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Poincar\'e recurrence theorem implies the density of recurrent points for volume-preserving dynamical systems on compact domains. The density of closed orbits in the non-wandering set is one of the essential properties of Axiom A and chaos.…

We characterize exponential systems on sets of finite measure that form a frame or a Riesz sequence at the critical density. Namely, they are precisely those systems for which the underlying point set admits a weak limit that yields a Riesz…

We give a unified proof of the existence of turbulence for some classes of continuous interval maps which include, among other things, maps with periodic points of odd periods > 1, some maps with dense chain recurrent points and densely…

We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated…

We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function \psi, there exists a unique…

We prove that if $f$ and $g$ are postcritically finite rational maps whose Julia sets $\mathcal{J}(f), \mathcal{J}(g)$, respectively, are Sierpi\'nski carpets, and if $\xi$ is a quasiregular map of the Riemann sphere $\widehat{\mathbb{C}}$…

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue $L^2$-space to the existence of analytic bounded point evaluations. Analogues to the complex plane…

In this paper, first we introduce a new mapping for finding a common fixed point of an infinite family of nonexpansive mappings then we consider iterative method for finding a common element of the set of fixed points of an infinite family…

In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their…

In the present article we study the periodic structure of some well-known classes of $C^1$ self-maps on the product of spheres of different dimensions: transversal maps, Morse-Smale diffeomorphisms and maps with all its periodic points…

We study the dynamics of the exponential maps $E_{\lambda}: \mathbb{C} \longrightarrow \mathbb{C}$ defined by $E_{\lambda}(z) = \lambda e^z$, where $\lambda > \frac{1}{e}$. We prove that for itineraries of a certain form, the set of all…

Let $-1<\lambda<1$ and $f:[0,1)\to\mathbb{R}$ be a piecewise $\lambda$-affine map, that is, there exist points $0=c_0<c_1<\cdots <c_{n-1}<c_n=1$ and real numbers $b_1,\ldots,b_n$ such that $f(x)=\lambda x+b_i$ for every $x\in…

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a graph of density strictly exceeding 1/2, and…

We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset $S$ of the Euclidean space $\mathbb{R}^{k}$. We assume the maps are monotone (with respect to a suitable partial…

We investigate the connection between the instability of rational maps and summability methods applied to the spectrum of a critical point on the Julia set of a given rational map.

We study polynomial random dynamical systems with complete connections on the Riemann sphere. In this framework, the choice of the next polynomial map is governed by a state-dependent rule with memory, extending both i.i.d. random dynamics…

Given a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps…

Let $L_d$ be the Latt\`es map associated to the multiplication-by-$d$ endomorphism of an elliptic curve $E$ defined over a finite field $\mathbb{F}_q$. We determine the density $\delta(L_d,q)$ of periodic points for $L_d$ in…

We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We concentrate on a specific two-parameter family, describing the dynamics arising from models in…

We discuss in detail the dynamics of maps $z\mapsto ae^z+be^{-z}$ for which both critical orbits are strictly preperiodic. The points which converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called…