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Dynamical systems are one of the important research topics that have applications in many disciplines. In particular, investigating whether a dynamical system is chaotic gives us important information about the dynamical system. In this paper, some dynamical systems are defined on the Cantor dust C x C inspired by the construction of Smale's horseshoe map. We also express these dynamical systems via ternary numbers and compute their periodic points. We thus obtain some dynamical systems which are chaotic in the sense of Devaney. Finally, we compare these dynamical systems according to whether they are topologically equivalent or not. [ABSTRACT FROM AUTHOR]

We will characterize topologically conjugate one-sided topological Markov shifts in terms of their subgroups of continuous full groups and subalgebras of Cuntz–Krieger algebras. [ABSTRACT FROM AUTHOR]

In this paper, we focus on some properties, calculations and estimations of topological entropy for a nonautonomous dynamical system (X , f 0 , ∞) generated by a sequence of continuous self-maps f 0 , ∞ = { f n } n = 0 ∞ on a compact uniform space X. We prove that (X , f 0 , ∞) and its k -th product system have the same entropy. We confirm that the entropy of (X , f 0 , ∞) equals to that of its n -th compositions system, and also equals to the entropy of f 0 , ∞ restricted to its non-wandering set if f 0 , ∞ is equi-continuous. We prove that the entropy of (X , f 0 , ∞) is less than or equal to that of its limit system (X , f) when f 0 , ∞ converges uniformly to f. We show that two topologically equi-semiconjugate systems have the same entropy if the equi-semiconjugacy is finite-to-one. Finally, we obtain the estimations of upper and lower bounds of entropy for an invariant subsystem of a coupled-expanding system associated with a transition matrix. [ABSTRACT FROM AUTHOR]

This paper characterizes a continuous map s such that a measure preserving interval map t exists to be topologically conjugate with s. We show that under mild assumptions this problem is reduced to a Markov chain problem. We derive the conditions for such a t to exist and be unique. [ABSTRACT FROM AUTHOR]

We show that every intermediate β -transformation is topologically conjugate to a greedy β -transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate β -transformation, complementing the result of Li, Sahlsten, Samuel and Steiner [22], and (2) obtain new metric and topological results on survivor sets of intermediate β -transformations with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [18]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate β -transformations one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [15], Tseng [35] and Färm, Persson and Schmeling [11]. [ABSTRACT FROM AUTHOR]

In this paper, it is aimed to construct two different dynamical systems on the Sierpinski tetrahedron. To this end, we consider the dynamical systems on a quotient space of {0, 1, 2, 3}N by using the code representations of the points on the Sierpinski tetrahedron. Finally, we compare the periodic points to investigate topological conjugacy of these dynamical systems and we conclude that they are not topologically equivalent. [ABSTRACT FROM AUTHOR]

We consider the real polynomials of degree d + 1 with a fixed point of multiplicity d = 2. Such polynomials are conjugate to fa,d(x) = axd(x - 1) + x, a ? R \ {0}. In this family, the point 0 is always a non-hyperbolic fixed point. We prove that for given d, d ', and a, where d and d ' are positive even numbers and a belongs to a special subset of R-, there is a ' < 0 such that fa,d is topologically conjugate to fa',d'. Then we extend the properties that we have studied in case d = 2 to this family for every even d > 2. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the paper, Matsumoto ['Subshifts, $\lambda $ -graph bisystems and $C^*$ -algebras', J. Math. Anal. Appl. 485 (2020), 123843]. A $\lambda $ -graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift $\Lambda $ , there exists a $\lambda $ -graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ with shift automorphism $\rho _{\mathcal {L}}$ from a $\lambda $ -graph bisystem $({\mathcal {L}}^-,{\mathcal {L}}^+)$ , and define a $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ by the crossed product. It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If $\lambda $ -graph bisystems come from two-sided subshifts, these $C^*$ -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ and the K-theory formulas of the $C^*$ -algebras ${\mathcal {F}}_{\mathcal {L}}$ and ${\mathcal R}_{\mathcal {L}}$. The K-group for the AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ is regarded as a two-sided extension of the dimension group of subshifts. [ABSTRACT FROM AUTHOR]

This paper mainly considers the complex dynamics of a discrete-time ring neural network with multiple time delays. By transforming the network system into a system of difference equations, one can obtain a chaotic neural network. More specifically, by projection method one can determine a closed invariant set of the system governed by some difference equations, and prove that some invariant subsystem is topologically conjugate to the two-sided symbolic dynamical system. Moreover, numerical simulations are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Topological conjugacy plays an important role in simplifying the systems and equations in the research of dynamical systems and functional equations. In this paper, we deal with increasing upper semi-continuous multifunctions with finitely many set-valued points. Some sufficient and necessary conditions are given for the topological conjugate relation between those multifunctions, and the methods of construction for all conjugacies are also presented. Moreover, as an application, our results are applied to study the problem of iterative roots for the multifunctions with a unique set-valued point. [ABSTRACT FROM AUTHOR]

We consider several computational problems related to conjugacy between subshifts of finite type, restricted to k -block codes: verifying a proposed k -block conjugacy, deciding if two shifts admit a k -block conjugacy, and reducing the representation size of a shift via a k -block conjugacy. We give a polynomial-time algorithm for verification, and show GI - and NP -hardness for deciding conjugacy and reducing representation size, respectively. Our approach focuses on 1-block conjugacies between vertex shifts, from which we generalize to k -block conjugacies and to edge shifts. We conclude with several open problems. [ABSTRACT FROM AUTHOR]

A natural class of expansive endomorphisms of the infinite-dimensional torus (the Cartesian product of countably many circles with the product topology) is considered. The endomorphisms in this class can be represented in the form of the sum of a linear expansion and a periodic addition. The following standard facts of hyperbolic theory are proved: the topological conjugacy of any expansive endomorphism from the class under consideration to a linear endomorphism of the torus, the structural stability of , and the topological mixing property of on . [ABSTRACT FROM AUTHOR]

A λ-graph system L is a labelled Bratteli diagram with shift operation. It is a generalized notion of finite labelled graph and presents a subshift. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and C ∗ -algebras constructed from λ-graph systems. [ABSTRACT FROM AUTHOR]

We give a global geometric decomposition of continuously differentiable vector fields on R n . More precisely, given a vector field of class C 1 on R n , and a geometric structure on R n , we provide a unique global decomposition of the vector field as the sum of a left (right) gradient-like vector field (naturally associated to the geometric structure) with potential function vanishing at the origin, and a vector field which is left (right) orthogonal to the Euler vector field, with respect to the geometric structure. As application, we provide a criterion to decide topological conjugacy of complete vector fields of class C 1 on R n based on topological conjugacy of the corresponding parts given by the associated geometric decompositions. [ABSTRACT FROM AUTHOR]

Topological conjugacy between multifunctions plays an important role in the research of dynamical systems. This paper deals with conjugacies of strictly increasing and upper semicontinuous multifunctions on a closed interval with finite set-valued points. All conjugacies between these two multifunctions are constructed with an extension method. The C 1 smoothness of the conjugacies is also given. Moreover, an example is presented to illustrate the applications of the main results. [ABSTRACT FROM AUTHOR]

We study the problem of topological conjugacy of Brouwer flows. We give a sufficient and necessary condition for Brouwer flows to be topologically conjugate. To obtain this result we use a cover of the plane by maximal parallelizable regions and relations between parallelizing homeomorphisms of these regions. We show that for topologically equivalent Brouwer flows there exists a one-to-one correspondence between such covers of the plane. [ABSTRACT FROM AUTHOR]

In the present paper, we define the concept of weak topological conjugacy and we establish sufficient conditions to obtain this kind of topological conjugacy between two limit sets. We use the character of recurrence to obtain the results. [ABSTRACT FROM AUTHOR]

In this paper we strengthen our previous results related to a construction of some type of one-sided cellular automata (CA) in the metric Cantor space Ãℕ(Ãℤ). We proved that the mentioned CA have the following properties. They are not positively expansive and like in positively expansive case in Ãℕ, they are right-closing, open, topologically mixing, strongly transitive, and their topological entropies form the set H = {log n : n ≥ 2, n ∈ ℕ}. Additionally, they show a significant divergence in the range of an another invariant of topological conjugacy, i.e., the cardinality of the set of m--periodic points. Their uniqueness lies also in the fact that they have continuum of 2--periodic points. Consequently, in this paper, we justify that they are not topologically conjugate to a wider set of subclasses of transitive CA in Ãℕ(Ãℤ) which are often viewed as ergodic or suspected to be ergodic systems with respect to the Borel uniform Bernoulli measure. We also prove that like in positively expansive case in Ãℕ(Ãℤ), the mentioned probability measure is a measure of maximal entropy for these CA. It broadens an answer to the question raised by F. Blanchard, A. Maass (1997> Is it possible to extend some of the results obtained for positively expansive CA in Ãℕ to a certain subclass of right-closing and non positively expansive CA in this space? [ABSTRACT FROM AUTHOR]

It is suggested in a literature of the subject to undertake a research on transitivity of cellular automata (CA) and its stronger variants in the metric Cantor space B N(B Z) as well as on possibilities of classification of transitive CA in this space up to topological conjugacy. In the response to these suggestions, we present an alternative, simple proof of the strong transitivity of surjective CA in B N with memory m > 0 according to the stronger variant of this notion. Then, generalising a known method, we prove that for any integer m > 0 a set of m--periodic points of this type CA is finite. We also prove that extensions, from B N to B Z, of positively expansive CA and constructions of CA which are bijective, expansive and topologically conjugate to two-sided full shifts, also have this property. Obtained in such a way one-sided CA in B Z and surjective CA in B N with memory m > 0 are topologically mixing and they are not topologically conjugate to one-sided open, topologically mixing and strongly transitive CA in B N(B Z) with a continuum of 2--periodic points constructed in our previous papers. [ABSTRACT FROM AUTHOR]

In previous work, a topological conjugacy between PM functions and their iterative roots, which are increasing on the characteristic interval, was considered. In the present paper, we continue to investigate those functions that are decreasing on the characteristic interval. The differences come from the fact that neither orientation nor monotonicity can be preserved under iteration in the decreasing case. Thus, we need to partition the whole interval with a different manner and then prove the existence of topological conjugacy on each sub-interval. We also give a construction method for the conjugacies. [ABSTRACT FROM AUTHOR]

The topological conjugacy classification of elementary cellular automata with majority memory (ECAMs) is studied under the framework of symbolic dynamics. In the light of the conventional symbolic sequence space, the compact symbolic vector space is identified with a feasible metric and topology. A slight change is introduced to present that all global maps of ECAMs are continuous functions, thereafter generating the compact dynamical systems. By exploiting two fundamental homeomorphisms in symbolic vector space, all ECAMs are furthermore grouped into 88 equivalence classes in the sense that different mappings in the same global equivalence are mutually topologically conjugate. [ABSTRACT FROM AUTHOR]

λ-Graph systems are labelled Bratteli diagrams with shift operations. They present subshifts. Their matrix presentations are called symbolic matrix systems. We define skew products of λ-graph systems and study extensions of subshifts by finite groups. We prove that two canonical symbolic matrix systems areG-strong shift equivalent if and only if their presented subshifts areG-conjugate. [ABSTRACT FROM AUTHOR]

The paper develops a more general notion of dichotomy, referred to as “nonuniform ( h , k , μ , ν ) -dichotomy”. The new notion unifies most versions of existing dichotomy in the literature, includes them in one comprehensive mathematics, and reveals more dichotomic behaviors of dynamical systems. Then we show that any linear nonautonomous system admits a nonuniform ( h , k , μ , ν ) -dichotomy if it has an ( h , k ) Lyapunov exponent with different signs or a Lyapunov function with different growth rates in the stable and unstable subspaces of solutions. This implies that the nonuniform ( h , k , μ , ν ) -dichotomy arises naturally and exists widely in the linear systems. Using the general notation, new versions of robustness, Hartman–Grobman theorem, and stable invariant manifold theorem for nonautonomous dynamical systems in Banach spaces will be presented. Especially, these new results not only generalize the corresponding ones in the literature, but also characterize the influences of different growth functions in the stable and unstable subspaces and in the uniform and nonuniform parts of a linear system for solving the above problems. This improves the application range of dichotomy in the nonautonomous systems. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the complexity of topological conjugacy of pointed Cantor minimal systems from the point of view of descriptive set theory. We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation $$\varDelta _{\mathbb {R}}^+$$ on $$\mathbb {R}^{{\mathbb {N}}}$$ defined by $$x \varDelta _{\mathbb {R}}^+y \Leftrightarrow \{x_i{:}\,i \in {\mathbb {N}}\}=\{y_i{:}\,i \in {\mathbb {N}}\}$$ . Moreover, we show that $$\varDelta _{\mathbb {R}}^+$$ is a lower bound for the Borel complexity of topological conjugacy of Cantor minimal systems. Finally, we interpret our results in terms of properly ordered Bratteli diagrams and discuss some applications. [ABSTRACT FROM AUTHOR]

Homeomorphisms allowing us to prove topological equivalences between one-parameter families of maps undergoing the same bifurcation are constructed in this paper. This provides a solution for a classical problem in bifurcation theory that was set out three decades ago and remained unexpectedly unpublished until now. [ABSTRACT FROM AUTHOR]

We characterize the symbolical dynamical systems which are topologically isomorphic to the Fibonacci dynamical system. We prove that there are infinitely many injective primitive substitutions generating a dynamical system in the Fibonacci conjugacy class. In this class there are infinitely many dynamical systems not generated by a substitution. An example is the system generated by doubling the 0's in the infinite Fibonacci word. [ABSTRACT FROM AUTHOR]

In this paper, some necessary and sufficient conditions are given for maps in Hausdorff spaces to be topologically conjugate or semi-conjugate to subshifts of finite type. These results not only extend some related existing results in metric spaces to Hausdorff spaces but also relax their conditions. In addition, it is shown that the strictA-coupled-expansion for a map in a Hausdorff space is a sufficient condition for it to be semi-conjugate to the subshift σAin Σ+m(A). Based on these results, two new criteria of chaos are established, in which the maps are shown to be chaotic either in the sense of both Devaney and Li-Yorke or in the sense of Li-Yorke. [ABSTRACT FROM PUBLISHER]

Topological conjugacy for continuous skew product flows on topological group bundles and semidirect products is studied resulting in a splitting theorem. The results are used to discuss fiber entropy. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general class of discrete multiple-time delayed recurrent neural networks with external inputs. By applying a new transformation, we transform an m-neuron network model into a parameterized map from to . A chaotic invariant set of the neural networks system is obtained by using a family of projections from onto . Furthermore, we prove that the dynamics of this neural networks system restricted to the chaotic invariant set is topologically conjugate to the dynamics of the full shift map with two symbols, which indicates that chaos occurs. Numerical simulations are presented to illustrate the theoretical outcomes. [ABSTRACT FROM AUTHOR]

This paper aims to characterize whether a multi-layer cellular neural network is of deep architecture; namely, when can an n -layer cellular neural network be replaced by an m -layer cellular neural network for m < n yet still preserve the same output phenomena? From a mathematical point of view, such characterization involves investigating whether the topological structure of two (or multiple) layers is conjugate. A decision procedure that addresses the necessary and sufficient condition for the topological conjugacy between two layers in a network is revealed. [ABSTRACT FROM AUTHOR]

In this paper, we study stability of non-autonomous discrete dynamical systems. For a two-sided non-autonomous systems with covering relations determined by a transition matrixA, we show that any smallC0perturbed system has a sequence of compact invariant sets restricted to which the system is topologically semi-conjugate to σA, the two-sided subshift of finite type induced byA. Together with Lyapunov condition of good rate, the semi-conjugacy will become conjugacy. Moreover, if the Lyapunov condition is strict and has perfect rate, then any smallC1perturbed systems is topological conjugate to σA. We also study topological chaos of one-sided systems and systems with limit functions. Lack of hyperbolicity and the time dependence of the rate prevent us from applying classical hyperbolic results or earlier works for autonomous systems: cone condition and Lyapunov function. Two examples are provided to demonstrate the existence of a non-trivial, non-hyperbolic invariant and essential of controlling rate. [ABSTRACT FROM AUTHOR]

This paper investigates conjugacies between asymmetric Bernoulli shifts. It is shown that there exist a unique increasing conjugacy and a unique decreasing conjugacy. We respectively construct a sequence of functions to approximate these two conjugacies, and give an estimation for the error of the approximation. We also present explicit formulae of these two conjugacies. It is shown that these two conjugacies are singular, Hölder continuous and not differentiable. [ABSTRACT FROM AUTHOR]

In this paper, we give two examples to show that an invertible mapping being Li-Yorke chaotic does not imply its inverse being Li-Yorke chaotic, in which one is an invertible bounded linear operator on an infinite dimensional Hilbert space and the other is a homeomorphism on the unit open disk. Moreover, we use the last example to prove that Li-Yorke chaos is not preserved under topological conjugacy. [ABSTRACT FROM AUTHOR]

We show that the family of the Smale-Williams solenoid attractors parameterized by its contraction rate can be characterized as solutions of a set of differential equations. The exact formula describing the attractor can be obtained by solving the differential equations subject to explicitly given initial conditions. Using the formula, we present in this note a simple and explicit proof of the result that the dynamics on the solenoid is topologically conjugate to the shift on the inverse limit space of the expanding map t ↦ mt mod 1 for some integer m ≥ 2 and to a suspension over the adding machine. [ABSTRACT FROM AUTHOR]

This paper investigates the conjugacy of any two skew tent maps. An explicit formula is given for the conjugacy. It is proved that the conjugacy is singular, Hölder continuous and not differentiable as well as its inverse. We calculate the arc-length of the conjugacy curve and the area under the conjugacy curve. We construct a sequence of functions to approximate the conjugacy, and give an estimation for the error of the approximation. [ABSTRACT FROM AUTHOR]

Let AC( M, SL( d,ℝ)) denote the pairs ( f, A) so that f ∈ A ⊂ Diff( M) is a C-Anosov transitive diffeomorphisms and A is an SL( d,ℝ) cocycle dominated with respect to f. We prove that open and densely in AC( M, SL( d,ℝ)), in appropriate topologies, the pair ( f, A) has simple spectrum with respect to the unique maximal entropy measure µ. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in Aut( M) × L( M, SL( d,ℝ)). [ABSTRACT FROM AUTHOR]

For a topological dynamical system (X, f), consisting of a continuous map f : X → X, and a (not necessarily compact) set Z ⊂ X, Bowen (1973), defined a dimension-like version of entropy, hX(f,Z). In the same work, he introduced a notion of entropy-conjugacy for pairs of invertible compact systems: the systems (X, f) and (Y, g) are entropy-conjugate if there exist invariant Borel sets X' ⊂ X and Y' ⊂ Y such that hX(f,X \ X') < hX(f,X), hY (g, Y \ Y') < hY (g, Y), and (X', f|X') is topologically conjugate to (Y', g|Y'). Bowen conjectured that two mixing shifts of finite type are entropy-conjugate if they have the same entropy. We prove that two mixing shifts of finite type with equal entropy and left ideal class are entropyconjugate. Consequently, in every entropy class Bowen’s conjecture is true up to finite index. [ABSTRACT FROM AUTHOR]

Floyd-Auslander systems form an essential class of minimal non-homogeneous systems, moreover, their dynamics is realised by homeomorphisms. We give a full classification up to topological conjugacy of Floyd-Auslander systems with fixed subdivision pattern. Besides the trivial identity conjugacy, any Floyd-Auslander system is conjugate only to the Floyd-Auslander system defined by the mirror subdivision pattern. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study distributional chaos for bounded linear operators. We show that distributional chaos of type k ∈ {1,2} is an invariant of topological conjugacy between two bounded linear operators. We give a necessary condition for distributional chaos of type 2 where it is possible to distinguish distributional chaos and Li-Yorke chaos. Following this condition, we compare distributional chaos with other well-studied notions of chaos for backward weighted shift operators and give an alternative proof to the one where strong mixing does not imply distributional chaos of type 2 (Martínez-Giménez F, Oprocha P, Peris A. Distributional chaos for operators with full scrambled sets. Math Z 2013; 274: 603-612.). Moreover, we also prove that there exists an invertible bilateral forward weighted shift operator such that it is DC1 but its inverse is not DC2. [ABSTRACT FROM AUTHOR]

The conjugacy problem is an important topic in the theory of dynamical systems and functional equations. In this paper, we investigate a class of piecewise monotone and continuous maps with nonmonotonicity height ≥1. We give a sufficient and necessary condition under which any two of these maps are topologically conjugate, and construct a topological conjugacy with an extension method if such a conjugacy exists. [ABSTRACT FROM AUTHOR]

For a dynamical system (X, f), the passage of various dynamical properties such as transitivity, total transitivity, weakly mixing, mixing, topological exactness, topological conjugacy, to the hyperspace C(X) of X consisting of nonempty closed connected subsets of X, and also to the hyperspace suspension HS(X) of X, have been considered and studied. [ABSTRACT FROM AUTHOR]

We show that if f : S1 × S1 → S1 × S1 is C2, with f(x, t) = (ft(x), t), and the rotation number of ft is equal to t for all t ∊ S1, then f is topologically conjugate to the linear Dehn twist of the torus (1 1 0 1). We prove a differentiability result where the assumption that the rotation number of ft is t is weakened to say that the rotation number is strictly monotone in t. [ABSTRACT FROM AUTHOR]

Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincaré map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. This work proposes a method for obtaining explicit Poincaré mappings by using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. Indeed, the enforcement of topological conjugacies is the critical neural network regularization for learning the coordinate and dynamics pairing. We provide expository applications of the method to low-dimensional systems such as the Rössler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto–Sivashinsky equation. • Machine learning is leveraged to obtain accurate and parsimonious conjugate Poincare mappings. • Method can understand and classify the topology of a chaotic attractor using the equivalence class of dynamical conjugacies. • Used to seed initial guesses for systematically extracting unstable periodic orbits embedded in the chaotic attractor. • Examples include the Rossler, Lorenz, and Kuramoto-Sivashinsky equations. [ABSTRACT FROM AUTHOR]

Topological conjugacy or conjugacy plays a fundamental role in studying the dynamics of continuous maps and functional equations. In this paper, we construct all homeomorphic solutions and continuously non-monotone solutions of the conjugacy equation, where,are two givenr-modal interval maps with non-monotonicity height equal to 1, andI,Jare closed intervals. Moreover, the sufficient conditions are presented for the existence ofhomeomorphic solutions andnon-monotone solutions, respectively. Finally, as an application, we give a result of iterative roots of anr-modal interval map with non-monotonicity height equal to 1, as well as one numerical example. [ABSTRACT FROM PUBLISHER]

In this paper, we study the flows of nonzero left invariant vector fields on Lie groups with respect to topological conjugacy. Using the fundamental domain method, we are able to show that on a simply connected nilpotent Lie group any such flows are topologically conjugate. Combining this result with the Iwasawa decomposition, we find that on a noncompact semisimple Lie group the flows of two nilpotent or abelian fields are topologically conjugate. Finally, for affine groups G = HV , V ≅ n, we show that the conjugacy class of a left invariant vector field does not depend on its Euclidean component. [ABSTRACT FROM AUTHOR]

In this paper two aspects of numerical dynamics are used for an artificial neural network (ANN) analysis. It is shown that topological conjugacy of gradient dynamical systems and both the shadowing and inverse shadowing properties have nontrivial implications in the analysis of a perceptron learning process. The main result is that, generically, any such process is stable under numerics and robust. Implementation aspects are discussed as well. The analysis is based on the theorem concerning global topological conjugacy of cascades generated by a gradient flow on a compact manifold without a boundary. [ABSTRACT FROM AUTHOR]