We show that, for an iterated function system fsjg n j1 of similitudes that satis. The idea of self similarity is one of the most fundamental in the modern mathematics. Wang sun yatsen university 2012, dec, cuhk, hong kong x. The second is a complete understanding of the structure of complex hyperbolic reflection triangle groups in cases where the angle is small. Random walks and induced dirichlet forms on selfsimilar sets. Fractalsiterations in the complex planeparameter plane. We show that, for an iterated function system sj n j1 of similitudes that satisfies the open set condition, there is a natural graph structure in the representing symbolic space to make it a hyperbolic graph, and the hyperbolic boundary is homeomorphic to the self similar set generated by sj n. For an iterated function system ifs of simillitidues, we define two graphs on the representing symbolic space. Lipschitz equivalence of selfsimilar sets and hyperbolic boundaries. Many objects in the real world, such as coastlines, are statistically self similar.
Topology of boundaries of hyperbolic groups problem 1 misha kapovich. This is a truly excellent book for introducing advanced undergraduates to hyperbolic geometry. Carnot groups, and spaces satisfying poincare inequalities. Citeseerx selfsimilar sets as hyperbolic boundaries. We show that if the self similar set \k\ has positive lebesgue measure or the ifs satisfies the weak separation condition, then the graphs are hyperbolic. In this study we establish results for the lipschitz equivalence of self similar sets with touching structures in with arbitrarily many branches. Lipschitz equivalence is an important property in fractal geometry because it preserves many key properties of fractal sets.
The premier application of the augmented trees is to use their hyperbolic boundaries to study the self similar sets. In 9 kaimanovich introduced the concept of augmented tree on the symbolic space of a self similar set. From wikibooks, open books for an open world boundaries of hyperbolic components. Author index ems european mathematical society publishing.
Luo, lipschitz equivalence of selfsimilar sets and hyperbolic boundaries ii. Scale invariance is an exact form of selfsimilarity where at any magnification there is a smaller piece of the object that is similar to the whole. In, two of the authors gave a study of lipschitz equivalence of selfsimilar sets through the augmented trees, a class of hyperbolic graphs introduced by kaimanovich in 9 and developed by lau and wang 10. Selfsimilar sets as hyperbolic boundaries semantic scholar. Journal of fractal geometry title index ems european. This extends a consideration of kigami, where he investigated random walks on the binary trees with cantor type sets as boundaries. As an illustration, we will give a brief discussion of two such applications in section 5. Lau lipschitz equivalence of self similar sets and hyperbolic boundaries. In \citelula, two of the authors initiated a study of lipschitz equivalence of self similar sets through the augmented trees, a class of hyperbolic graphs introduced by kaimanovich \citeka03 and developed by lau and wang \citelawa09. Raincoast books all these boundaries africa, asia, malaysia, america are set by men. In order to describe the selfsimilar set k, we need more edges.
I used this text for an extracurricular undergraduate reading group. Self similarity and lipschitz equivalence are two basic and important properties of fractal sets. The following theorem is for lipschitz equivalence on the hyperbolic boundaries, it is the crucial step to establish the equivalence for the selfsimilar sets. Graphs induced by iterated function systems springerlink. Self similar sets, simple augmented trees, and their lipschitz equivalence luo, jun jason. The aim of this book is to introduce hyperbolic geometry and its ap. Selfsimilar sets, simple augmented trees, and their. In \citelula, two of the authors initiated a study of lipschitz equivalence of selfsimilar sets through the augmented trees, a class of hyperbolic. Key to our study is the introduction of a geometric condition for self similar sets called substitutable. Geometric measures one knows that self similar sets e. C and mv for more general setting of conformal cantor sets, for hyperbolic polynomials the mixing is exponentially fast. Bibtex entry for this abstract preferred format for this abstract see preferences find similar abstracts. The mandelbrot set is the set of complex numbers c \displaystyle c c for which the function f c. It is hyperbolic in the sense of gromov, and it was shown in that under the open set condition, a self similar set can be identified with the hyperbolic boundary of the tree.
Kaimanovich introduced the concept of augmented tree on the symbolic space of a self similar set. For a given pcf self similar fractal, a certain network weighted graph is constructed whose ideal boundary is homeomorphic to the fractal. Self help alliance better boundaries 2010 3 content page number. In an accessible and straightforward manner, richard evan schwartz also presents a large amount of useful information on complex hyperbolic. Topological structure of fractal squares mathematical. On the equivalence of homogeneous iterated function. The notion of renormalization group, which plays an essential role in quantum field theory, statistical physics and dynamical systems, is related to it. We then apply this to consider the lipschitz equivalence of self. Z fnu all the examples we looked at are locally maximal the structure of hyperbolic sets p. In mathematics, a self similar object is exactly or approximately similar to a part of itself i. Relationships test our boundaries, so its important to know where to draw the line. The little copies of the mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set. It follows from the definition of a self similar action that for every m g 9jt and h g g the compositions m. It is well known that the above limit exists and is independent of x0 2rd.
Lipschitz equivalence of selfsimilar sets with touching. We give a necessary and sufficient condition that the union is a self similar set. In this paper we provide an uptodate survey on the study of lipschitz equivalence of self similar sets. The bimodule associated to a self similar action is defined as the set vjl of transformations v i. Quasisymmetric structure of self similar spaces, such as boundaries of hyperbolic groups. No matter what anyone else tells you, if it feels bad, it is bad for.
We like to identify the similar sets, the hyperbolic boundaries and the martin boundaries, which allows us to. Lipschitz equivalence of selfsimilar sets and hyperbolic boundaries ii guotai deng, kasing lau,1 and jun jason luo2 abstract. More generally, w ang 14 extended these results to the weak sep aration c ondition. What spaces can arise as boundaries of hyperbolic groups. There is a discrete potential theory and harmonic function theory on the boundary doob. Selfsimilar sets as hyperbolic boundaries 1779 graph x. My main recommendation assuming you have some college level math knowledge is that if what you are interested in is specifically hyperbolic geo. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. In 9 kaimanovich introduced the concept of augmented tree on the symbolic space of a selfsimilar set. The hitting distribution of the chain is also obtained. The book is very accessible and presents a reasonable range of exercises for undergrads although not for grad students.
Throughout the paper, we assume a self similar set k is generated by an iterated function system ifs s i i 1 m on r d where 2. So far, all the known results only cover self similar sets in with no more than three branches. Hyperbolic geometry, surfaces, and 3manifolds bruno martelli dipartimento di matematica \tonelli, largo pontecorvo 5. We show that, for an iterated function system sj n j1 of similitudes that satisfies the open set condition, there is a natural graph structure in the representing symbolic space to make it a hyperbolic graph, and the hyperbolic boundary is. We also show that as a semigroup is finitely generated. Boundaries of hyperbolic groups department of mathematics. Well it depends on your level of mathematical sophistication, but there are several good books.
Geometric group theory closely interacts with lowdimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry. Selfsimilar sets as hyperbolic boundaries request pdf. We discuss work of volodymyr nekrashevych and give a reexposition of his theorem and proof that julia sets. In \citelula, two of the authors initiated a study of lipschitz equivalence of selfsimilar sets through the augmented trees, a class of hyperbolic graphs introduced by kaimanovich \citeka03 and developed by lau and wang \citelawa09. Let o1odenote the quotient space over the equivalent relation oioj. We show that, for an iterated function system sj n j1 of similitudes that satisfies the open set condition, there is a natural graph structure in the representing symbolic space to make it a hyperbolic graph, and the hyperbolic boundary is homeomorphic to the selfsimilar set generated by sj n j1. In other words, the boundary of the mandelbrot set is a fractal curve.
Even though the boundaries were defined as sets of equivalence classes, they. This connection is established through the central topics of the book, which are the notions of the iterated monodromy. Hyperbolic graphs, fractal boundaries, and graph limits. On hyperbolic graphs induced by iterated function systems. The nth tensor power of a selfsimilar action is the action on the set. Lipschitz equivalence of selfsimilar sets and hyperbolic. We characterize the family of homogeneous iterated function systems of similarities on that satisfy the strong separation condition and generate the same self similar set f. Rigidity for quasiisometries of symmetric spaces and euclidean buildings.
For an iterated function system ifs of similitudes, we define two graphs on the representing symbolic space. Pdf hyperbolic spaces from selfsimilar group actions. The hyperbolic boundary of the selfsimilarity graph of the. Selfsimilar fractals as boundaries of networks the network nf associated to a fractal f compacti. It also has nice examples and proofs that are written with. What are the best introductory books on hyperbolic geometry.
Many objects in the real world, such as coastlines, are statistically selfsimilar. In this paper, we consider those properties of the union of cantor set and its translate. Structure of the class of iterated function systems that generate the same. This will be relevant for limit sets, as a hyperbolic manifold has a deformation space consisting of new hyper. Leung connectedness of planar self affine sets associated with noncollinear digit sets, geometriae dedicata, 2015 175. This book studies the self similarity phenomenon in group theory and shows its intimate relationship with dynamical systems and more classical self similar structures, such as fractals, julia sets, and self affine tilings. Hyperbolic graph martin boundary boundary theory and selfsimilar set x. Not every hyperbolic component can be reached by a sequence of direct. Gromov hyperbolic space, julia set, self similar group. Lipschitz equivalence of selfconformal sets journal. We show that the martin boundary, the hyperbolic boundary and the selfsimilar set \k\ are homeomorphic. Chapter 1 includes all the preliminaries we will need, all the material in the subsequent chapters is self contained. For which k do kdimensional stable menger spaces appear as boundaries. Lau, lipschitz equivalence of self similar sets and hyperbolic boundaries, adv.
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