Nonwandering Research Articles

Inverse shadowing and related measures

We study various weaker forms of inverse shadowing property for discrete dynamical systems on a smooth compact manifold. First, we introduce the so-called Ergodic Inverse Shadowing property (Birhhoff averages of continuous functions along the exact trajectory and the approximating one are close). We demonstrate that this property implies continuity of the set of invariant measures in Hausdorff metrics. We show that the class of systems with Ergodic Inverse Shadowing is quite broad, it includes all diffeomorphisms with hyperbolic nonwandering sets. Secondly, we study the so-called Individual Inverse Shadowing (any exact trajectory can be traced by approximate ones but this shadowing is not uniform with respect to selection of the initial point of the trajectory). We demonstrate that this property is closely related to Structural Stability and $\Omega$-stability of diffeomorphisms.

Any closed 3-manifold supports A-flows with 2-dimensional expanding attractors

Abstract We prove that given any closed 3-manifold M 3, there is an A-flow f t on M 3 such that the non-wandering set NW (f t ) consists of 2-dimensional non-orientable expanding attractor and trivial basic sets.

On Limit Sets of Monotone Maps on Dendroids

Abstract Let X be a dendrite, f : X → X be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point x ∈ X has the next properties: (1) ω ( x , f ) ⊆ Per ( f ) ¯ \omega (x,f) \subseteq \overline {Per(f)} , where Per( f ) is the set of periodic points of f ; (2) ω(x, f ) is either a periodic orbit or a minimal Cantor set. In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that (3) Ω ( f ) = Per ( f ) ¯ \Omega (f) = \overline {Per(f)} , where Ω( f ) is the set of non-wandering points of f. The aim of this note is to show that the above results (1) – (3) do not hold for monotone maps on dendroids.

Supporting manifolds for high-dimensional Morse-Smale diffeomorphisms with few saddles

We describe the topological structure of closed manifolds of dimension no less than four which admit Morse-Smale diffeomorphisms such that its non-wandering set contains any number of sink periodic points, and any number of source periodic points, and few saddle periodic points.

The centre and the depth of the centre for continuous maps on dendrites with unique branch point

Let D be a dendrite with unique branch point and f:D→D be continuous. Denote by R(f) and Ω(f) the set of recurrent points and the set of non-wandering points of f, respectively. Let Ω0(f)=D and Ωk(f)=Ω(f|Ωk−1(f)) for any positive integer k. The minimal k such that Ωk(f)=Ωk+1(f) is called the depth of f, where k is a positive integer or ∞. In this note, we show that Ω2(f)=R(f)‾ and the depth of f is at most 2.

Spectral decomposition and Ω-stability of flows with expanding measures

In this paper we present a measurable version of the classical spectral decomposition theorem for flows. More precisely, we prove that if a flow ϕ on a compact metric space X is invariantly measure expanding on its chain recurrent set CR(ϕ) and has the invariantly measure shadowing property on CR(ϕ) then ϕ has the spectral decomposition, i.e. the nonwandering set Ω(ϕ) is decomposed by a disjoint union of finitely many invariant and closed subsets on which ϕ is topologically transitive. Moreover we show that if ϕ is invariantly measure expanding on CR(ϕ) then it is invariantly measure expanding on X. Using this, we characterize the measure expanding flows on a compact C∞ manifold via the notion of Ω-stability.

On Two-Dimensional Expanding Attractors of A-Flows

We prove that given any closed $n$-manifold $M^n$, $n\geq 4$, there is an A-flow $f^t$ on $M^n$ such that the non-wandering set $NW(f^t)$ consists of 2-dimensional expanding attractor (the both, orientable and non-orientable) and trivial basic sets. For 3-manifolds, we prove that given any closed 3-manifold $M^3$, there is an A-flow $f^t$ on $M^3$ such that the non-wandering set $NW(f^t)$ consists of a non-orientable 2-dimensional expanding attractor and trivial basic sets. Moreover, there is a nonsingular A-flow $f^t$ on a 3-sphere such that the non-wandering set $NW(f^t)$ consists of an orientable 2-dimensional expanding attractor and trivial basic sets (isolated periodic trajectories).

Mean dimensions for partial actions

In this paper, we introduce the definitions of mean dimensions and upper metric mean dimensions for partial -actions. Our theorem provides a natural and intrinsic characterization of mean dimensions and upper metric mean dimensions for partial -actions, that mean dimension for a partial action is bounded by the upper metric mean dimension of its globalization and for certain systems the upper metric mean dimension is concentrated on the non-wandering set.

Topological chain and shadowing properties of dynamical systems on uniform spaces

This paper discusses topological shadowing property, chain transitivity, total chain transitivity, and chain mixing property for dynamical systems on uniform spaces and characterizes some topological chain properties for dynamical systems on compact uniform spaces. In particular, it is proved that a compact dynamical system is topologically chain mixing if and only if it is totally topologically chain transitive. Moreover, some basic properties of topological shadowing and non-wandering points on uniform spaces are obtained.

Dynamics and topological entropy of 1D Greenberg–Hastings cellular automata

In this paper we analyse the non-wandering set of one-dimensional Greenberg–Hastings cellular automaton models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$.

Strange attractors and wandering domains near a homoclinic cycle to a bifocus

In this paper, we explore the three-dimensional chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions. We extend previous results on the field and we show that the first return map to a given cross section may be approximated by a map exhibiting heteroclinic tangencies associated to two periodic orbits. Under an additional hypothesis about the coexistence of two heteroclinically related periodic points (one without dominated splitting into one-dimensional sub-bundles), the heteroclinic tangencies can be slightly modified in order to satisfy Tatjer's conditions for a generalized tangency of codimension two. This configuration may be seen as the organizing center, by which one can obtain Bogdanov-Takens bifurcations and therefore, strange attractors, infinitely many sinks and non-trivial contracting wandering domains.

Eventually Shadowable Points

We study the eventually shadowable points namely points for which every pseudo orbit passing through then can be eventually shadowed (Good and Meddaugh in Ergod Theory Dyn Syst 38(1):143–154, 2018). We will prove the following results: the set of eventually shadowable points of a surjective continuous map of a compact metric space is invariant (possibly empty or noncompact) and the map has the eventual shadowing property if and only if every point is eventually shadowable. The chain recurrent and nonwandering sets coincide when every chain recurrent point is eventually shadowable. A surjective continuous map of a compact metric space has the eventual shadowing property if and only if the set of eventually shadowable points has a full measure with respect to every ergodic invariant probability measure. If there is an eventually shadowable point for which the associated Li–Yorke set equals the whole space, then the map has the eventual shadowing property. Proximal or transitive maps with eventually shadowable points have the eventual shadowing property. The eventually shadowable and shadowable points coincide for surjective equicontinuous maps on compact metric spaces. In particular, a surjective equicontinuous map of a compact metric space has the eventual shadowing property if and only if it has the shadowing property.

The depths and the attracting centres for continuous maps on local dendrites with the number of branch points being finite

Let Y be a local dendrite with the number of branch points being finite and f:Y→Y be continuous. Denote by P(f), R(f), Ω(f) and ω(x,f) the set of periodic points of f, the set of recurrent points of f, the set of non-wandering points of f and the set of ω-limit points of x under f, respectively. Write ω(f)=∪x∈Yω(x,f) and ωn+1(f)=∪x∈ωn(f)ω(x,f) and Ωn+1(f)=Ω(f|Ωn(f)) for any positive integer n. In this paper, we show that Ω3(f)=R(f)‾ and the depth of f is at most 3, and ω3(f)=ω2(f). Furthermore, we show that R(f)‾=R(f)∪P(f)‾.

Scenario of a Simple Transition from a Structurally Stable 3-Diffeomorphism with a Two-Dimensional Expanding Attractor to a DA Diffeomorphism

The Smale surgery on the three-dimensional torus allows one to obtain a so-called DA diffeomorphism from the Anosov automorphism. The nonwandering set of a DA diffeomor-phism consists of a single two-dimensional expanding attractor and a finite number of source periodic orbits. As shown by V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, the dynamics of an arbitrary structurally stable 3-diffeomorphism with a two-dimensional expanding attrac-tor generalizes the dynamics of a DA diffeomorphism: such a 3-diffeomorphism exists only on the three-dimensional torus, and the two-dimensional attractor is its unique nontrivial basic set, but its nonwandering set may contain isolated saddle periodic orbits together with source periodic orbits. In the present study, we describe a scenario of a simple transition (through elementary bifurcations) from a structurally stable diffeomorphism of the three-dimensional torus with a two-dimensional expanding attractor to a DA diffeomorphism. A key moment in the construction of the arc is the proof that the closure of the separatrices of boundary periodic points of a nontrivial attractor and of isolated saddle periodic points are tamely embedded. This result demonstrates the fundamental difference of the dynamics of such diffeomorphisms from the dynamics of three-dimensional Morse-Smale diffeomorphisms, in which the closure of the separatrices of saddle periodic points may be wildly embedded.

Spectral decomposition for rescaling expansive flows with rescaled shadowing

In this paper, we introduce the concepts of rescaled expansiveness and the rescaled shadowing property for flows on metric spaces which are dynamical properties, and present a spectral decomposition theorem for flows. More precisely, we prove that if a flow is rescaling expansive and has the rescaled shadowing property on a locally compact metric space, then it admits the spectral decomposition. Moreover, we show that if a flow on locally compact metric space has the rescaled shadowing property then its restriction on nonwandering set also has the rescaled shadowing property.

Dynamics of homeomorphisms of regular curves

In this paper, we prove first that the space of minimal sets of any homeomorphisms $f:X\to X$ of a regular curve $X$ is closed in the hyperspace $2^X$ of closed subsets of $X$ endowed with the Hausdorff metric, and the non-wandering set $\Omega (f)$ is equal to the set of recurrent points of $f$. Second, we study the continuity of the map $\omega_f:X\to 2^X;x\mapsto \omega_f (x) $, we show for instance the equivalence between the continuity of $\omega_f $ and the equality between the $\omega$-limit set and the $\alpha $-limit set of every point in $X $. Finally, we prove that there is only one (infinite) minimal set when there is no periodic point.

Higher Prolongations of Control Affine Systems: Absolute Stability and Generalized Recurrence

This manuscript deals with higher prolongations and higher prolongational limit sets of control affine systems. The higher prolongations extend the positive semiorbit and determine a great number of stability concepts, indexed by the ordinal numbers. The highest stability is named absolute stability and can be characterized by a continuous Lyapunov functional. It is proved that compact positively invariant uniform attractors and positive semiorbits of dispersive control systems are absolutely stable sets. The higher prolongational limit sets determine the generalized recurrence, which extends the recursive concepts of Poincaré recurrence and nonwandering points. The notion of chain prolongation is introduced in order to discuss the generalized recurrence. The main result shows that the chain prolongation is the largest extension of the positive semiorbit, and then the chain recurrence is the more general concept of recurrence.

Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

We consider order-preserving $C^3$ circle maps with a flat piece, irrational rotation number and critical exponents $(l_1, l_2)$. We detect a change in the geometry of the system. For $(l_1, l_2) \in [1, 2]^2$ the geometry is degenerate and becomes bounded for $(l_1, l_2) \in [2, \infty)^2 \backslash \{(2, 2)\}$. When the rotation number is of the form $[abab \ldots]$; for some $a, b \in \mathbb{N}^*$, the geometry is bounded for $(l_1, l_2)$ belonging above a curve defined on $]1, +\infty[^2$. As a consequence, we estimate the Hausdorff dimension of the nonwandering set $K_f=\mathcal{S}^1\backslash \bigcup^\infty_{i=0}f^{-i}(U)$. Precisely, the Hausdorff dimension of this set is equal to zero when the geometry is degenerate and it is strictly positive when the geometry is bounded.

The Topological Classification of Diffeomorphisms of the Two-Dimensional Torus with an Orientable Attractor

This paper is devoted to the topological classification of structurally stable diffeomorphisms of the two-dimensional torus whose nonwandering set consists of an orientable one-dimensional attractor and finitely many isolated source and saddle periodic points, under the assumption that the closure of the union of the stable manifolds of isolated periodic points consists of simple pairwise nonintersecting arcs. The classification of one-dimensional basis sets on surfaces has been exhaustively obtained in papers by V. Grines. He also obtained a classification of some classes of structurally stable diffeomorphisms of surfaces using combined algebra-geometric invariants. In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic differentiating invariants.

On the shadowing and limit shadowing properties

We study the relation between the shadowing property and the limit shadowing property. We prove that if a continuous self-map $f$ of a compact metric space has the limit shadowing property, then the restriction of $f$ to the non-wandering set satisfies the shadowing property. As an application, we prove the equivalence of the two shadowing properties for equicontinuous maps.