Topological Entropy Research Articles

Generalized Fibonacci shifts in the Lorenz attractor

In this article we deal with symmetric Lorenz attractors having a homoclinic loop that exhibits a well ordered orbit. We show the symmetry implies a very regular behaviour on the dynamic in the topological and metric sense. Let ([−1,1],f) be the one-dimensional reduction Lorenz map satisfying a well ordered orbit and ([−1,0],f̃) be the quotient map, given by the equivalence relation x∼−x, the dynamic of f˜ is described explicitly as a subshift of finite type which generalizes the Fibonacci shifts and this fact is used to compute topological entropy of f.Moreover we show that in general ([−1,0],f̃) is related to a factor of the k-bonacci shift. In particular we found that the 1-dimensional Lorenz map replicates an interesting duplicating behaviour of the k-bonacci shift found in Sirvent (1996, 2011).

Topological dynamics of Markov multi-maps of the interval

We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we define the forward trajectory system of a Markov multi-map, and we investigate whether it has properties such as topological transitivity, topological mixing, density of periodic points, and specification. To each Markov multi-map we associate a shift of finite type (SFT), and our main results relate the properties of the SFT with those of the forward trajectory system. Under a general coding condition, we establish necessary and sufficient conditions for topological transitivity and topological mixing of the forward trajectory system. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.

Parametrized quantum circuit for weight-adjustable quantum loop gas

Motivated by the recent success of realizing the topologically ordered ground state of the exactly solvable toric code model by a quantum circuit on the real quantum device [K. J. Satzinger {\it et al}., Science \textbf{374}, 1237 (2021)], here we propose a parametrized quantum circuit (PQC) with the same real-device-performable optimal structure to represent quantum loop gas states with adjustably weighted loop configurations. Combining such a PQC with the variational quantum eigensolver, we obtain the accurate quantum circuit representation for the toric code model in an external magnetic field with any field strength, where the system is not exactly solvable. The topological quantum phase transition in this system is further observed in the optimized circuits by measuring the magnetization and topological entanglement entropy.

Topological Entropy and Metric Entropy for Regular Impulsive Semiflows

Topological Entropy and Metric Entropy for Regular Impulsive Semiflows

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.

Chaotic light scattering around extremal black holes

We show that the scattering of light in the field of Nge 3 static extremal black holes is chaotic in the planar case. The relativistic dynamics of such extremal objects reduce to that of a classical Hamiltonian system. Certain values of the dilaton coupling then allow one to apply techniques from symbolic dynamics and classical potential scattering. This results in a lower bound on the topological entropy of order log (N-1), thus proving the emergence of chaotic scattering for N ge 3 black holes.

Relative topological conditional entropy and a Ledrappier's type variational principle for it

In this paper we provide a sufficient condition for the existence of invariant measures with maximal relative measure-theoretic entropy, by introducing a new invariant for any factor map between topological dynamical systems, the concept of relative topological conditional entropy. It is proved a Ledrappier's type variational principle concerning relative topological conditional entropy. Consequently, if the factor map has zero relative topological conditional entropy (such a factor map is called asymptotically $ h $-expansive), then there exist invariant measures with maximal relative measure-theoretic entropy (i.e., whose relative measure-theoretic entropy equals exactly relative topological conditional entropy of the factor map). We explore further properties of relative topological conditional entropy, for example, the addition formula for a product factor map, the estimation of it for composition of two factor maps, and so on. We also interpret relative topological conditional entropy, respectively, using relative topological entropy of subsets and Bowen's dimensional entropy.

An analogue of topological sequence entropy for Markov hom tree-shifts

In this article, an analogue $h^S_{\rm top}$ of topological sequence entropy is defined for Markov hom tree-shifts. We explore various aspects of $h^S_{\rm top}$, including the existence of the limit in the definition, its relationship to topological entr

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.

Symbolic Extensions of Amenable Group Actions and the Comparison Property

In topological dynamics, the Symbolic Extension Entropy Theorem (SEET) (Boyle and Downarowicz, 2004) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift on finitely many symbols. The theorem gives a precise estimate on the entropy of such a symbolic extension (and hence on the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov–Sinai entropy serves as an estimate in an analogous problem, in the topological setup the task reaches beyond the classical theory of measure-theoretic and topological entropy. Necessary are tools from an extended theory of entropy, the theory of entropy structures developed in Downarowicz (2005). The main goal of this paper is to prove the analog of the SEET for actions of (discrete infinite) countable amenable groups: Let a countable amenable group G G act by homeomorphisms on a compact metric space X X and let M G ( X ) \mathcal {M}_{G}(X) denote the simplex of all G G -invariant Borel probability measures on X X . A function E A {E}_{\mathsf {A}} on M G ( X ) \mathcal {M}_{G}(X) equals the extension entropy function h π h^\pi of a symbolic extension π : ( Y , G ) → ( X , G ) \pi :(Y,G)\to (X,G) , where h π ( μ ) = sup { h ν ( Y , G ) : ν ∈ π − 1 ( μ ) } h^\pi (\mu )=\sup \{h_\nu (Y,G): \nu \in \pi ^{-1}(\mu )\} ( μ ∈ M G ( X ) \mu \in \mathcal {M}_{G}(X) ), if and only if E A {E}_{\mathsf {A}} is a finite affine superenvelope of the entropy structure of ( X , G ) (X,G) . Of course, the statement is preceded by the presentation of the concepts of an entropy structure and its superenvelopes, adapted from the case of Z \mathbb {Z} -actions. In full generality we are able to prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of several earlier works (e.g. Downarowicz and Huczek (2018) and Downarowicz, Huczek, and Zhang (2019)) and in this paper we review and complement the theory developed there. The full version of the SEET (with genuine symbolic extensions) is proved for groups which are either residually finite or enjoy the so-called comparison property. In order to describe the range of our theorem more clearly, we devote a large portion of the paper to studying the comparison property. Our most important result in this aspect is showing that all subexponential groups have the comparison property (and thus satisfy the SEET). To summarize, the heart of the paper is the presentation of the following four major topics and the interplay between them: Symbolic extensions, Entropy structures, Tiling systems (and their encodability), The comparison property.

Slow entropy for some Anosov-Katok diffeomorphisms

<p style='text-indent:20px;'>The Anosov-Katok method is one of the most powerful tools of constructing smooth volume-preserving diffeomorphisms of entropy zero with prescribed ergodic or topological properties. To measure the complexity of systems with entropy zero, invariants like slow entropy have been introduced. In this article we develop several mechanisms facilitating computation of topological and measure-theoretic slow entropy of Anosov-Katok diffeomorphisms.</p>

Formulae of volume growth for random Anosov diffeomorphisms

For $ C^2 $ random Anosov diffeomorphisms, we give a formula describing the volume growth by measure-theoretic fiber entropy and Lyapunov exponents. Furthermore, the topological fiber entropy coincides with the volume growth for $ C^2 $ fiber hyperbolic systems, which was for $ C^{\infty} $ random dynamical systems in [15] and $ C^2 $ Axiom A systems in [10]. We establish two tools, a volume lemma and a variational principle for random radii in $ C^2 $ random Anosov diffeomorphisms.

Review on contraction analysis and computation of contraction metrics

<p style='text-indent:20px;'>Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the solutions under consideration. Using an appropriate metric, with respect to which the distance is contracting, one can show convergence to a unique equilibrium or, if attraction only occurs in certain directions, to a periodic orbit.</p><p style='text-indent:20px;'>Contraction analysis was originally considered for ordinary differential equations, but has been extended to discrete-time systems, control systems, delay equations and many other types of systems. Moreover, similar techniques can be applied for the estimation of the dimension of attractors and for the estimation of different notions of entropy (including topological entropy).</p><p style='text-indent:20px;'>This review attempts to link the references in both the mathematical and the engineering literature and, furthermore, point out the recent developments and algorithms in the computation of contraction metrics.</p>

Topological equicontinuity and topological uniform rigidity for dynamical system

In this paper, we study topological equicontinuity, topological uniform rigidity and their properties. For a dynamical system, on a compact, T3 space, we study relations among the set of recurrent points of the map, the set of non-wandering points of the map and the intersection of the range sets of all iterations of the map. We define topological version of uniform rigidity and show that on a compact and T3 space any dynamical system is topologically uniformly rigid if it is first countable, almost topologically equicontinuous and transitive or it is second countable, topologically equicontinuous and has a dense set of periodic points. We show that a topologically uniformly rigid dynamical system, on a compact, Hausdorff space, has zero topological entropy. Moreover, we provide necessary examples and counterexamples.

Set of definitely attained values of the topological entropy of continuous mappings of the Cantor set

Установлено, что в любой окрестности каждого непрерывного отображения совершенного множества Кантора найдется отображение с заданной топологической энтропией.

Rigorous numerical study of the Colpitts oscillator with an exponential nonlinearity.

The dynamics of the Colpitts oscillator with an exponential nonlinearity is investigated using rigorous interval arithmetic based tools. The existence of various types of periodic attractors is proved using the interval Newton method. The main results involve the chaotic case for which a trapping region for the associated return map is constructed and a rigorous lower bound for the value of the topological entropy is computed, thus proving that the system is chaotic in the topological sense. A systematic search for unstable periodic orbits embedded in the chaotic attractor is carried out and the results are used to obtain an accurate approximation of the topological entropy of the system.

A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems

Abstract This article is devoted to the study of the Bowen topological entropy for nonautonomous dynamical systems, which is an extension of the classical definition of Bowen topological entropy. We show that the Bowen topological entropy can be determined by the local entropies of measures for nonautonomous dynamical systems, which extends Ma and Wen’s result.

Topological entropy of pseudo-Anosov maps on punctured surfaces vs. homology of mapping tori

We investigate the relation between the topological entropy of pseudo-Anosov maps on surfaces with punctures and the rank of the first homology of their mapping tori. On the surface \(S\) of genus \(g\) with \(n\) punctures, we show that the minimal entropy of a pseudo-Anosov map is bounded from above by \(\dfrac{(k+1)\log(k+3)}{|\chi(S)|}\) up to a constant multiple when the rank of the first homology of the mapping torus is \(k+1\) and \(k, g, n\) satisfy a certain assumption. This is a partial generalization of precedent works of Tsai and Agol-Leininger-Margalit.

Revisiting the dynamic of q-deformed logistic maps

We consider the logistic family and apply the q-deformation ϕq(x)=1−qx1−q. We study the stability regions of the fixed points of the q-deformed logistic map and the regions where the dynamic is complex through topological entropy and Lyapunov exponents. Our results show that the dynamic of this deformed family is richer than that of the q-deformed family studied in Cánovas (2022).

Topological characterization and entropy measures of large cavity cycloarene oligomers

Topological characterization and entropy measures of large cavity cycloarene oligomers