Positive Topological Entropy Research Articles

Simple braids tend toward positive entropy

A simple braid is a positive braid that can be drawn so that any two strands cross at most once. We prove that as [Formula: see text], the proportion of simple braids on [Formula: see text] strands that have positive topological entropy tends toward 100%. Notably, such braids are either pseudo-Anosov or reducible with a pseudo-Anosov component. Our proof involves a method of reduction from simple braids to non-simple three-strand braids that may be of independent interest.

The Bowen-Franks type theorems for multivalued maps and impulsive differential equations

The aim of the present paper is to extend the well known Bowen-Franks type theorems to interval and circle multivalued maps. In this way, a positive topological entropy, including its lower estimate, can be implied by the existence of periodic orbits whose orders differ from a power of 2, or provided the topological degree of given circle maps is absolutely greater than 1. Simple applications are also given to impulsive differential equations.

Random expansive measures

The notion of expansivity and its generalizations (measure expansive, measure positively expansive, continuum-wise expansive, countably-expansive) are well known for deterministic systems and can be a useful property for studying significant type of behavior, such as chaotic one. This study aims to extend these notions into a random context and prove a relationship between relative positive entropy and random expansive measures and apply it to show that if a random dynamical system has positive relative topological entropy then the w-stable classes have zero measure for the conditional measures. We also prove that there exists a probability measure that is both invariant and expansive. Moreover, we obtain a relation between the notions of random expansive measures and random countably-expansive systems.

Generic properties of $3$-dimensional Reeb flows: Birkhoff sections and entropy

In this paper we use broken book decompositions to study Reeb flows on closed 3 -manifolds. We show that if the Liouville measure of a non-degenerate contact form can be approximated by periodic orbits, then there is a Birkhoff section for the associated Reeb flow. In view of Irie’s equidistribution theorem, this is shown to imply that the set of contact forms whose Reeb flows have a Birkhoff section contains an open and dense set in the C^{\infty} -topology. We also show that the set of contact forms whose Reeb flows have positive topological entropy is open and dense in the C^{\infty} -topology.

On topological entropy (chaos) realized far from fixed point

In many papers, the concept of chaos is associated with positive topological entropy. The points around which chaos is concentrated are also often examined. In this paper we distinguish places of chaos generation and its realization. Moreover, we will consider the possibility of approximation of continuous function by functions having full entropy point x0 (there is chaos generated by any neighbourhood of it) but simultaneously x0 is not a focal entropy point (chaos is realized at some distance from the considered point).

First-time Sensitive Homeomorphisms

AbstractWe introduce first-time sensitivity for a homeomorphism of a compact metric space, that is a condition on the first increasing times of open balls of the space. Continuum-wise expansive homeomorphisms, the shift map on the Hilbert cube, and also some partially hyperbolic diffeomorphisms satisfy this condition. We prove the existence of local unstable continua satisfying similar properties with the local unstable continua of continuum-wise expansive homeomorphisms, but assuming first-time sensitivity. As a consequence we prove that first-time sensitivity (with some additional technical assumptions) implies positive topological entropy.

Topological entropy, sets of periods, and transitivity for circle maps

UDC 517.9 Transitivity, the existence of periodic points, and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that, for every graph that is not a tree and any ε > 0 , there exist (complicated) totally transitive maps (then with cofinite set of periods) such that the topological entropy is smaller than ε (simplicity). To numerically measure the complexity of the set of periods, we introduce a notion of the <em>boundary of cofiniteness.</em> Larger boundary of cofiniteness corresponds to a simpler set of periods. We show that, for any continuous degree one circle maps, every totally transitive (and, hence, robustly complicated) map with small topological entropy has arbitrarily large (simplicity) boundary of cofiniteness.

Density of periodic measures and large deviation principle for generalised mod one transformations

We introduce a new class of piecewise monotonic maps, called generalised mod one transformations, which include all (α,β) and generalised β -transformations, and prove that all transitive generalised mod one transformations with positive topological entropy satisfy the level-2 large deviation principle with a unique measure of maximal entropy. This is obtained by our result on the density of periodic measures for more general class of piecewise mototonic maps, which is a partial answer to the open problem posed by Hofbauer and Raith (1998 Fund. Math. 157 221–34); Raith (www.math.iupui.edu/~mmisiure/open/PR1.pdf).

Some properties of dynamical systems having shadowing

Let X be a compact metric space and f:X→X be a continuous map having the shadowing property. In this paper, first, we prove that if X is connected, then ωf is continuous if, and only if, f is minimal. Second, we prove that the continuity of ωf yield CR(f)=R(f)=AP(f). Third, it is shown that if any point is recurrent then the collection of minimal sets is closed in (2X,dH). Finally, we show that the existence of a Li-Yorke pair is sufficient to have positive topological sequence entropy and if furthermore, f has s-shadowing property then f is Li-Yorke chaotic.

Computer‐assisted analysis of chaos in a three‐species food chain model

AbstractIn this paper, a three‐species food chain model with Holling type IV and Beddington–DeAngelis functional responses is formulated. Numerical simulations show that this system can generate chaos for some parameter values. But the mechanism behind chaos is still unclear only through numerical simulations. Then, using the topological horseshoe theories and Conley–Moser conditions, we present a computer‐assisted analysis to show the chaoticity of this system in the topological sense, that is, it has positive topological entropy. We prove that the Poincaré map of this model possesses a closed uniformly hyperbolic chaotic invariant set, and it is topologically conjugate to a 2‐shift map. At last, we consider the impact of fear on this three‐species model. It is an important factor in controlling chaos in biological models, which has been validated in other models.

Parametric Topological Entropy for Multivalued Maps and Differential Inclusions with Nonautonomous Impulses

The main purpose of this paper is to investigate a parametric topological entropy for impulsive differential inclusions on tori. In this way, besides other matters, we would like to extend our recent results concerning impulsive differential equations as well as those on “nonparametric” topological entropy to impulsive differential inclusions. Parametric topological entropy, which is usually called a topological entropy for nonautonomous dynamical systems, is considered here via the compositions of associated multivalued Poincaré translation operators with the single-valued time-dependent impulsive maps. On compact polyhedra and, in particular on tori, parametric topological entropy for families of admissible multivalued maps can be estimated from below by means Ivanov-type inequality in terms of the asymptotic Nielsen and Lefschetz numbers which are, unlike the topological entropy, homotopy invariants. In the scalar case, an effective criterion for a positive parametric topological entropy can be given by topological degree arguments for equi-continuous impulsive maps. In a single-valued nonparametric case, a positive topological entropy usually signifies topological chaos. Some simple illustrative examples are provided.

Parametric topological entropy of families of multivalued maps in topological spaces and induced hyperspace maps

A positive parametric topological entropy is examined whether or not it is preserved from given continuous maps to the induced hyperspace maps. An established affirmative answer generalizes all the related results obtained in this field so far, because the continuous maps under consideration can be multivalued and nonautonomous (time-dependent) in compact topological Hausdorff spaces. A variety of definitions of parametric topological entropy for multivalued maps is investigated, together with their properties. Several simple illustrative examples are supplied.

Remarks on minimal sets on dendrites and finite graphs

ABSTRACT Let X be a dendrite and let be a continuous map that has an infinite minimal set M in X. In the article we obtain conditions on the structure of a dendrite X and the set M, under which f has an arc horseshoe, and, hence, f has a positive topological entropy. We show that this result is not correct both for continuous maps with an empty set of periodic points, and for continuous maps with a non-empty set of periodic points on finite graphs.

Parametric topological entropy and differential equations with time–dependent impulses II: Multivalued case

The main aim of this article is to establish an effective criterion for a positive parametric topological entropy to differential equations with multivalued nonautonomous impulses on tori. The crucial tool for this goal consists in developing an appropriate Ivanov-like inequality for the lower estimate of a new kind of entropy by means of the logarithm of asymptotic Nielsen numbers for the compositions of a one-parameter family of multivalued admissible maps. This inequality is then applied to impulsive differential equations on tori via the associated Poincaré translation operators along their trajectories. The obtained results generalize in a significant way those in our recent papers, especially in the one with the same title (whence the indication by II), into a multivalued setting. Another nontrivial generalization can be regarded with respect to their simple reduction into a “nonparametric” case.

Topological chaos and statistical triviality

While transferring one flow ϕ with fixed points homeomorphically to its equivalent flow ψ, we show that positive topological entropy degenerates to zero (such degeneracy happens for equivalent continuous flows [3] and equivalent differential flows [6]) if and only if all ergodic invariant measures with positive measure-theoretic entropy degenerate to fixed points. Whenever ϕ is assumed to be topological transitive, the measure degeneracy implies that the resulted equivalent flow ψ is topologically chaotic but statistically trivial, meaning that all ergodic invariant measures are supported on fixed points. Using different approaches in different areas people constructed examples of topological chaotic but statistical trivial systems, see [3] for C0 flows, see [6] for Cr, r≥1, flows, see [1][11] for C0 homeomorphisms, see time one map in [6] for Cr, r≥1 diffeomorphisms. We point out it is non-hyperbolic singularity causes the degeneracy while changing one flow equivalently to another.

Proper extensions of the 2-sphere’s conformal group present entropy and are 4-transitive

AbstractIn this paper, we prove using elementary techniques that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. As an important consequence, we derive that any such group must always contain an element of positive topological entropy. We also provide a self-contained characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.

A Proof of Chaos for a Seasonally Perturbed Version of Goodwin Growth Cycle Model: Linear and Nonlinear Formulations

We show the existence of complex dynamics for a seasonally perturbed version of the Goodwin growth cycle model, both in its original formulation and for a modified formulation, encompassing nonlinear expressions of the real wage bargaining function and of the investment function. The need to deal with a modified formulation of the Goodwin model is connected with the economically sensible position of orbits, which have to lie in the unit square, in contrast to what occurs in the model’s original formulation. In proving the existence of chaos, we follow the seminal idea by Goodwin of studying forced models in economics. Namely, the original and the modified formulations of Goodwin model are described by Hamiltonian systems, characterized by the presence of a nonisochronous center, and the seasonal variation of the parameter, representing the ratio between capital and output, which is common to both frameworks, is empirically grounded. Hence, exploiting the periodic dependence on time of that model parameter we enter the framework of Linked Twist Maps. The topological results valid in this context allow us to prove that the Poincaré map, associated with the considered systems, is chaotic, focusing on sets that lie in the unit square, and also when dealing with the original version of the Goodwin model. Accordingly, the trademark features of chaos follow, such as sensitive dependence on initial conditions and positive topological entropy.

Induced dynamics of nonautonomous dynamical systems

Let f0,∞={fn}n=0∞ be a sequence of continuous self-maps on a compact metric space X. The nonautonomous dynamical system (X,f0,∞) induces the set-valued system (K(X),f¯0,∞) and the fuzzified system (F(X),f˜0,∞). We prove that under some natural conditions, positive topological entropy of (X,f0,∞) implies infinite entropy of (K(X),f¯0,∞) and (F(X),f˜0,∞), respectively; and zero entropy of (S1,f0,∞) implies zero entropy of some invariant subsystems of (K(S1),f¯0,∞) and (F(S1),f˜0,∞), respectively. We confirm that (K(I),f¯) and (F(I),f˜) have infinite entropy for any transitive interval map f. In contrast, we construct a transitive nonautonomous system (I,f0,∞) such that both (K(I),f¯0,∞) and (F(I),f˜0,∞) have zero entropy. We obtain that (K(X),f¯0,∞) is chain weakly mixing of all orders if and only if (F1(X),f˜0,∞) is so, and chain mixing (resp. h-shadowing and multi-F-sensitivity) among (X,f0,∞), (K(X),f¯0,∞) and (F1(X),f˜0,∞) are equivalent, where (F1(X),f˜0,∞) is the induced normal fuzzification.

Topological entropy and differential equations

On the background of a brief survey panorama of results on the topic in the title, one new theorem is presented concerning a positive topological entropy (i.e. topological chaos) for the impulsive differential equations on the Cartesian product of compact intervals, which is positively invariant under the composition of the associated Poincaré translation operator with a multivalued upper semicontinuous impulsive mapping.

Depinning transition of travelling waves for particle chains

In this paper we apply Aubry–Mather theory for equilibria of 1D Hamiltonian lattice systems and the theory of invariant ordered circles to investigate the depinning transition of travelling waves for particle chains. Assume A < B are two critical values such that the particle chain has three homogeneous equilibria if the driving force . It is already known that there exist transition thresholds of the driving force such that the particle chain has stationary fronts but no travelling fronts for and travelling fronts but no stationary fronts if or . The novelty of our approach is that we prove the transition threshold () coincides with the upper (lower) limit of the upper (lower) depinning force as the rotation number tends to zero from the right. Based on this conclusion, we demonstrate that when the driving force , besides stationary fronts there are various kinds of equilibria with rotation numbers close to zero such that the spatial shift map has positive topological entropy on the set of equilibria. Furthermore, we give a necessary and sufficient condition for the absence of propagation failure, i.e. , in terms of a minimal foliation. Finally we show that are continuous with respect to potential functions in C 1 topology.