Entropy of categorical dynamics is defined by Dimitrov–Haiden–Katzarkov–Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov–Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.

We show that if a finitely generated group $G$ has a nonelementary WPD action on a hyperbolic metric space $X$ , then the number of $G$ -conjugacy classes of $X$ -loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$ . As an application we prove that for $N\geq 3$ the number of distinct $\text{Out}(F_{N})$ -conjugacy classes of fully irreducible elements $\unicode[STIX]{x1D719}$ from an $R$ -ball in the Cayley graph of $\text{Out}(F_{N})$ with $\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$ of the order of $R$ grows exponentially in $R$ .

Many fundamental properties of graph C*-algebras may be determined directly from the structure of the underlying graph, and because of this, they have been celebrated as C*-algebras that can be seen. This paper shows how permutative endomorphisms of graph C*-algebras can be represented by labelled directed multigraphs that give visual representations of the endomorphisms and facilitate computations. This formalism provides a useful calculus for permutative automorphisms and allows efficient exhaustive construction of such automorphisms.

In a general setting of an ergodic dynamical system, we give a more accurate calculation of the speed of the recurrence of a point to itself (or to a fixed point). Precisely, we show that for a certain $\unicode[STIX]{x1D709}$ depending on the dimension of the space, $\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,x)=0$ almost everywhere and $\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,y)=0$ for almost all $x$ and $y$ . This is done by assuming the exponential decay of correlations and making a weak assumption on the invariant measure.

We examine dynamical systems which are ‘nonchaotic’ on a big (in the sense of Lebesgue measure) set in each neighbourhood of a fixed point $x_{0}$ , that is, the entropy of this system is zero on a set for which $x_{0}$ is a density point. Considerations connected with this family of functions are linked with functions attracting positive entropy at $x_{0}$ , that is, each mapping sufficiently close to the function has positive entropy on each neighbourhood of $x_{0}$ .

We consider stable and almost stable points of autonomous and nonautonomous discrete dynamical systems defined on the closed unit interval. Our considerations are associated with chaos theory by adding an additional assumption that an entropy of a function at a given point is infinite.

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

Ott, Tomforde and Willis proposed a useful compactification for one-sided shifts over infinite alphabets. Building from their idea, we develop a notion of two-sided shift spaces over infinite alphabets, with an eye towards generalizing a result of Kitchens. As with the one-sided shifts over infinite alphabets, our shift spaces are compact Hausdorff spaces but, in contrast to the one-sided setting, our shift map is continuous everywhere. We show that many of the classical results from symbolic dynamics are still true for our two-sided shift spaces. In particular, while for one-sided shifts the problem about whether or not any $M$ -step shift is conjugate to an edge shift space is open, for two-sided shifts we can give a positive answer for this question.

We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period $p,p\geq 3$ , for a member of the family.

A classical article by Misiurewicz and Ziemian (J. Lond. Math. Soc.40(2) (1989), 490–506) classifies the elements in Homeo$_{0}(\mathbf{T}^{2})$ by their rotation set $\unicode[STIX]{x1D70C}$, according to wether $\unicode[STIX]{x1D70C}$ is a point, a segment or a set with nonempty interior. A recent classification of nonwandering elements in Homeo$_{0}(\mathbf{T}^{2})$ by Koropecki and Tal was given in (Invent. Math.196 (2014), 339–381), according to the intrinsic underlying ambient space where the dynamics takes place: planar, annular and strictly toral maps. We study the link between these two classifications, showing that, even abroad the nonwandering setting, annular maps are characterized by rotation sets which are rational segments. Also, we obtain information on the sublinear diffusion of orbits in the—not very well understood—case that $\unicode[STIX]{x1D70C}$ has nonempty interior.

This paper is devoted to probabilistic cellular automata (PCAs) on N,Z or Z / nZ, depending on two neighbors with a general alphabet E (finite or infinite, discrete or not). We study the following question: under which conditions does a PCA possess a Markov chain as an invariant distribution? Previous results in the literature give some conditions on the transition matrix (for positive rate PCAs) when the alphabet E is finite. Here we obtain conditions on the transition kernel of a PCA with a general alphabet E. In particular, we show that the existence of an invariant Markov chain is equivalent to the existence of a solution to a cubic integral equation. One of the difficulties in passing from a finite alphabet to a general alphabet comes from the problem of measurability, and a large part of this work is devoted to clarifying these issues.

We prove that if $G_{P}$ is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group $P$ of pattern size $d$, $d\geq 2$, and if $G_{P}$ has maximal Hausdorff dimension (equal to $1-1/2^{d-1}$), then $G_{P}$ is not topologically finitely generated. We describe precisely all essential pattern groups $P$ that yield finitely constrained groups with maximal Hausdorff dimension. For a given size $d$, $d\geq 2$, there are exactly $2^{d-1}$ such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree of depth $d$.

For a finite alphabet ${\mathcal{A}}$ and shift $X\subseteq {\mathcal{A}}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group $\text{Aut}(X)$. For such systems, we show that every finitely generated subgroup of $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then $\text{Aut}(X)$ is virtually $\mathbb{Z}$; if $X$ has dense aperiodic points, then $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$. We also classify all finite groups that arise as the automorphism group of a shift.

We provide a computation of the Čech cohomology of the pinwheel tiling using the Anderson–Putnam complex. A border-forcing version of the pinwheel tiling is produced that allows an explicit construction of the complex for the quotient of the continuous hull by the circle. The cohomology of the continuous hull is given using a spectral sequence argument of Barge, Diamond, Hunton and Sadun.

We introduce a family of infinite nonamenable discrete groups as an interpolation of the Higman–Thompson groups by using the topological full groups of the groupoids defined by $\beta $-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras, and realized as groups of piecewise-linear functions on the unit interval in the real line if the $\beta $-expansion of $1$ is finite or ultimately periodic. We also classify them by a number-theoretical property of $\beta $.

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

The class of $\lambda $-synchronizing subshifts generalizes the class of irreducible sofic shifts. A $\lambda $-synchronizing subshift can be presented by a certain $\lambda $-graph system, called the $\lambda $-synchronizing $\lambda $-graph system. The $\lambda $-synchronizing $\lambda $-graph system of a $\lambda $-synchronizing subshift can be regarded as an analogue of the Fischer cover of an irreducible sofic shift. We will study algebraic structure of the ${C}^{\ast } $-algebra associated with a $\lambda $-synchronizing $\lambda $-graph system and prove that the stable isomorphism class of the ${C}^{\ast } $-algebra with its Cartan subalgebra is invariant under flow equivalence of $\lambda $-synchronizing subshifts.

We study recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. In particular the transience–recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of noncompact type are transient. We show that we also have ‘harmonic transience’, that is, local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli’s Lévy–Khinchine formula.

We consider a hybrid model, created by coupling a continuum and an agent-based model of infectious disease. The framework of the hybrid model provides a mechanism to study the spread of infection at both the individual and population levels. This approach captures the stochastic spatial heterogeneity at the individual level, which is directly related to deterministic population level properties. This facilitates the study of spatial aspects of the epidemic process. A spatial analysis, involving counting the number of infectious agents in equally sized bins, reveals when the spatial domain is nonhomogeneous.

There is an unfortunate error in Theorem 4.1 of our paper. However, the statement of the theorem remains true with a correct construction of adding a tail to enlarge the dynamical system.