Fractal sets satisfying the strong open set condition in complete metric spaces

• Autorzy: Goodman G. S.
• Języki publikacji: EN

Abstrakty

EN

Let K be a Hutchinson fractal in a complete metric space X, invariant under the action S of the union of a finite number of Lipschitz contractions. The Open Set Condition states that X has a non-empty subinvariant bounded open subset V, whose images under the maps are disjoint. It is said to be strong if V meets K. We show by a category argument that when K ⊄ V and the restrictions of the contractions to V are open, the strong condition implies that [formula] termed the core of V, is non-empty. In this case, it is an invariant, proper, dense, subset of K, made up of points whose addresses are unique. Conversely, [formula] implies the SOSC, without any openness assumption.

Słowa kluczowe

EN

address; Baire category; fractal; scaling function; scaling operator; strong open set condition

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2008
• Tom: Vol. 28, no. 4
• Strony: 463--470
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Goodman G. S.

• via Dazzi, 11 50141 Firenze, Italy,

Bibliografia

• [1] S. Graf, On Bandt’s tangential distribution for self-similar measures, Monatsh. Math. 120 (1995) 3–4, 223–246.
• [2] G.S. Goodman, Invariant measures whose supports possess the strong open set property, Opusc. Math. 28 (2008) 4, 471–480.
• [3] J.E. Hutchinson, Fractals and self-similarity, Indiana J. Math. 30 (1981), 713–747.
• [4] S.P. Lalley, The packing and covering dimensions of some self-similar fractals, Indiana Univ. Math. J. 37 (1988), 699–709.
• [5] A. Lasota, J. Myjak, T. Szarek, Markov operators with a unique invariant measure, J. Math. Anal. Appl. 276 (2002), 343–356.
• [6] P.A.P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Camb. Phil. Soc. 42 (1946), 15–23.
• [7] Y. Peres, M. Rams, K. Simon, B. Solomyak, Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets, Proc. Amer. Math. Soc. 129 (2001) 9, 2689–2699.
• [8] A. Schief, Separation Properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1991) 1, 101–115.
• [9] A. Schief, Self-similar sets in complete metric spaces, Proc. Amer. Math. Soc. 124 (1996) 2, 481–490.