# Dynamical boundary of a self-similar set

Fundamenta Mathematicae (1999)

- Volume: 160, Issue: 1, page 1-14
- ISSN: 0016-2736

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topMorán, Manuel. "Dynamical boundary of a self-similar set." Fundamenta Mathematicae 160.1 (1999): 1-14. <http://eudml.org/doc/212378>.

@article{Morán1999,

abstract = {Given a self-similar set E generated by a finite system Ψ of contracting similitudes of a complete metric space X we analyze a separation condition for Ψ, which is obtained if, in the open set condition, the open subset of X is replaced with an open set in the topology of E as a metric subspace of X. We prove that such a condition, which we call the restricted open set condition, is equivalent to the strong open set condition. Using the dynamical properties of the forward shift, we find a canonical construction for the largest open set V satisfying the restricted open set condition. We show that the boundary of V in E, which we call the dynamical boundary of E, is made up of exceptional points from a topological and measure-theoretic point of view, and it exhibits some other boundary-like properties. Using properties of subself-similar sets, we find a method which allows us to obtain the Hausdorff and packing dimensions of the dynamical boundary and the overlapping set in the case when X is the n-dimensional Euclidean space and Ψ satisfies the open set condition. We show that, in this case, the dimension of these sets is strictly less than the dimension of the set E.},

author = {Morán, Manuel},

journal = {Fundamenta Mathematicae},

keywords = {self-similar sets; Hausdorff dimension; open set condition; packing dimension; separation condition; restricted open set condition; strong open set condition; dynamical boundary},

language = {eng},

number = {1},

pages = {1-14},

title = {Dynamical boundary of a self-similar set},

url = {http://eudml.org/doc/212378},

volume = {160},

year = {1999},

}

TY - JOUR

AU - Morán, Manuel

TI - Dynamical boundary of a self-similar set

JO - Fundamenta Mathematicae

PY - 1999

VL - 160

IS - 1

SP - 1

EP - 14

AB - Given a self-similar set E generated by a finite system Ψ of contracting similitudes of a complete metric space X we analyze a separation condition for Ψ, which is obtained if, in the open set condition, the open subset of X is replaced with an open set in the topology of E as a metric subspace of X. We prove that such a condition, which we call the restricted open set condition, is equivalent to the strong open set condition. Using the dynamical properties of the forward shift, we find a canonical construction for the largest open set V satisfying the restricted open set condition. We show that the boundary of V in E, which we call the dynamical boundary of E, is made up of exceptional points from a topological and measure-theoretic point of view, and it exhibits some other boundary-like properties. Using properties of subself-similar sets, we find a method which allows us to obtain the Hausdorff and packing dimensions of the dynamical boundary and the overlapping set in the case when X is the n-dimensional Euclidean space and Ψ satisfies the open set condition. We show that, in this case, the dimension of these sets is strictly less than the dimension of the set E.

LA - eng

KW - self-similar sets; Hausdorff dimension; open set condition; packing dimension; separation condition; restricted open set condition; strong open set condition; dynamical boundary

UR - http://eudml.org/doc/212378

ER -

## References

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