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The Legendre transform is applied to the attractors of linear randomized systems of iterated functions X 1 and X 2, construction using the ordered (F1) and disordered (F2) algorithms. The original attractors X 1 and X 2 have identical similarity dimension d S (X 1) = d S (X 2), but applying the Legendre transform to X 1 generates a set X 1 * with a linearly ordered structure and similarity dimension 1 < d S ( X 1 ∗ ) < d S ( X 1 ) , and applying the Legendre transform to X 2 generates a set X 2 * with a disordered structure and similarity dimension d S ( X 2 ) < d S ( X 2 ∗ ) < 2 .
The Legendre transform is applied to the attractors of linear randomized systems of iterated functions X 1 and X 2, construction using the ordered (F1) and disordered (F2) algorithms. The original attractors X 1 and X 2 have identical similarity dimension d S (X 1) = d S (X 2), but applying the Legendre transform to X 1 generates a set X 1 * with a linearly ordered structure and similarity dimension 1 < d S ( X 1 ∗ ) < d S ( X 1 ) , and applying the Legendre transform to X 2 generates a set X 2 * with a disordered structure and similarity dimension d S ( X 2 ) < d S ( X 2 ∗ ) < 2 .
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