fixed-point measures has a basin of attraction of full Hausdorff dimension 1.