Related papers: Some dynamical properties related to polynomials

This note considers the maximal positively invariant set for polynomial discrete time dynamics subject to constraints specified by a basic semialgebraic set. The note utilizes a relatively direct, but apparently overlooked, fact stating…

A time-dependent monic polynomial in the z variable with N distinct roots such that exactly one root has multiplicity m>=2 is considered. For k=1,2, the k-th derivatives of the N roots are expressed in terms of the derivatives of order j<=…

Let (X,Z) be a minimal dynamical system on a compact metric X and k an integer such that mdim X< k. We show that (X,Z) admits an equivariant embedding in the shift (D^k)^Z where D is a superdendrite.

We show that an $R^d$-topological dynamical system equipped with an invariant ergodic measure has discrete spectrum if and only it is $\mu$-mean equicontinuous (proven for $Z^d$ before). In order to do this we introduce mean equicontinuity…

Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…

The notion of $\Delta$-weakly mixing set is introduced, which shares similar properties of weakly mixing sets. It is shown that if a dynamical system has positive topological entropy, then the collection of $\Delta$-weakly mixing sets is…

In this work, on the one hand, we survey and amplify old results concerning tame dynamical systems and, on the other, prove some new results and exhibit new examples of such systems. In particular, we study tame symbolic systems and…

In this paper we discuss a couple of observations related to polynomial convexity. More precisely, (i) We observe that the union of finitely many disjoint closed balls with centres in $\cup_{\theta\in[0,\pi/2]}e^{i\theta}V$ is polynomially…

For every bivariate polynomial $p(z_1, z_2)$ of bidegree $(n_1, n_2)$, with $p(0,0)=1$, which has no zeros in the open unit bidisk, we construct a determinantal representation of the form $$p(z_1,z_2)=\det (I - K Z),$$ where $Z$ is an…

Let $\cD$ be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function $f\in\cD$ to be {\em cyclic}, i.e. for $\{pf:…

Consider a class of probability distributions which is dense in the space of all probability distributions on $\mathbb{R}^{d}$ with respect to weak convergence, for every $d\in\mathbb{N}$. Then, we construct various explicit classes of…

We tersely review a recently introduced technique to identify systems of two nonlinearly-coupled Ordinary Di{\S}erential Equations (ODEs) solvable by algebraic operations; and we report some specifc examples of this kind, namely systems of…

We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice - the d-fold direct product of the primes - of positive…

Let $P_1,\dots, P_n$ and $Q_1,\dots, Q_n$ be convex polytopes in $\mathbb{R}^n$ such that $P_i\subset Q_i$. It is well-known that the mixed volume has the monotonicity property: $V(P_1,\dots,P_n)\leq V(Q_1,\dots,Q_n)$. We give two criteria…

We show that there exists endotactic and strongly endotactic dynamical systems that are not weakly reversible and possess infinitely many steady states. We provide a few examples in two dimensions and an example in three dimensions that…

Let $p_1,...,p_L\in Z[x_1,...,x_d]$ be non-constant polynomials with zero constant term. The ergodic theoretical proofs of the polynomial and the IP-polynomial Szemeredi theorems as well as some of the ergodic-theoretical and combinatorial…

We show that functions of type $X_n = P[Z^n]$, where $P[t]$ is a periodic function and $Z$ is a generic real number, can produce sequences such that any string of values $X_{s}, X_{s+1}, ...,X_{s+m}$ is deterministically independent of past…

S\'ark\"ozy proved that dense sets of integers contain two elements differing by a $k$th power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of S\'ark\"ozy's theorem…

We identify pairs of positive integers $(t, d)$ with the property that the integer sequence with general term $\lfloor{n^t/d\rfloor}$ contains at most finitely many primes.

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal…