Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations

The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H.     

Omega-limit sets in hereditarily locally connected continua

In this paper we give a topological characterization of ω-limit sets in hereditarily locally connected continua. Moreover, we characterize also orbit-enclosing ω-limit sets in these spaces.     

On the centre and the set of ω-limit points of continuous maps on dendrites

It is well known that for dynamical systems generated by continuous maps of a graph, the centre of the dynamical system is a subset of the set of ω-limit points.In this paper we provide an example of a continuous self-map f₁ of a dendrite such that ω(f₁) is a proper subset of C(f₁).The second example is a continuous self-map f₂ of a dendrite having a strictly increasing sequence of ω-limit sets which is not contained in any maximal one. Again, this is impossible for continuous maps on graphs.     

A triangular map with homoclinic orbits and no infinite ω-limit set containing periodic points

Recently, Forti, Paganoni and Smítal constructed an example of a triangular map of the unite square, F(x,y)=(f(x),g(x,y)), possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. In this note we show that the above quoted map F has a homoclinic orbit. As a consequence, we answer in the negative the problem presented by A.N. Sharkovsky in the eighties whether, for a triangular map of the square, existence of a homoclinic orbit implies the existence of an infinite ω-limit set containing a periodic point. It is well known that, for a continuous map of the interval, the answer is positive.     

Measure of minimal sets for certain discrete dynamical systems

Let X be a separable metric space, μ a complete Borel measure on X that is finite on balls, and f a closed discrete dynamical system on X that preserves μ and has the diameters of all orbits bounded. We prove that almost every point in X (in the sense of measure μ) has its orbit contained in its ω-limit set.     

On the existence of maximal ω-limit sets for dendrite maps

For interval maps and also for graph maps, every ω-limit set is a subset of a maximal one. In this note we construct a continuous map on a dendrite with no maximal ω-limit set. Moreover, the set of branch points is nowhere dense, every ω-limit set of the map is nowhere dense, the set of periodic points and the set of recurrent points are equal and the set of ω-limit points is not closed (an example with the last property was constructed by the authors already in [Ko?an Z, Kornecká-Kurková V, Málek M. On the centre and the set of omega-limit points of continuous maps on dendrites. Topol Appl 2009;156:2923-2931]).     

Almost periodic points and minimal sets in ω-regular spaces

In this paper a notion of ω-regular space is raised, which is an extension of that of regular space, and several known results concerning almost periodic points and minimal sets of maps are generalized from regular spaces to ω-regular spaces.     

On totally periodic ω-limit sets in regular continua

Let f be a continuous self-mapping of a compact metric space X, an ω-limit set of f is said to be totally periodic if it is composed of periodic points. We prove that a totally periodic ω-limit set of one-to-one continuous self mapping of regular continuum is finite. In the other hand, we built a continuous self-mapping (not one-to-one) of a dendrite having a totally periodic ω-limit set with unbounded periods.     

Characterizing omega-limit sets which are closed orbits

Let X be a vector field in a compact n-manifold M, n?2. Given Σ⊂M we say that q∈M satisfies (P)_Σ if the closure of the positive orbit of X through q does not intersect Σ, but, however, there is an open interval I with q as a boundary point such that every positive orbit through I intersects Σ. Among those q having saddle-type hyperbolic omega-limit set ω(q) the ones with ω(q) being a closed orbit satisfy (P)_Σ for some closed subset Σ. The converse is true for n=2 but not for n?4. Here we prove the converse for n=3. Moreover, we prove for n=3 that if ω(q) is a singular-hyperbolic set [C. Morales, M. Pacifico, E. Pujals, On C¹ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris Sér. I 26 (1998) 81-86], [C. Morales, M. Pacifico, E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2) 160 (2) (2004) 375-432], then ω(q) is a closed orbit if and only if q satisfies (P)_Σ for some Σ closed. This result improves [S. Bautista, Sobre conjuntos hiperbólicos-singulares (On singular-hyperbolic sets), thesis Uiversidade Federal do Rio de Janeiro, 2005 (in Portuguese)] and [C. Morales, M. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001) 359-378].     

Strong distributional chaos and minimal sets

In the class T of triangular maps of the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [B. Schweizer, J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994) 737-854] for continuous maps of the interval. We show that there is a DC1 homeomorphism F∈T such that any ω-limit set contains unique minimal set. This homeomorphism is constructed such that it is increasing on some fibres, and decreasing on the other ones. Consequently, F has zero topological entropy. Similar behavior is impossible when F is nondecreasing on the fibres, as shown by Paganoni and Smítal [L. Paganoni, J. Smítal, Strange distributionally chaotic triangular maps, Chaos Solitons Fractals 26 (2005) 581-589]. This result contributes to the solution of an old problem of Sharkovsky concerning classification of triangular maps but it is interesting by itself since it implies interesting open problems concerning relations between DC1 and minimality.     

On box products

We prove two theorems about box products. The first theorem says that the box product of countable spaces is pseudonormal, i.e. any two disjoint closed sets one of which is countable can be separated by open sets. The second theorem says that assuming CH a certain uncountable box product is normal (i.e. <_ω1?□_α<ω1X_α where each Xα is a compact metric space).     

Shift spaces, ω-chaos and specification property

The main aim of the present paper is to describe some relations between specification property and ω-chaos. In particular, we study how properties of factor maps can be used to transfer this kind of chaos by a semiconjugacy with a shift space.     

The triangular maps with closed sets of periodic points

In a recent paper we provided a characterization of triangular maps of the square, i.e., maps given by F(x,y)=(f(x),gx(y)), satisfying condition (P1) that any chain recurrent point is periodic. For continuous maps of the interval, there is a list of 18 other conditions equivalent to (P1), including (P2) that there is no infinite ω-limit set, (P3) that the set of periodic points is closed and (P4) that any regularly recurrent point is periodic, for instance. We provide an almost complete classification among these conditions for triangular maps, improve a result given by C. Arteaga [C. Arteaga, Smooth triangular maps of the square with closed set of periodic points, J. Math. Anal. Appl. 196 (1995) 987-997] and state an open problem concerning minimal sets of the triangular maps. The paper solves partially a problem formulated by A.N. Sharkovsky in the eighties. The mentioned open problem, the validity of (P4) ⇒ (P3), is related to the question whether some regularly recurrent point lies in the fibres over an f-minimal set possessing a regularly recurrent point. We answered this question in the positive for triangular maps with nondecreasing fiber maps. Consequently, the classification is completed for monotone triangular maps.     

A pair of spaces of upper semi-continuous maps and continuous maps

For a Tychonoff space X, we use ↓USC(X) and ↓C(X) to denote the families of the regions below all upper semi-continuous maps and of the regions below all continuous maps from X to I=[0,1], respectively. In this paper, we consider the spaces ↓USC(X) and ↓C(X) topologized as subspaces of the hyperspace Cld(X×I) consisting of all non-empty closed sets in X×I endowed with the Vietoris topology. We shall prove that ↓USC(X) is homeomorphic (≈) to the Hilbert cube Q=ω[−1,1] if and only if X is an infinite compact metric space. And we shall prove that (↓USC(X),↓C(X))≈(Q,c₀), where , if and only if ↓C(X)≈c₀ if and only if X is a compact metric space and the set of isolated points is not dense in X.     

Choquet simplices as spaces of invariant probability measures on post-critical sets

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak^∗ topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures on the post-critical set of a logistic map. Here, the post-critical set of a logistic map is the ω-limit set of its unique critical point. In fact we show the logistic map f can be taken in such a way that its post-critical set is a Cantor set where f   is minimal, and such that each invariant probability measure on this set has zero Lyapunov exponent, and is an equilibrium state for the potential −ln|f^′|$-\ln |f^{\prime }|$.     

Integral invariants and limit sets of planar vector fields

We show that if a planar system of differential equations admits an inverse integrating factor V defined in a neighborhood of a singular point with exactly one zero eigenvalue then V vanishes along any separatrix of the singular point. Additionally we prove that if K is a compact α- or ω-limit set that contains a regular point (or has an elliptic or parabolic sector if not), and if V is defined on a neighborhood of K, then V vanishes at at least one point of K (and on all of K if V is real analytic or Morse).     

Distributional chaos and irregular recurrence

For a continuous map φ:X→X of a compact metric space, we study relations between distributional chaos and the existence of a point which is quasi-weakly almost periodic, but not weakly almost periodic. We provide an example showing that the existence of such a point does not imply the strongest version of distributional chaos, DC1. Using this we prove that, even in the class of triangular maps of the square, there are no relations to DC1. This result, among others, contributes to the solution of a problem formulated by A.N. Sharkovsky in the eighties.     

New basic result in classical descriptive set theory: Preservation of completeness

The aim of this note is to prove the following result:Assume that f is a continuous function from the space of irrationals ωω onto Y such that the image f(U) of every set U which is open and closed in ωω is the union of one open and one closed set. Then Y is a completely metrizable space.