Related papers: Automatic sequences are orthogonal to aperiodic mu…

We prove that the automorphism group $\mathrm{Aut}(X)$ of an affine spherical variety $X$ acts transitively on the set of smooth points $X^{reg}.$ If every invertible regular function on $X$ is constant, we prove that $X$ is flexible, i.e.,…

Let $X=(V\!X,E\!X)$ be an infinite, locally finite, connected graph without loops or multiple edges. We consider the edges to be oriented, and $E\!X$ is equipped with an involution which inverts the orientation. Each oriented edge is…

We show that every automorphism $\alpha$ of a free group $F_k$ of finite rank $k$ has {\it asymptotically periodic} dynamics on $F_k$ and its boundary $\partial F_k$: there exists a positive power $\alpha^q$ such that every element of the…

A continuum $X$ is a dendrite if it is locally connected and contains no simple closed curve, a self mapping $f$ of $X$ is called monotone if the preimage of any connected subset of $X$ is connected. If $X$ is a dendrite and $f:X\to X$ is a…

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or…

Moraga and Yeong conjectured that for a smooth complex projective variety $X$ of dimension $n$, an ample line bundle $A$ on $X$ and an integer $m \ge 3 n + 1$, very general elements of the adjoint linear system $|\omega_{X} \otimes…

In this paper we study the asymptotic behaviour of two relatively new complexity functions defined on infinite words and their relationship to periodicity. Given a factor $u$ of an infinite word $x$, we say $u$ is closed if it is a letter…

An automaton is universal if it accepts every possible input. We study the notion of u-universality, which asserts that the automaton accepts every input starting with u. Universality and u-universality are both EXPTIME-hard for…

We show that the (morphic) sequence $(-1)^{s_\varphi(n)}$ is asymptotically orthogonal to all bounded multiplicative functions, where $s_\varphi$ denotes the Zeckendorf sum-of-digits function. In particular we have $\sum_{n<N}…

Cellular automata are fully-discrete, spatially-extended dynamical systems that evolve by simultaneously applying a local update function. Despite their simplicity, the induced global dynamic produces a stunning array of richly-structured,…

Distributed automata are finite-state machines that operate on finite directed graphs. Acting as synchronous distributed algorithms, they use their input graph as a network in which identical processors communicate for a possibly infinite…

Let $X$ be an Archimedean vector lattice. We investigate subalgebras of $\mathscr{L}(X)$ consisting of regular operators that contain all rank-one operators of the form $a \otimes \varphi_b$, where $a$ and $b$ are atoms of $X$ and…

We study the synchronous and asynchronous automatic structures on the fundamental group of a graph of groups in which each edge group is finite. Up to a natural equivalence relation, the set of biautomatic structures on such a graph product…

We prove that any Hamiltonian diffeomorphism of a closed symplectic manifold equipped with an atoroidal symplectic form has simple non-contractible periodic orbits of arbitrarily large period, provided that the diffeomorphism has a…

Denote by $\lambda(n)$ Liouville's function concerning the parity of the number of prime divisors of $n$. Using a theorem of Allouche, Mend\`es France, and Peyri\`ere and many classical results from the theory of the distribution of prime…

Let $K$ be the Cantor set. We prove that arbitrarily close to a homeomorphism $T:K\rightarrow K$ there exists a homeomorphism $\widetilde T:K\rightarrow K$ such that the $\alpha$-limit and the $\omega$-limit of every orbit is a periodic…

In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof…

We show that, for each finite algebra A, either it has symmetric term operations of all arities or else some finite algebra in the variety generated by A has two automorphisms without a common fixed point. We also show this two-automorphism…

A set $X\subseteq\mathbb N$ is S-recognizable for an abstract numeration system S if the set $\rep_S(X)$ of its representations is accepted by a finite automaton. We show that the growth function of an S-recognizable set is always either…

Let $A$ be a rational function of degree $n\geq 2$. Let us denote by $ G(A)$ the group of M\"obius transformations $\sigma$ such that $ A\circ \sigma=\nu_{\sigma} \circ A$ for some M\"obius transformations $\nu_{\sigma}$, and by $\Sigma(A)$…