This paper concerns the self-similarity of topological spaces, in the sense defined in math.DS/0411344. I show how to recognize self-similar spaces, or more precisely, universal solutions of self-similarity systems. Examples include the standard simplices (self-similar by barycentric subdivision) and solutions of iterated function systems. Perhaps surprisingly, every compact metrizable space is self-similar in at least one way. From this follow the classical results on the role of the Cantor set among compact metrizable spaces.

2004-11-15math.CTmath.DSmath.GN

A general theory of self-similarity I

Consider a self-similar space X. A typical situation is that X looks like several copies of itself glued to several copies of another space Y, and Y looks like several copies of itself glued to several copies of X, or the same kind of thing with more than two spaces. Thus, the self-similarity of X is described by a system of simultaneous equations. Here I formalize this idea and the notion of a `universal solution' of such a system. I determine exactly when a system has a universal solution and, when one does exist, construct it. A sequel (math.DS/0411345) contains further results and examples…

2004-11-15math.CTmath.DSmath.GN

The Similarity Boundary of a Self-Similar Set

We define the similarity boundary of a self-similar set and use it to analyze the properties of self-similar sets in the general setting of any complete metric space. The similarity boundary is an attempt at extending the concept of the topological boundary in a way that is consistent with our intuitive understanding of the term boundary. We also show how with the analysis of the similarity boundary, we can restrict ourselves to the space $K$ and disregard the ambient space.

1999-07-16math.DSmath.GN

气泡的自相似性

Self-similarity of bubbles

本文研究与圆周微分同胚相关的分形集气泡（Arnold舌的复类似物），证明了其近似自相似性。

2019-01-08math.DS

On the self-similarity problem for smooth flows on orientable surfaces

On each compact, connected, orientable surface of genus greater than one we construct a class of flows without self-similarities.

2011-06-02math.DS

Graphs, groups and self-similarity

We study a family of C*-algebras generalizing both Katsura algebras and certain algebras introduced by Nekrashevych in terms of self-similar groups.

2013-07-03math.DSmath.GRmath.OA

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On the self-similarity problem for Gaussian-Kronecker flows

It is shown that a countable symmetric multiplicative subgroup $G=-H\cup H$ with $H\subset\mathbb{R}_+^\ast$ is the group of self-similarities of a Gaussian-Kronecker flow if and only if $H$ is additively $\mathbb{Q}$-independent. In particular, a real number $s\neq\pm1$ is a scale of self-similarity of a Gaussian-Kronecker flow if and only if $s$ is transcendental. We also show that each countable symmetric subgroup of $\mathbb{R}^\ast$ can be realized as the group of self-similarities of a simple spectrum Gaussian flow having the Foias-Stratila property.

2012-05-21math.DS

Self-similarity of Jankins-Neumann ziggurat

Primarily having emerged from a topological question, Jankins-Neumann ziggurat also appears in the theory of dynamical systems on the circle. It describes an answer to the following question: given the rotation numbers of two orientation-preserving circle homeomorphisms, what can be said about the rotation number of their composition? In this paper, we consider a formula, proved by Calegari and Walker, as definition of ziggurat. Using it, we establish its self-similarity. Also, we propose a short proof of equivalence between Calegary-Walker and Jankins-Neumann's descriptions of the ziggurat.

2015-03-10math.DS

On the self similarity of generalized Cantor sets

We consider the self-similar structure of the class of generalized Cantor sets $$Γ_{\mathcal{D}}=\Big\{\sum_{n=1}^\infty d_nβ^{n}: d_n\in D_n, n\ge 1\Big\},$$ where $0<β<1$ and $D_n, n\ge 1,$ are nonempty and finite subsets of $\mathbb{Z}$. We give a necessary and sufficient condition for $Γ_{\mathcal{D}}$ to be a homogeneously generated self similar set. An application to the self-similarity of intersections of generalized Cantor sets will be given.

2013-05-23math.CAmath.DS

Self-similarity and limit spaces of substitution tiling semigroups

We show that Kellendonk's tiling semigroup of an FLC substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk and Nekrashevych. We extend the notion of the limit space of a self-similar group to the setting of self-similar semigroups, and show that it is homeomorphic to the Anderson--Putnam complex for such substitution tilings, with natural self-map induced by the substitution. Thus, the inverse limit of the limit space, given by the limit solenoid of the self-similar semigroup, is homeomorphic to the translational hull of the tiling.

2021-12-14math.DS

Continuous self-similarity in parametric piecewise isometries

We exhibit two distinct renormalization scenarios in many-parameter families of piecewise isometries (PWI) of a rhombus. The rotational component, defined over the quadratic field $\mathbb{K}=\mathbb{Q}(\sqrt{5})$, is fixed. The translations are specified by affine functions of the parameters, with coefficients in $\mathbb{K}$. In each case the parameters range over a convex domain. In one scenario the PWI is self-similar if and only if one parameter belongs to $\mathbb{K}$, while the other is free. Such a continuous self-similarity is due to the possibility of merging adjacent atoms of an ind…

2015-08-24math.DS

Self-similarity for the tricorn

We discuss self-similar property of the tricorn, the connectedness locus of the anti-holomorphic quadratic family. As a direct consequence of the study on straightening maps by Kiwi and the author, we show that there are many homeomorphic copies of the Mandelbrot sets. With help of rigorous numerical computation, we also prove that the straightening map is not continuous for the candidate of a "baby tricorn" centered at the airplane, hence not a homeomorphism.

2016-02-19math.DS

Numerical evidences of universality and self-similarity in the Forced Logistic Map

We explore different families of quasi-periodically Forced Logistic Maps for the existence of universality and self-similarity properties. In the bifurcation diagram of the Logistic Map it is well known that there exist parameter values $s_n$ where the $2^n$-periodic orbit is superattracting. Moreover these parameter values lay between one period doubling and the next. Under quasi-periodic forcing, the superattracting periodic orbits give birth to two reducibility-loss bifurcations in the two dimensional parameter space of the Forced Logistic Map, both around the points $s_n$. In the present w…

2011-12-18math.DS

Self similarity and attraction in stochastic nonlinear reaction-diffusion systems

Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of stochastic PDEs in a form to which stochastic centre manifold theory may be applied we resolve these issues in this class. For definiteness, a first example of self-similarity of the Burgers' equation driven by some stochastic forced is studied. Under…

2011-11-06math.APmath.DS

Equation-Free Dynamic Renormalization: Self-Similarity in Multidimensional Particle System Dynamics

We present an equation-free dynamic renormalization approach to the computational study of coarse-grained, self-similar dynamic behavior in multidimensional particle systems. The approach is aimed at problems for which evolution equations for coarse-scale observables (e.g. particle density) are not explicitly available. Our illustrative example involves Brownian particles in a 2D Couette flow; marginal and conditional Inverse Cumulative Distribution Functions (ICDFs) constitute the macroscopic observables of the evolving particle distributions.

2005-05-17math.DS

V-Variable Fractals: Fractals with Partial Self Similarity

We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V-variable fractal sets or measures. These V-variable fractals can be obtained from the points on the attractor of a single deterministic iterated function system. Existence, uniqueness and approximation results are established under average contractive assumptions. We also obtain extensions of some basic results concerning iterated function syst…

2008-02-01math.DSmath.PR