We study the dynamics of iterated function systems generated by a pair of circle diffeomorphisms close to rotations in the $C^{1+\text{bv}}$-topology. We characterize the obstruction to minimality and describe the limit set. In particular, there are no invariant minimal Cantor sets, which can be seen as a Denjoy/Duminy type theorem for iterated systems on the circle.

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive $(1,1)$-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.

We study invariant Fatou components for holomorphic endomorphisms in $\mathbb{P}^{2}$. In the recurrent case these components were classified by Fornæss and Sibony [Classification of recurrent domains for some holomorphic maps. Math. Ann. 301(4) (1995), 813–820]. Ueda [Holomorphic maps on projective spaces and continuations of Fatou maps. Michigan Math J.56(1) (2008), 145–153] completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and Peters [Classification of invariant Fatou components for dissipative Hénon maps. Preprint] classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of $\mathbb{C}^{2}$ with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets.

We define the notion of ${\it\varepsilon}$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits ${\it\varepsilon}$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that an ${\it\varepsilon}$-perturbation to an ${\it\varepsilon}$-flexible point allows us to change it to a stable index one periodic point whose (one-dimensional) stable manifold is an arbitrarily chosen $C^{1}$-curve. We also show that the existence of flexible points is a general phenomenon among systems with a robustly non-hyperbolic two-dimensional center-stable bundle.

Let $(X,f)$ be a topological dynamical system and ${\mathcal{F}}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an ${\mathcal{F}}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}:f^{n}(x)\in U\}$, is in ${\mathcal{F}}$; the system $(X,f)$ is called ${\mathcal{F}}$-point transitive if there exists some ${\mathcal{F}}$-transitive point. In this paper, we first discuss the connection between ${\mathcal{F}}$-point transitivity and ${\mathcal{F}}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by ${\mathcal{F}}$-point transitivity, completing results in [Transitive points via Furstenberg family. Topology Appl. 158 (2011), 2221–2231]. We also show that multi-transitivity, ${\rm\Delta}$-transitivity and multi-minimality can be characterized by ${\mathcal{F}}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates. Ergod. Th. & Dynam. Sys. 32 (2012), 1661–1672].

In this paper we consider the set of all extremal Volterra quadratic stochastic operators defined on a unit simplex $S^{4}$ and show that such operators can be reinterpreted in terms of zero-sum games. We show that an extremal Volterra operator is non-ergodic and an appropriate zero-sum game is a rock-paper-scissors game if either the Volterra operator is a uniform operator or for a non-uniform Volterra operator $V$ there exists a subset $I\subset \{1,2,3,4,5\}$ with $|I|\leq 2$ such that $\sum _{i\in I}(V^{n}\mathbf{x})_{i}\rightarrow 0,$ and the restriction of $V$ on an invariant face ${\rm\Gamma}_{I}=\{\mathbf{x}\in S^{m-1}:x_{i}=0,i\in I\}$ is a uniform Volterra operator.

This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works on well-chosen examples (such as Gambaudo and Tresser [Some difficulties generated by small sinks in the numerical study of dynamical systems: two examples. Phys. Lett. A 94(9) (1983), 412–414]) show that the dynamical behaviours of dynamical systems and of their discretizations can be quite different. We are interested in generic homeomorphisms of compact manifolds. So our aim is to tackle the following question: can the dynamical properties of a generic homeomorphism be detected on the spatial discretizations of this homeomorphism? We will prove that the dynamics of a single discretization of a generic conservative homeomorphism does not depend on the homeomorphism itself, but rather on the grid used for the discretization. Therefore, dynamical properties of a given generic conservative homeomorphism cannot be detected using a single discretization. Nevertheless, we will also prove that some dynamical features of a generic conservative homeomorphism (such as the set of the periods of all periodic points) can be read on a sequence of finer and finer discretizations.

The aim of this paper is to study the spectrum of the $L^{p}$ Laplacian and the dynamics of the $L^{p}$ heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in the rank-one case, it turns out that the $L^{p}$ heat semigroup on $M$ has a certain chaotic behavior if $p\in (1,2)$, whereas for $p\geq 2$ such chaotic behavior never occurs.

In this paper, we use abstract Lyapunov graphs as a combinatorial tool to obtain a complete classification of Smale flows on $\mathbb{S}^{2}\times \mathbb{S}^{1}$. This classification gives necessary and sufficient conditions that must be satisfied by an (abstract) Lyapunov graph in order for it to be associated to a Smale flow on $\mathbb{S}^{2}\times \mathbb{S}^{1}$.

A Herman–Avila–Bochi type formula is obtained for the average sum of the top $d$ Lyapunov exponents over a one-parameter family of $\mathbb{G}$-cocycles, where $\mathbb{G}$ is the group that leaves a certain, non-degenerate Hermitian form of signature $(c,d)$ invariant. The generic example of such a group is the pseudo-unitary group $\text{U}(c,d)$ or, in the case $c=d$, the Hermitian-symplectic group $\text{HSp}(2d)$ which naturally appears for cocycles related to Schrödinger operators. In the case $d=1$, the formula for $\text{HSp}(2d)$ cocycles reduces to the Herman–Avila–Bochi formula for $\text{SL}(2,\mathbb{R})$ cocycles.

We show that any subset $A\subset \mathbb{N}$ with positive upper Banach density contains the pattern $\{m,m+[n{\it\alpha}],\dots ,m+k[n{\it\alpha}]\}$, for some $m\in \mathbb{N}$ and $n=p-1$ for some prime $p$, where ${\it\alpha}\in \mathbb{R}\setminus \mathbb{Q}$. Making use of the Furstenberg correspondence principle, we do this by proving an associated recurrence result in ergodic theory along the shifted primes. We also prove the convergence result for the associated averages along primes and indicate other applications of these methods.

In a previous paper the author proved a distributional convergence for the Birkhoff sums of functions of null average defined over a dynamical system with an infinite, invariant, ergodic measure, akin to a central limit theorem. Here we extend this result to larger classes of observables, with milder smoothness conditions, and to larger classes of dynamical systems, which may no longer be mixing. A special emphasis is given to continuous time systems: semi-flows, flows, and $\mathbb{Z}^{d}$-extensions of flows. The latter generalization is applied to the geodesic flow on $\mathbb{Z}^{d}$-periodic manifolds of negative sectional curvature.

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\leq n,1\leq p<(n/2)-1$) on $\mathbb{S}^{2}$, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group of $F_{n,p}$. We also establish that $F_{n,p}$ and $F_{n^{\prime },p^{\prime }}$ are quasisymmetrically equivalent if and only if $(n,p)=(n^{\prime },p^{\prime })$.