A study of rational maps of the real or complex projective plane of degree two or more, concentrating on those which map an elliptic curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of exotic dynamical behaviors which are perhaps familiar to the applied dynamics community but not to specialists in several complex variables. For example, we describe smooth attractors with riddled or intermingled attracting basins, and we observe ``blowout'' bifurcations when the transverse Lyapunov exponent for the invariant curve changes sign. In the complex case, the elliptic curve (a topological torus) can never have a trapping neighborhood, yet it can have an attracting basin of large measure (perhaps even of full measure). We also describe examples where there appear to be Herman rings (that is topological cylinders mapped to themselves with irrational rotation number) with open attracting basin. In some cases we provide proofs, but in other cases the discussion is empirical, based on numerical computation.
Let f be a rational self-map of P^2 which leaves invariant an elliptic curve C with strictly negative transverse Lyapunov exponent. We show that C is an attractor, i.e. it possesses a dense orbit and its basin is of strictly positive measure.
We show that resonance zones near an elliptic periodic point of a reversible map must, generically, contain asymptotically stable and asymptotically unstable periodic orbits, along with wild hyperbolic sets.
Let $f$ be a holomorphic endomorphism of $\mathbb P^k$ of degree $d.$ For each quasi-attractor of $f$ we construct a finite set of currents with attractive behaviors. To every such an attracting current is associated an equilibrium measure which allows for a systematic ergodic theoretical approach in the study of quasi-attractors of $\mathbb P^k.$ As a consequence, we deduce that there exist at most countably many quasi-attractors, each one with topological entropy equal to a multiple of $\log d.$ We also show that the study of these analytic objects can initiate a bifurcation theory for attra…
We apply the dynamical approach to the study of the second order semi-linear elliptic boundary value problem in a cylindrical domain with a small parameter at the second derivative with respect to the "time" variable corresponding to the axis of the cylinder. We prove that, under natural assumptions on the nonlinear interaction function and the external forces, this problem possesses the uniform attractors and that these attractors tend to the attractor of the limit parabolic equation. Moreover, in case where the limit attractor is regular, we give the detailed description of the structure of …
Let $f$ be a holomorphic endomorphism of $\mathbb{CP}^k$ having an attracting set $A$. In this paper, we address the question of the "size" of $A$ in a pluripolar sense. We introduce a conceptually simple framework to have non-algebraic attracting sets. We prove that adding a dimensional condition, these sets support a closed positive current with bounded quasi-potential (which answers a question from T.C. Dinh). Therefore, they are not pluripolar. Moreover, the examples are abundant on $\mathbb{CP}^k$.
For any compact set $K \subseteq \mathbb{R}^3$ we define a number $r(K)$ that is either a nonnegative integer or $\infty$. Intuitively, $r(K)$ provides some information on how wildly $K$ sits in $\mathbb{R}^3$. We show that attractors for discrete or continuous dynamical systems have finite $r$ and then prove that certain arcs, balls and spheres cannot be attractors by showing that their $r$ is infinite.
Topology of global attractors for homeomorphisms with the topological shadowing property in $\mathbb{R}^m$
Centro Universitario Regional Este, Universidad de la República（乌拉圭共和国大学东方地区中心）；Instituto de Matemática y Estadística “Rafael Laguardia”, Facultad de Ingeniería, Universidad de la República（乌拉圭共和国大学工程学院拉斐尔·拉瓜迪亚数学与统计研究所）
We prove that almost every finite collection of matrices in $GL_d(\mathbb{R})$ and $SL_d(\mathbb{R})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2(\mathbb{R})$ matrices induces a (generalized) iterated function system on the projective line $\mathbb{RP}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.
We propose a theoretical framework for an explanation of the numerically discovered phenomenon of the attractor-repeller merger. We identify regimes which are observed in dynamical systems with attractors as defined in a work by Ruelle and show that these attractors can be of three different types. The first two types correspond to th ewell-known types of chaotic behavior - conservative and dissipative, while the attractors of the third type, the reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regi…
Asymptotic random dynamics of weak solutions for a damped stochastic wave equation with the nonlinearity of arbitrarily large exponent and the additive noise on $\mathbb{R}^n$ is investigated. The existence of a pullback random attractor is proved in a parameter region with a breakthrough in proving the pullback asymptotic compactness of the cocycle with the quasi-trajectories defined on the integrable function space of arbitrary exponent and on the unbounded domain of arbitrary dimension.
As a first step to understand how complicated attractors for dynamical systems can be, one may consider the following realizability problem: given a continuum $K \subseteq \mathbb{R}^3$, decide when $K$ can be realized as an attractor for a homeomorphism of $\mathbb{R}^3$. In this paper we introduce toroidal sets as those continua $K \subseteq \mathbb{R}^3$ that have a neighbourhood basis comprised of solid tori and, generalizing the classical notion of genus of a knot, give a natural definition of the genus of toroidal sets and study some of its properties. Using these tools we exhibit knots …
In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for both discrete and continuous dynamical systems globally defined in $\mathbb{R}^3$. We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of $\mathbb{R}^3$ that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of $\mathbb{S}^3$ which arise naturally when considering toroidal sets.
We study dynamical systems generated by skew products: $$T: [0,1)\times\mathbb{C}\to [0,1)\times\mathbb{C} \quad\quad T(x,y)=(bx\mod1,γy+ϕ(x))$$ where integer $b\ge2$, $γ\in\mathbb{C}$ such that $0<|γ|<1$, and $ϕ$ is a real analytic $\mathbb{Z}$-periodic function. Let $Δ\in[0,1) $ such that $γ=|γ|e^{2πiΔ}$. For the case $Δ\notin\mathbb{Q}$ we prove the following dichotomy for the solenoidal attractor $K^ϕ_{b,\,γ}$ for $T$: Either $K^ϕ_{b,\,γ}$ is a graph of real analytic function, or the Hausdorff dimension of $K^ϕ_{b,\,γ}$ is equal to $\min\{3,1+\frac{\log b}{\log1/|γ|}\}$. Furthermore, given…
In this paper we focus on compacta $K \subseteq \mathbb{R}^3$ which possess a neighbourhood basis that consists of nested solid tori $T_i$. We call these sets toroidal. In \cite{hecyo1} we defined the genus of a toroidal set as a generalization of the classical notion of genus from knot theory. Here we introduce the self-geometric index of a toroidal set $K$, which captures how each torus $T_{i+1}$ winds inside the previous $T_i$. We use this index in conjunction with the genus to approach the problem of whether a toroidal set can be realized as an attractor for a flow or a homeomorphism of $\…
We show that a strong partially hyperbolic diffeomorphism of $\mathbb{T}^3$ isotopic to Anosov has a unique quasi-attractor. Moreover, we study the entropy of the diffeomorphism restricted to this quasi-attractor.
In this paper, we prove the existences of pullback attractors in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ for stochastic Fitzhugh-Nagumo system driven by both additive noises and deterministic non-autonomous forcings. The nonlinearity is polynomial like growth with exponent $p-1$. The asymptotic compactness for the cocycle in $L^{p}(\mathbb{R}^N)\times L^{2}(\mathbb{R}^N)$ is proved by using asymptotic a priori method, where the plus and minus signs of the nonlinearity at large value are not required.
Consider a dynamical system $T:\mathbb{T}\times \mathbb{R}^{d} \rightarrow \mathbb{T}\times \mathbb{R}^{d} $ given by $ T(x,y) = (E(x), C(y) + f(x))$, where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear contracting map of $\mathbb{R}^d$ and $f$ is in $C^2(\mathbb{T},\mathbb{R}^d)$. We prove that if $T$ is volume expanding and $u\geq d$, then for every $E$ there exists an open set $\mathcal{U}$ of pairs $(C,f)$ for which the corresponding dynamic $T$ admits an absolutely continuous invariant probability. A geometrical characteristic of transversality between self-intersections …
We study the continuity of pullback and uniform attractors for non-autonomous dynamical systems with respect to perturbations of a parameter. Consider a family of dynamical systems parameterised by a complete metric space $Λ$ such that for each $λ\inΛ$ there exists a unique pullback attractor $\mathcal A_λ(t)$. Using the theory of Baire category we show under natural conditions that there exists a residual set $Λ_*\subseteqΛ$ such that for every $t\in\mathbb R$ the function $λ\mapsto\mathcal A_λ(t)$ is continuous at each $λ\inΛ_*$ with respect to the Hausdorff metric. Similarly, given a family…
In this paper, we mainly study the regularity of pullback $\mathcal{D}$-attractors for a nonautonomous nonclassical diffusion equation with delay term $b(t,u_t)$ which contains some hereditary characteristics. Under a critical nonlinearity $f$, a time-dependent force $g(t,x)$ with exponential growth and a delayed force term $b(t,u_t)$, we prove that there exists a pullback $\mathcal{D}$-attractor $\mathcal{A}=\{A(t):t \in \mathbb{R}\}$ in $\mathbb{K}^1=H_0^1(Ω) \times L^2((-h,0);L^2(Ω))$ to problem \eqref{ine01} and for each $t \in \mathbb{R}$, $A(t)$ is bounded in $\mathbb{K}^2=H^2(Ω) \cap H_…
We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose density are regular. The maps we consider are given by $T(x,y) = (E (x), C(y) + f(x) )$, where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear contracting map of $\mathbb{R}^d$, $f$ is in $C^r(\mathbb{T}^u,\mathbb{R}^d)$ and $r \geq 2$. We prove that if $|(\det C)(\det E)| \|C^{-1}\|^{-2s}>1$ for some $s<r-(\frac{u+d}{2}+1)$ and $T$ satisfies a certain transversality condition, then the density of the SRB measure of $T$ is contained in the Sobolev space $…
We study two-digit attractors (2-attractors) in $\mathbb{R}^d$ which are self-affine compact sets defined by two contraction affine mappings with the same linear part. They are widely studied in the literature under various names: twindragons, two-digit tiles, 2-reptiles, etc., due to many applications in approximation theory, in the construction of multivariate Haar systems and other wavelet bases, in the discrete geometry, and in the number theory. We obtain a complete classification of isotropic 2-attractors in $\mathbb{R}^d$ and show that they are all homeomorphic but not diffeomorphic. In…
This article concerns the long-term random dynamics in regular spaces for a non-autonomous Navier-Stokes equation defined on a bounded smooth domain $\mathcal{O}$ driven by multiplicative and additive noise. For the two kinds of noise driven equations, we demonstrate the existence of a unique pullback attractor which is backward compact and asymptotically autonomous in $\mathbb{L}^2(\mathcal{O})$ and $\mathbb{H}_0^1(\mathcal{O})$, respectively. The backward-uniform flattening property of the solution is used to prove the backward-uniform pullback asymptotic compactness of the non-autonomous ra…
In this paper, we study the upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that, the pullback attractor $\{A_\varepsilon(t)\}_{t\in\mathbb R}$} of Eq.(1.1) with $\varepsilon\in[0,1]$ satisfies that for any $[a,b]\subset\mathbb R$ and $\varepsilon_0\in[0,1]$, $\lim_{\varepsilon\to\varepsilon_0} \sup_{t\in[a,b]} \mathrm{dist}_{H_0^1\times L^2} (A_\varepsilon(t), A_{\varepsilon_0}(t))=0$, and $\cup_{t\in[a,b]} \cup_{\varepsilon\in[0,1]} A_\varepsilon(t)$ is precompact in $H_0^1 (Ω) \times L^2(Ω)$.
Using shape theory and the concept of cellularity, we show that if $A$ is the global attractor associated with a dissipative partial differential equation in a real Hilbert space $H$ and the set $A-A$ has finite Assouad dimension $d$, then there is an ordinary differential equation in ${\mathbb R}^{m+1}$, with $m >d$, that has unique solutions and reproduces the dynamics on $A$. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor $X$ arbitrarily close to $LA$, where $L$ is a homeomorphism from $A$ into ${\mathbb R}^{m+1}$.
There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of vector fields unfolding an attracting heteroclinic network, linking two saddle-foci with $ (\mathbb{SO}(2) \oplus \mathbb{Z}_2)$-symmetry. The vector field is the restriction to $\mathbb{S}^3$ of a polynomial vector field in $\mathbb{R}^4$. We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a phenomenon called Torus-Breakdown theory.…
We analyze the dynamical properties of a tetrahedron transformation on the space of non-degenerate tetrahedra which can be identified with the non-compact globally symmetric $8$-dimensional space $\mbox{Sl}(3,\mathbb{R}) / \mbox{So}(3,\mathbb{R})$. We establish the existence of a local attractor which coincides with the set of regular tetrahedra and identify conditions which imply that the basin of attraction is the entire space. In numerical tests, these conditions are fulfilled for a large set of random tetrahedra.
It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom $A$ and the topology of the ambient manifold. In the given article, this statement is considered for the class $\mathbb G(M^2)$ of $A$-diffeomorphisms of closed orientable surfaces such that their non-wandering set consists of $k_f\geq 2$ connected components of one-dimensional basic sets (attractors and repellers). We prove that the ambient surface of every diffeomorphism $f\in \mathbb G(M^2)$ is homeomorphic to the connected sum of $k_f$ closed orientable surfaces and $l_f$ two-di…
A topological classification of many classes of dynamical systems with regular dynamics in low dimensions is often reduced to combinatorial invariants. In dimension 3 combinatorial invariants are proved to be insufficient even for simplest Morse-Smale diffeomorphisms. The complete topological invariant for the systems with a single saddle point on the 3-sphere is the embedding of the homotopy non-trivial knot into the manifold $\mathbb S^2\times\mathbb S^1$. If a diffeomorphism has several saddle points their unstable separatrices form arcs frames in the basin of the sink and circles frame in …
Suppose that a closed surface $S \subseteq \mathbb{R}^3$ is an attractor, not necessarily global, for a discrete dynamical system. Assuming that its set of wild points $W$ is totally disconnected, we prove that (up to an ambient homeomorphism) it has to be contained in a straight line. Using this result and a modification of the classical construction of a wild sphere due to Antoine we show that there exist uncountably many different $2$--spheres in $\mathbb{R}^3$ none of which can be realized as an attractor for a homeomorphism.
In this paper, we prove the existence of a compact global attractor for the flow generated by equation $$ \frac{\partial u}{\partial t}(x,t)+u(x,t)= \int_{\mathbb{R}^{N}}J(x-y)(f( u(y,t))dy+ h, \quad h > 0, \quad x\in \mathbb{R}^{N}, \quad t\in\mathbb{R}_{+} $$ in the weight space $L^{p}(\mathbb{R}^{N}, ρ)$. We also give uniform estimates on the size of the attractor and we exhibit a Lyapunov functional to the flow generated by this equation
We show that the manifold $(\mathbb{S}^2 \times \mathbb{S}^2) \operatorname{\#} (\mathbb{S}^2 \times \mathbb{S}^2)$ does not admit a non-constant non-injective uniformly quasiregular self-map. This answers a question of Martin, Mayer, and Peltonen, and provides the first example of a quasiregularly elliptic manifold which is not uniformly quasiregularly elliptic. To obtain the result, we introduce conformally formal manifolds, which are closed smooth $n$-manifolds $M$ admitting a measurable conformal structure $[g]$ for which the $(n/k)$-harmonic $k$-forms of the structure $[g]$ form an algebr…
Let $E$ be a linear space and suppose that $A$ is the global attractor of either (i) a homeomorphism $F:E\rightarrow E$ or (ii) a semigroup $S(\cdot)$ on $E$ that is injective on $A$. In both cases $A$ has trivial shape, and the dynamics on $A$ can be described by a homeomorphism $F:A\rightarrow A$ (in the second case we set $F=S(t)$ for some $t>0$). If the topological dimension of $A$ is finite we show that for any $ε>0$ there is an embedding $e:A\rightarrow{\mathbb R}^k$, with $k\sim{\rm dim}(A)$, and a (dynamical) homeomorphism $f:\R^k\rightarrow\R^k$ such that $F$ is conjugate to $f$ on $A…
Consider two objects associated to the Iterated Function System (IFS) $\{1+λz,-1+λz\}$: the locus $\mathcal{M}$ of parameters $λ\in\mathbb{D}\setminus\{0\}$ for which the corresponding attractor is connected; and the locus $\mathcal{M}_0$ of parameters for which the related attractor contains $0$. The set $\mathcal{M}$ can also be characterized as the locus of parameters for which the attractor of the IFS $\{1+λz, λz, -1+λz\}$ contains $λ^{-1}$. Exploiting the asymptotic similarity of $\mathcal{M}$ and $\mathcal{M}_0$ with the respective associated attractors, we give sufficient conditions on …
The paper deals with topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finitedimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large "chaotic sea" with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. I…
Let $M$ be a locally compact metric space endowed with a continuous flow $ϕ: M \times \mathbb{R} \longrightarrow M$. Frequently an attractor $K$ for $ϕ$ exists which is of interest, not only in itself but also the dynamics in its basin of attraction $\mathcal{A}(K)$. In this paper the class of {\sl attractors with no external explosions}, which is intermediate between the well known {\sl stable attractors} and the extremely wild {\sl unstable attractors}, is studied. We are mainly interested in their cohomological properties, as well as in the strong relations which exist between their shape (…
In 1987, J. H. Elton, has proved the first fundamental result in convergence of IFS, the Elton's Ergodic Theorem. In this work we prove the natural extension of this theorem to the projected Hutchinson measure $μ_α$ associated to a GIFSpdp $\mathcal{S}=\left(X, (ϕ_j:X^{m} \to X)_{j=0,1, ..., n-1}, (p_j)_{j=0,1, ..., n-1}\right),$ in a compact metric space $(X,d)$. More precisely, the average along of the trajectories $x_{n}(a)$ of the GIFS, starting in any initial points $x_0, ..., x_{m-1} \in X$ satisfies, for any $f \in C(X , \mathbb{R})$, $$\lim_{N\to +\infty} \frac{1}{N}\sum_{n=0 }^{N-1} f…
