A standard way to parametrize the boundary of a connected fractal tile T is proposed. The parametrization is Hölder continuous from R/Z to ∂T and fixed points of ∂T have algebraic preimages. A class of planar tiles is studied in detail as sample cases and a relation with the recurrent set method by Dekking is discussed. When the tile T is a topological disk, this parametrization is a bi-Hölder homeomorphism.

We consider a class of planar self-affine tiles T = M −1 a∈D (T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:We give a parametrization S 1 → ∂T of the boundary of T with the following standard properties. It is Hölder continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on ∂T and have algebraic preimages. We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.

No abstract

A Peano continuum means a locally connected continuum. A compact metric space is called a Peano compactum if all its components are Peano continua and if for any constant C > 0 all but finitely many of its components are of diameter less than C. Given a compact set K ⊂ C, there usually exist several upper semi-continuous decompositions of K into subcontinua such that the quotient space, equipped with the quotient topology, is a Peano compactum. We prove that one of these decompositions is finer than all the others and call it the core decomposition of K with Peano quotient. This core decomposition gives rise to a metrizable quotient space, called the Peano model of K, which is shown to be determined by the topology of K and hence independent of the embedding of K into C. We also construct a concrete continuum K ⊂ R 3 such that the core decomposition of K with Peano quotient does not exist. For specific choices of K ⊂ C, the above mentioned core decomposition coincides with two models obtained recently, namely the locally connected model for unshielded planar continua (like connected Julia sets of polynomials) and the finitely Suslinian model for unshielded planar compact sets (like polynomial Julia sets that may not be connected). The study of such a core decomposition provides partial answers to several questions posed by Curry in 2010. These questions are motivated by other works, including those by Curry and his coauthors, that aim at understanding the dynamics of a rational map f :Ĉ →Ĉ restricted to its Julia set.In this paper, a compact metric space is called a compactum and a connected compactum is called a continuum. If K, L are two compacta, a continuous onto map π : K → L such that the preimage of every point in L is connected is called monotone [19]. We are interested in compacta in the complex plane C or in the Riemann sphereĈ. Given a compactum K ⊂ C, an upper semi-continuous decomposition D of K is a partition of K such that for every open set B ⊂ K the union of all d ∈ D with d ⊂ B is open in K (see [11]). Let π be the natural projection sending x ∈ K to the unique element of D that contains x. Then a set A ⊂ D is said to be open in D if and only if π −1 (A) is open in K. This defines the quotient topology on D. If all the elements of such a decomposition D are compact the equivalence on K corresponding to D is a closed subset of K × K and the quotient topology on D is metrizable [9, p.148, Theorem 20]. Equipped with an appropriate metric compatible with the quotient topology, the quotient space D is again a compactum.An upper semi-continuous decomposition of a compactum K ⊂ C is monotone if each of its elements is a subcontinuum of K. In this case, the mapping π described above is a monotone map. Let D and D ′ be two monotone decompositions of a compactum K ⊂ C, with projections π and π ′ , and suppose that D and D ′ both satisfy a topological property (T ). We say that D is finer than D ′ with respect to (T ) if there is a map g : D → D ′ such that π ′ = g • π. If a monotone decomposition of a com...

We consider the self-affine tiles with collinear digit set defined as follows. Let A; B 2 Z satisfy jAj B 2 and M 2 Z 22 be an integral matrix with characteristic polynomial x 2 C Ax C B. Moreover, let D D ¹0; v; 2v; : : : ; .B 1/vº for some v 2 Z 2 such that v; M v are linearly independent. We are interested in the topological properties of the self-affine tile T defined by M T D S d 2D .T C d /. Lau and Leung proved that T is homeomorphic to a closed disk if and only if 2jAj B C2. In particular, T has no cut point. We prove here that T has a cut point if and only if 2jAj B C 5. For 2jAj B 2 ¹3; 4º, the interior of T is disconnected and the closure of each connected component of the interior of T is homeomorphic to a closed disk.