Consider the cellular automata (CA) of $\mathbb{Z}^{2}$-action $Φ$ on the space of all doubly infinite sequences with values in a finite set $\mathbb{Z}_{r}$, $r \geq 2$ determined by cellular automata $T_{F[-k, k]}$ with an additive automaton rule $F(x_{n-k},...,x_{n+k})=\sum\limits_{i=-k}^{k}a_{i}x_{n+i}(mod r)$. It is investigated the concept of the measure theoretic directional entropy per unit of length in the direction $ω_{0}$. It is shown that $h_μ(T_{F[-k,k]}^{u})=uh_μ(T_{F[-k,k]})$, $h_μ(Φ^{u})=uh_μ(Φ)$ and $h_{\vec{v}}(Φ^{u})=uh_{\vec{v}}(Φ)$ for $\vec{v} \in \mathbb{Z}^{2}$ where $h$ is the measure-theoretic entropy.
We introduce and study the notion of a directional complexity and entropy for maps of degree 1 on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a combinatorial machinery to obtain exact formulas for the directional entropy, to find the maximal directional entropy, and to show that it equals the topological entropy of the map. Keywords: Rotation interval, Space-time window, Directional complexity, Directional entropy;
In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb{Z}^q$-systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, direcitonal discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb{Z}^q$-system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb{Z}^q$-system has directional discrete spectrum along $q$ linearly independent directions if and only if it has discrete spectrum.
In this short paper, we compute the directional sequence entropy for of $\mathbb{Z}_+\times \mathbb{Z}$-actions generated by cellular automata and the shift map. Meanwhile, we study the directional dynamics of this system. As a corollary, we prove that there exists a sequence such that for any direction, some of the systems above have positive directional sequence entropy. Moreover, with help of mean ergodic theory for directional weak mixing systems, we obtain a result of number theory about combinatorial numbers.
We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal $\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$. We study the structural properties of systems that satisfy the so called unique closing parallelepiped property and we characterize them in several ways. In the distal case, we build the maximal factor of a $\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$ that satisfies this property by taking the quotient with respect to the directional regionally proximal relation. Finally, we completely describe distal $\mathbb{Z}^d$…
We study directional mean dimension of $\mathbb{Z}^k$-actions (where $k$ is a positive integer). On the one hand, we show that there is a $\mathbb{Z}^2$-action whose directional mean dimension (considered as a $[0,+\infty]$-valued function on the torus) is not continuous. On the other hand, we prove that if a $\mathbb{Z}^k$-action is continuum-wise expansive, then the values of its $(k-1)$-dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamoto's theorem on mean dimension and expansive multiparameter actions) of a classical re…
We define notions of direction $L$ ergodicity, weak mixing, and mixing for a measure preserving $\mathbb Z^d$ action $T$ on a Lebesgue probability space $(X,μ)$, where $L\subseteq\mathbb R^d$ is a linear subspace. For $\mathbb R^d$ actions these notions clearly correspond to the same properties for the restriction of $T$ to $L$. For $\mathbb Z^d$ actions $T$ we define them by using the restriction of the unit suspension $\widetilde T$ to the direction $L$ and to the subspace of $L^2(\widetilde X,\widetilde μ)$ perpendicular to the suspension rotation factor. We show that for $\mathbb Z^d$ acti…
Given $q\in\mathbb{N}$, let $(X,T)$ be a $\mathbb{Z}^q$-system, $\vec{v}\in\mathbb{R}^q\setminus\{\vec{0}\}$ be a direction vector and $\textbf{b}\in\mathbb{R}_+^{q-1}$. We study $(X,T)$ that has bounded complexity with respect to three kinds of metrics defined along direction $\vec{v}$: the directional Bowen metric $d_k^{\vec{v},\textbf{b}}$, the directional max-mean metric $\hat{d}_k^{\vec{v},\textbf{b}}$ and the directional mean metric $\bar{d}_k^{\vec{v},\textbf{b}}$. It is shown that $(X,T)$ has bounded topological complexity with respect to $\{d_k^{\vec{v},\textbf{b}}\}_{k=1}^{\infty}$ (…
LLet $f$ be a holomorphic endomorphism of $\mathbb P^ 2$ of degree $d \geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $ν$. We infer several applications. The first one shows that the currents $S$ containing a measure of entropy $h\_ν> \log d$ have a directional dimension $>2$, which answers a question by de Thélin-Vigny. The second application asserts that the Dujardin's semi-extremal endomorphisms are close to suspensions of one-dimensional Lattès maps. Finally, we obtain an upper bound for the dimension of the eq…
In this paper, two types of Lyapunov exponents: random Lyapunov exponents and directional Lyapunov exponents, and the corresponding entropies: random entropy and directional entropy, are considered for smooth $\mathbb{Z}^k$-actions. The close relation among these quantities are investigated and the formulas of them are given via the Lyapunov exponents of the generators.
Let $d(δ)$ denote the Hausdorff dimension of the Julia set of the polynomial $f_δ(z)=z^2-2+δ$. In this paper we will study the directional derivative of the function $d$ along directions landing at the parameter $0$, which corresponds to $-2$ in the case of family $p_c(z)=z^2+c$. We will consider all directions, except the one $δ\in\mathbb{R}^+$, which is inside the Mandelbrot set. We will prove asymptotic formula for the directional derivative of $d$. Moreover, we will see that the derivative is negative for all directions in the closed left half-plane. Computer calculations show that it is n…
Let $d(\varepsilon)$ and $\mathcal D(δ)$ denote the Hausdorff dimension of the Julia sets of the polynomials $p_\varepsilon(z)=z^2+1/4+\varepsilon$ and $f_δ(z)=(1+δ)z+z^2$ respectively. In this paper we will study the directional derivative of the functions $d(\varepsilon)$ and $\mathcal D(δ)$ along directions landing at the parameter $0$, which corresponds to $1/4$ in the case of family $z^2+c$. We will consider all directions, except the one $\varepsilon\in\mathbb{R}^+$ (or two imaginary directions in the $δ$ parametrization) which is outside the Mandelbrot set and is related to the paraboli…
In this paper we introduce the definition of entropy for a partial $\mathbb{Z}$-action. We show that the definition of partial entropy is an extension of the definition of topological entropy for a $\mathbb Z$-action. We also prove that the partial topological entropy is concentrated on the non-wandering set.
In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $\mathbb{Z}^k$-actions which are generated by random compositions of the generators of $\mathbb{Z}^k$-actions. Applying Pesin's theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of $C^{2}$ random $\mathbb{Z}^k$-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random $\mathbb{Z}^k$(or $\mathbb{Z}_+^k$ )-actions generated by more general maps, such as Lipschitz maps, continuous maps on fi…
Let $Γ$ be a sofic group with a copy of $\mathbb{Z}$ in its center. We construct an uncountable family of pairwise nonisomorphic measure-preserving $Γ$ actions with completely positive entropy, none of which is a factor of a Bernoulli shift. Our construction shows that the relation of isomorphism among completely positive entropy $Γ$ actions is not smooth, in contrast with the relation of isomorphism among Bernoulli shifts.
For a circle map $f\colon\mathbb{S}\to\mathbb{S}$ with zero topological entropy, we show that a non-diagonal pair $\langle x,y\rangle\in \mathbb{S}\times \mathbb{S}$ is non-separable if and only if it is an IN-pair if and only if it is an IT-pair. We also show that if a circle map is topological null then the maximal pattern entropy of every open cover is of polynomial order.
In this work we study the dynamical behavior Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $ E_L^{-1}(c)$ (i.e $ c> c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale propriety in $ E_L^{-1}(c)$ (i.e, all closed orbit with energy $c$ are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_L^{-1}(c)$). The proof requires the use of well-known results in Aubry-Mather's Theory.
In this paper, we first concentrate on the possible values and dense property of entropies for isotropic and anisotropic axial products of subshifts of finite type (SFTs) on $\mathbb{N}^d$ and $d$-tree $\mathcal{T}_d$. We prove that the entropies of isotropic and anisotropic axial products of SFTs on $\mathbb{N}^d$ are dense in $[0,\infty)$, and the same result also holds for anisotropic axial products of SFTs on $\mathcal{T}_d$. However, the result is no longer true for isotropic axial products of SFTs on $\mathcal{T}_d$. Next, motivated by the work of Johnson, Kass and Madden [16], and Schra…
We study the topological entropy of hom tree-shifts and show that, although the topological entropy is not a conjugacy invariant for tree-shifts in general, it remains invariant for hom tree higher block shifts. In doi:10.1016/j.tcs.2018.05.034 and doi:10.3934/dcds.2020186, Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and $h(\mathcal{T}_X) \geq h(X)$, where $\mathcal{T}_X$ is the hom tree-shift derived from $X$. We characterize a necessary and sufficient condition when the equality holds for the case where $X$ is a shift of finite type. In addition, two…
In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{Hölder}}$ diffeomorphisms of compact manifolds. We construct intermediate and coarse unstable manifolds for the action and establish controls on their local geometry. In the second part we consider the relationship between entropy, Lyapunov exponents, and the geometry of conditional measures for rank-1 systems given by a number of generalizations of the Ledrappier--You…
Given a dynamical system $(X,G)$, the centralizer $C(G)$ denotes the group of all homeomorphisms of $X$ which commute with the action of $G$. This group is sometimes called the automorphism group of the dynamical system $(X,G)$. In this note, we generalize the construction of Bulatek and Kwiatkowski (1992) to $\mathbb Z^d$-Toepltiz systems by identifying a class of $\mathbb Z^d$-Toeplitz systems that have trivial centralizers. We show that this class of $\mathbb Z^d$-Toeplitz with trivial centralizers contains systems with positive topological entropy.
In this paper we provide a concrete construction of Furstenberg entropy values of $τ$-boundaries of the group $\mathbb{Z}[\frac{1}{p_1},\ldots,\frac{1}{p_{l}}]\rtimes \{p_1^{n_1}\cdots p_{l}^{n_{l}} \, : \, n_i\in\mathbb{Z}\}$ by choosing an appropriate random walk $τ$. We show that the boundary entropy spectrum can be realized as the subsum-set for any given finite sequence of positive numbers.
The monotonicity of entropy is investigated for real quadratic rational maps on the real circle $\mathbb{R}\cup\{\infty\}$ based on the natural partition of the corresponding moduli space $\mathcal{M}_2(\mathbb{R})$ into its monotonic, covering, unimodal and bimodal regions. Utilizing the theory of polynomial-like mappings, we prove that the level sets of the real entropy function $h_\mathbb{R}$ are connected in the $(-+-)$-bimodal region and a portion of the unimodal region in $\mathcal{M}_2(\mathbb{R})$. Based on the numerical evidence, we conjecture that the monotonicity holds throughout th…
We prove that if two free p.m.p. $\mathbb{Z}$-actions are Shannon orbit equivalent then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein-Weiss and the entropy invariance results of Austin and Kerr-Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also …
We extend a result of Ledrappier, Hochman, and Solomyak on exact dimensionality of stationary measures for $\text{SL}_2(\mathbb{R})$ to disintegrations of stationary measures for $\text{GL}(\mathbb{R}^d)$ onto the one dimensional foliations of the space of flags obtained by forgetting a single subspace. The dimensions of these conditional measures are expressed in terms of the gap between consecutive Lyapunov exponents, and a certain entropy associated to the group action on the one dimensional foliation they are defined on. It is shown that the entropies thus defined are also related to simpl…
We show that the automorphism group of every zero entropy infinite shift admits a "drift" homomorphism to $(\mathbb{R},+)$ that maps the shift map to 1. This homomorphism arises as the expectation, under an invariant measure, of a cocycle defined on a space of asymptotic pairs.
We construct Z^2-SFTs at every computable level of the hierarchy of topological completely positive entropy (TCPE), answering Barbieri and García-Ramos, who asked if there was one at level 3. Furthermore, we show the property of TCPE in Z^2-SFTs is coanalytic complete. Thus there is no simpler description of TCPE in Z^2-SFTs than in the general case.
We partially generalize Peters' formula on modules over the group ring ${\mathbb F} Γ$ for a given finite field ${\mathbb F}$ and a sofic group $Γ$. It is also discussed that how the values of entropy are related to the zero divisor conjecture.
In 1974, M. Shub stated Topological Entropy Conjecture, that is, the inequality $\logρ\leq ent(f)$ is valid or not, where $f$ is a continuous self-map on a compact manifold $M$, $ent(f)$ is the topological entropy of $f$ and $ρ$ is the maximum absolute eigenvalue of $f_*$ which is the linear transformation induced by $f$ on the homology group $H_{*}(M;\mathbb{Z})=\bigoplus\limits_{i=0}^n{H_{i}(M;\mathbb{Z})}$. In 1986, A. B. Katok gave a counterexample such that the inequality $\logρ\leq ent(f)$ is invalid. In this paper, we define $f$-Čech homology group $\check{H}_{i}(X,f;\mathbb{Z})$ and to…
