The commonly used spatial entropy hr(U) of the multi-dimensional shift space U is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space Zd, d ≥ 2. This work studies spatial entropy hΩ(U) of shift space U on general expanding system Ω = {Ω(n)}∞ n=1 where Ω(n) is increasing finite sublattices and expands to Zd. Ω is called genuinely d-dimensional if Ω(n) contains no lower-dimensional part whose size is comparable to that of its d-dimensional part. We show that hr(U) is the supremum of hΩ(U) for all genuinely two-dimensional Ω. Furthermore, when Ω is genuinely d-dimensional and satisfies certain conditions, then hΩ(U) = hr(U). On the contrary, when Ω(n) contains a lower-dimensional part, then hr(U) < hΩ(U) for some U. Therefore, hr(U) is appropriate to be the d-dimensional spatial entropy.

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We show that given $${n \geqslant 3}$$ , $${q \geqslant 1}$$ , and a finite set $${\{y_1, \ldots, y_q \}}$$ in $${\mathbb{R}^n}$$ there exists a quasiregular mapping $${\mathbb{R}^n\to \mathbb{R}^n}$$ omitting exactly points $${y_1, \ldots, y_q}$$ .

We study the effective fronts of first order front propagations in two dimensions (n=2) in the periodic setting. Using PDE-based approaches, we show that for every α∈(0,1), the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts for given front speeds in C^{1,α}(T^2,(0,∞)). This result can also be formulated in the language of stable norms corresponding to periodic metrics in T^2. Similar results were known long time ago when n≥3 for front speeds in C^∞(T^n,(0,∞)). Due to topological restrictions, the two dimensional case is much more subtle. In fact, the effective front is C^1, which cannot be a polygon, for given C^{1,1}(T^2,(0,∞)) front speeds. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions.

We study the periodic homogenization of first order front propagations. Based on PDE methods, we provide a simple proof that, for n ≥ 3, the class of centrally symmetric polytopes with rational coordinates and nonempty interior is admissible as effective fronts, which was also established by I. Babenko and F. Balacheff [Sur la forme de la boule unit\'e de la norme stable unidimensionnelle, Manuscripta Math. 119 (2006) 347--358] and M. Jotz [Hedlund metrics and the stable norm, Diff. Geometry Appl. 27 (2009) 543--550] in the form of stable norms as an extension of G. A. Hedlund's classical result [Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math. 33 (1932) 719--739]. Besides, we obtain the optimal convergence rate of the homogenization problem for this class.

We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every non-trivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms.

This investigation studies nonemptiness problems of plane edge coloring with three colors. In the edge coloring (or Wang tiles) of a plane, unit squares with colored edges that have one of p colors are arranged side by side such that the touching edges of the adjacent tiles have the same colors. Given a basic set B of Wang tiles, the nonemptiness problem is to determine whether or not Σ(B) 6= ∅, where Σ(B) is the set of all global patterns on Z2 that can be constructed from the Wang tiles in B. Wang’s conjecture is that for any B of Wang tiles, Σ(B) 6= ∅ if and only if P(B) 6= ∅, where P(B) is the set of all periodic patterns on Z2 that can be generated by the tiles in B. When p ≥ 5, Wang’s conjecture is known to be wrong. When p = 2, theconjecture is true. This study proves that when p = 3, the conjecture is also true. If P(B) 6= ∅, then B has a subset B′ of minimal cycle generators such that P(B′) 6= ∅ and P(B′′) = ∅ for B′′ $ B′. This study demonstrates that the set C(3) of all minimal cycle generators contains 787, 605 members that can be classified into 2, 906 equivalence classes. N(3) is the set of all maximal non-cycle generators : if B ∈ N(3), then P(B) = ∅ and P(B ˜) 6= ∅ for B ˜ % B. Wang’s conjecture is shown to be true by proving that B ∈ N(3) implies Σ(B) = ∅.

This paper studies the dynamics of a network-based SIS epidemic model with nonmonotone incidence rate. This type of nonlinear incidence can be used to describe the psychological effect of certain diseases spread in a contact network at high infective levels. We first find a threshold value for the transmission rate. This value completely determines the dynamics of the model and interestingly, the threshold is not dependent on the functional form of the nonlinear incidence rate. Furthermore, if the transmission rate is less than or equal to the threshold value, the disease will die out. Otherwise, it will be permanent. Numerical experiments are given to illustrate the theoretical results. We also consider the effect of the nonlinear incidence on the epidemic dynamics.

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 $E$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_H X$ 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 $(X,E)$. 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\"im kernel in terms of the Gromov product. Moreover, the Naïm 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.

To study arithmetic structures of natural numbers, we introduce a notion of entropy of arithmetic functions, called anqie entropy. This entropy possesses some crucial properties common to both Shannon's and Kolmogorov's entropies. We show that all arithmetic functions with zero anqie entropy form a C*-algebra. Its maximal ideal space defines our arithmetic compactification of natural numbers, which is totally disconnected but not extremely disconnected. We also compute the $K$-groups of the space of all continuous functions on the arithmetic compactification. As an application, we show that any topological dynamical system with topological entropy $\lambda$, can be approximated by symbolic dynamical systems with entropy less than or equal to $\lambda$.