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For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is H\"older equivalent to the attractor $K$. This setup was taken up to study the probabilistic potential theory on $K$, and the bi-Lipschitz equivalence on $K$. In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics, bounded degree property, metric doubling property, and H\"older equivalence). We also study a new setup of "weighted" IFS and investigate its connection with the self-similar energy form in the analysis of fractals., Comment: 32 pages, 9 figures

We extend our study of random walks and induced Dirichlet forms on self-similar sets [arXiv:1604.05440, 1612.01708] to compact spaces of homogeneous type $(K, \rho ,\mu)$. A successive partition on $K$ brings a natural augmented tree structure $(X, E)$ that is Gromov hyperbolic, and the hyperbolic boundary is H\"older equivalent to $K$. We then introduce a class of transient reversible random walks on $(X, E)$ with return ratio $\lambda$. Using Silverstein's theory of Markov chains, we prove that the random walk induces an energy form on $K$ with $$ {\mathcal E}_K [u] \asymp \iint_{K\times K \setminus \Delta} \frac{|u(\xi) - u(\eta)|^2}{V(\xi, \eta)\rho (\xi, \eta)^\beta} d\mu(\xi) d\mu(\eta), $$ where $V(\xi, \eta)$ is the $\mu$-volume of the ball centered at $\xi$ with radius $\rho (\xi, \eta)$, $\Delta$ is the diagonal, and $\beta$ depends on $\lambda$. In particular, for an $\alpha$-set in ${\mathbb R}^d$, the kernel of the energy form is of order $\frac{1}{|\xi-\eta|^{\alpha +\beta}}$. We also discuss conditions for this energy form to be a non-local regular Dirichlet form., Comment: 21 pages, no figures

Let $\Gamma_n$ denote the $n$-th level Sierpi\'nski graph of the Sierpi\'nski gasket $K$. We consider, for any given conductance $(a_0, b_0, c_0)$ on $\Gamma_0$, the Dirchlet form ${\mathcal E}$ on $K$ obtained from a recursive construction of compatible sequence of conductances $(a_n, b_n, c_n)$ on $\Gamma_n, n\geq 0$. We prove that there is a dichotomy situation: either $a_0= b_0 =c_0$ and ${\mathcal E}$ is the standard Dirichlet form, or $a_0 >b_0 =c_0$ (or the two symmetric alternatives), and ${\mathcal E}$ is a non-self-similar Dirichlet form independent of $a_0, b_0$. The second situation has also been studied in [Hattori et al 1994][Hambley et al 2002] as a one-dimensional asymptotic diffusion process on the Sierpi\'nski gasket. For the spectral property, we give a sharp estimate of the eigenvalue distribution of the associated Laplacian, which improves a similar result in [Hambley et al 2002]., Comment: 20 pages, 7 figures

Let $B^{\sigma}_{2, \infty}$ denote the Besov space defined on a compact set $K \subset {\Bbb R}^d$ which is equipped with an $\alpha$-regular measure $\mu$. The {\it critical exponent} $\sigma^*$ is the supremum of the $\sigma$ such that $B^{\sigma}_{2, \infty} \cap C(K)$ is dense in $C(K)$. It is well-known that for many standard self-similar sets $K$, $B^{\sigma^*}_{2, \infty}$ are the domain of some local regular Dirichlet forms. In this paper, we explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We will restrict our consideration on the p.c.f. sets. We first develop a technique of quotient networks to study the general theory of these critical exponents. We then construct two asymmetric p.c.f. sets, and use them to illustrate the theory and examine the function properties of the associated Besov spaces at the critical exponents; the various Dirichlet forms on these fractals will also be studied., Comment: 34 pages,15 figures

In a previous paper [arXiv:1604.05440], we studied certain random walks on the hyperbolic graphs $X$ associated with the self-similar sets $K$, and showed that the discrete energy ${\mathcal E}_X$ on $X$ has an induced energy form ${\mathcal E}_K$ on $K$ that is a Gagliardo-type integral. The domain of ${\mathcal E}_K$ is a Besov space $\Lambda^{\alpha, \beta/2}_{2,2}$ where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ is a parameter determined by the "return ratio" of the random walk. In this paper, we study the functional relationship of ${\mathcal E}_X$ and ${\mathcal E}_K$. In particular, we investigate the critical exponents of the $\beta$ in the domain $\Lambda^{\alpha, \beta/2}_{2,2}$ in order for ${\mathcal E}_K$ to be a regular Dirichlet form. We provide some criteria to determine the critical exponents through the effective resistance of the random walk on $X$, and make use of certain electrical network techniques to calculate the exponents for some concrete examples., Comment: 40 pages, 10 figures

For any contractive iterated function system (IFS, including the Moran systems), we show that there is a natural hyperbolic graph on the symbolic space, which yields the H\"{o}lder equivalence of the hyperbolic boundary and the invariant set of the IFS. This completes the previous studies (\cite {[Ka]}, \cite{[LW1]}, \cite{[W]}) by eliminating superfluous conditions, and admits more classes of sets (e.g., the Moran sets). We also show that the bounded degree property of the graph can be used to characterize certain separation properties of the IFS (open set condition, weak separation condition); the bounded degree property is particularly important when we consider random walks on such graphs. This application and the other application to Lipschitz equivalence of self-similar sets will be discussed., Comment: 22page,7figures

Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure ${\mathfrak E}$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_HX$ with the Gromov metric is H\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0< \lambda <1$ on $(X, {\mathfrak E})$. We show that the Martin boundary ${\mathcal M}$ can be identified with $\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain precise estimates of the Martin kernel and the Na\"{i}m kernel in terms of the Gromov product. Moreover, the Na\"{i}m kernel turns out to be a jump kernel satisfying the estimate $\Theta (\xi, \eta) \asymp |\xi-\eta|^{-(\alpha+ \beta)}$, where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ depends on $\lambda$. For suitable $\beta$, the kernel defines a regular non-local Dirichlet form on $K$. This extends the results of Kigami concerning random walks on certain trees with Cantor-type sets as boundaries., Comment: 33 pages with 2 figures

In \cite{LuLa13}, 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 \cite{Ka03} and developed by Lau and Wang \cite{LaWa09}. In this paper, we continue such investigation. We remove a major assumption in the main theorem in \cite{LuLa13} by using a new notion of quasi-rearrangeable matrix, and show that the hyperbolic boundary of any simple augmented tree is Lipschitz equivalent to a Cantor-type set. We then apply this result to consider the Lipschitz equivalence of certain totally disconnected self-similar sets as well as their unions., Comment: 22 pages, 6 figures, accepted by Journal of Fractal Geometry

For $0<\rho<1$ and $N>1$ an integer, let $\mu$ be the self-similar measure defined by $\mu(\cdot)=\sum_{i=0}^{N-1}\frac 1N\mu(\rho^{-1}(\cdot)-i)$. We prove that $L^2(\mu)$ has an exponential orthonormal basis if and only if $\rho=\frac 1q$ for some $q>0$ and $N$ divides $q$. The special case is the Cantor measure with $\rho =\frac 1{2k}$ and $N=2$ \cite {JP}, which was proved recently to be the only spectral measure among the Bernoulli convolutions with $0<\rho<1$ \cite {D}.

We call a set $K \subset {\mathbb R}^s$ with positive Lebesgue measure a {\it spectral set} if $L^2(K)$ admits an exponential orthonormal basis. It was conjectured that $K$ is a spectral set if and only if $K$ is a tile (Fuglede's conjecture). Despite the conjecture was proved to be false on ${\mathbb R}^s$, $s\geq 3$ ([T], [KM2]), it still poses challenging questions with additional assumptions. In this paper, our additional assumption is self-similarity. We study the spectral properties for the class of self-similar tiles $K$ in ${\mathbb R}$ that has a product structure on the associated digit sets. We show that any strict product-form tiles and the associated modulo product-form tiles are spectral sets. As for the converse question, we give a pilot study for the self-similar set $K$ generated by arbitrary digit sets with four elements. We investigate the zeros of its Fourier transform due to the orthogonality, and verify Fuglede's conjecture for this special case., Comment: 22pages

Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In our previous paper, we classified such tile digit sets ${\mathcal D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into product of cyclotomic polynomials. In this paper, we first show that a tile digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal D}\oplus{\mathcal L} = {\Bbb Z}^s$ for some discrete set ${\mathcal L}$). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on ${\Bbb R}^1$ together with our previous results to characterize explicitly all tile digit sets ${\mathcal D}\subset {\Bbb Z}$ with $A = p^{\alpha}q$ ($p, q$ distinct primes) as {\it modulo product-form} of some order, an advance of the previously known results for $A = p^\alpha$ and $pq$.

Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq {0,1,...,n-1}^2$, there is a self-similar set $F \subset {\Bbb R}^2$ satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a periodic extension $H= F+ {\mathbb Z}^2$, we classify $F$ into three types according to their topological properties. We also provide some simple criteria for such classification., Comment: 17 pages, 12 figures

In [9] Kaimanovich introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown in [13] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently., Comment: Advances in Mathematics, accepted (2012). 29 pages, 10 figures

We study the structure of the digit sets ${\mathcal D}$ for the integral self-similar tiles $T(b,{\mathcal{D}})$ (we call such ${\mathcal D}$ a {\it tile digit set} with respect to $b$). So far the only available classes of such tile digit sets are the complete residue sets and the product-forms. Our investigation here is based on the spectrum of the mask polynomial $P_{\mathcal D}$, i.e., the zeros of $P_{\mathcal D}$ on the unit circle. By using the Fourier criteria of self-similar tiles of Kenyon and Protasov, as well as the algebraic techniques of cyclotomic polynomial, we characterize the tile digit sets through some product of cyclotomic polynomials (kernel polynomials), which is a generalization of the product-form to higher order.

Let $\mu$ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in $L^2(\mu)$. We show that if $L^2(\mu)$ admits an exponential frame, then $\mu$ must be of pure type. We also classify various $\mu$ that admits either kind of exponential bases, in particular, the discrete measures and their connection with integer tiles. By using this and convolution, we construct a class of singularly continuous measures that has an exponential Riesz basis but no exponential orthonormal basis. It is the first of such kind of examples.

Let $(\Sigma_A, \sigma)$ be a subshift of finite type and let $M(x)$ be a continuous function on $\Sigma_A$ taking values in the set of non-negative matrices. We extend the classical scalar pressure function to this new setting and prove the existence of the Gibbs measure and the differentiability of the pressure function. We are especially interested on the case where $M(x)$ takes finite values $M_1, ..., M_m$. The pressure function reduces to $P(q):=\lim_{n\to \infty}\frac{1}{n} \log \sum_{J \in \sum_{A, n}} \|M_J\|^q$. The expression is important when we consider the multifractal formalism for certain iterated function systems with overlaps.

The coding and noncoding length sequences constructed from a complete genome are characterised by multifractal analysis. The dimension spectrum $D_{q}$ and its derivative, the 'analogous' specific heat $C_{q}$, are calculated for the coding and noncoding length sequences of bacteria, where $q$ is the moment order of the partition sum of the sequences. From the shape of the $% D_{q}$ and $C_{q}$ curves, it is seen that there exists a clear difference between the coding/noncoding length sequences of all organisms considered and a completely random sequence. The complexity of noncoding length sequences is higher than that of coding length sequences for bacteria. Almost all $D_{q}$ curves for coding length sequences are flat, so their multifractality is small whereas almost all $D_{q}$ curves for noncoding length sequences are multifractal-like. We propose to characterise the bacteria according to the types of the $C_{q}$ curves of their noncoding length sequences., Comment: 15 pages with 5 figures, Latex, Accepted for publication in Physica A

This paper introduces the notion of measure representation of DNA sequences. Spectral analysis and multifractal analysis are then performed on the measure representations of a large number of complete genomes. The main aim of this paper is to discuss the multifractal property of the measure representation and the classification of bacteria. From the measure representations and the values of the $D_{q}$ spectra and related $C_{q}$ curves, it is concluded that these complete genomes are not random sequences. In fact, spectral analyses performed indicate that these measure representations considered as time series, exhibit strong long-range correlation. For substrings with length K=8, the $D_{q}$ spectra of all organisms studied are multifractal-like and sufficiently smooth for the $C_{q}$ curves to be meaningful. The $C_{q}$ curves of all bacteria resemble a classical phase transition at a critical point. But the 'analogous' phase transitions of chromosomes of non-bacteria organisms are different. Apart from Chromosome 1 of {\it C. elegans}, they exhibit the shape of double-peaked specific heat function., Comment: 12 pages with 9 figures and 1 table

This paper develops a theory for characterisation of DNA sequences based on their measure representation. The measures are shown to be random cascades generated by an infinitely divisible distribution. This probability distribution is uniquely determined by the exponent function in the multifractal theory of random cascades. Curve fitting to a large number of complete genomes of bacteria indicates that the Gamma density function provides an excellent fit to the exponent function, and hence to the probability distribution of the complete genomes., Comment: 16 pages with 4 figures and 1 table, To appear in J. Phys. A: Math. Gene