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Monounary algebras are the most simple type of an algebraic structure. Oriented graphs with one outgoing arrow from every vertex represent them. The aim of this paper is to point out the interdisciplinary relationships concerning this structure. Bernoulli shift is a paradigmatic mapping in dynamical systems. It is also called dyadic, bit shift, doubling or sawtooth. We offer a look at the properties of this mapping via monounary algebras. [ABSTRACT FROM AUTHOR]

We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition. [ABSTRACT FROM AUTHOR]

We study a generalized Hénon map in two-dimensional space. We find a region of the phase space where the nonwandering set exists, specify parameter values for which this nonwandering set is hyperbolic, and prove that our map when restricted to a specific invariant subset is topologically conjugate to the Bernoulli three-shift. Coupling two such maps, as a result, we obtain a map in four-dimensional space and show that Bernoulli shifts also exist in this map. [ABSTRACT FROM AUTHOR]

We study a generalized Hénon map in two-dimensional space. We find a region of the phase space where the nonwandering set exists, specify parameter values for which this nonwandering set is hyperbolic, and prove that our map when restricted to a specific invariant subset is topologically conjugate to the Bernoulli three-shift. Coupling two such maps, as a result, we obtain a map in four-dimensional space and show that Bernoulli shifts also exist in this map. [ABSTRACT FROM AUTHOR]

The algebraic dimension of a Polish permutation group Q ≤ Sym (N) is the size of the largest A ⊆ N with the property that the orbit of every a ∈ A under the pointwise stabilizer of A ∖ { a } is infinite. We study the Bernoulli shift P ↷ R N for various Polish permutation groups P and we provide criteria under which the P -shift is generically ergodic relative to the injective part of the Q -shift, when Q has algebraic dimension ≤ n. We use this to show that the sequence of pairwise ⁎-reduction-incomparable equivalence relations defined in [18] is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal ℵ 1 + which strongly resembles the equivalence relation of pinned cardinal ℵ 1 + from [25] , but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions. Our proofs rely on the study of symmetric models whose symmetries come from the group Q. We show that when Q is "locally finite"—e.g. when Q = Aut (M) , where M is the Fraïssé limit of a Fraïssé class satisfying the disjoint amalgamation property—the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model. [ABSTRACT FROM AUTHOR]

We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy), then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory. [ABSTRACT FROM AUTHOR]

Inverse limit spaces of compacta with upper semi-continuous compact set-valued functions are studied via shift maps and their variants. We give a representation of such spaces as the limits of ordinary inverse sequences, which allows us to prove some known results and their extensions in a unified scheme. Next we present a scheme to construct shift dynamics on the inverse limit space with various dynamical features. In particular we construct an inverse sequence over [ 0 , 1 ] with a single upper semi-continuous function f as its bonding function such that (i) the inverse limit space [ 0 , 1 ] f is homeomorphic to the Cantor set and (ii) the shift map σ f : [ 0 , 1 ] f → [ 0 , 1 ] f is topologically conjugate to a minimal subshift of a Bernoulli full shift. Also we study local/global connectivity of the inverse limit space over a compactum with a single upper semi-continuous bonding function in terms of homotopy/(co)homology groups, again via shift maps and their variants. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to locate trajectories of a perturbed system, which is known to behave chaotically. The unperturbed system is assumed to have the origin as a hyperbolic fixed point, and to admit a trajectory homoclinic to the origin. This homocline is assumed to lie in a prescribed region having the origin in its border. Using a Mel'nikov type approach, we introduce natural conditions ensuring that all the chaotic trajectories of the perturbed system, given by classical results, lie in the same region. The applicability of our results is illustrated in two examples. In the first one, we find positive radial solutions for a class of P.D.E.'s, obtaining new results in the case of critical equations ruled by Laplacian with Hardy potentials. In the other one, we show that under certain conditions one of two weakly coupled pendula moves in one direction only. [ABSTRACT FROM AUTHOR]

We prove a result on the topological entropy of a large class of Hamiltonian systems. This result is obtained variationally by constructing “multibump” homoclinic solutions. [ABSTRACT FROM AUTHOR]

We show that for any infinite countable group G and for any free Borel action G ↷ X there exists an equivariant class-bijective Borel map from X to the free part Free ( 2 G ) of the 2-shift G ↷ 2 G . This implies that any Borel structurability which holds for the equivalence relation generated by G ↷ Free ( 2 G ) must hold a fortiori for all equivalence relations coming from free Borel actions of G . A related consequence is that the Borel chromatic number of Free ( 2 G ) is the maximum among Borel chromatic numbers of free actions of G . This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic G -equivariant map to 2 G lands in the free part. As a corollary we obtain that for every ϵ > 0 , every free p.m.p. action of G has a free factor which admits a 2-piece generating partition with Shannon entropy less than ϵ . This generalizes a result of Danilenko and Park. [ABSTRACT FROM AUTHOR]

A measure-preserving action of a countably infinite group Γ is called totally ergodic if every infinite subgroup of Γ acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of Γ is totally ergodic, then there exists a finite normal subgroup N of Γ such that the stabilizer of almost every point is equal to N. Surprisingly, the proof relies on the group theoretic fact (proved by Hall and Kulatilaka, as well as by Kargapolov) that every infinite locally finite group contains an infinite abelian subgroup, of which all known proofs rely on the Feit-Thompson theorem. As a consequence, we deduce a group theoretic characterization of countable groups whose non-trivial Bernoulli factors are all free: these are precisely the groups that possess no finite normal subgroup other than the trivial subgroup. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to explore complex dynamics in elementary cellular automaton rule 26. Firstly, we study the dynamics of this automaton through its interval map and its subsystem respectively. The positive topological entropy explored in its subsystem indicates that this automaton is topologically transitive on its subsystem. After that, we also study its shift representation under the framework of general symbolic dynamics. We show that rule 26 can be expressed by the shift on symbolic space with two symbols. The obtained results indicate that this automaton exhibits rich dynamical behavior. [ABSTRACT FROM AUTHOR]

We show that for a given finitely generated group, its Bernoulli shift space can be equivariantly embedded as a subset of a space of pointed trees with Gromov–Hausdorff metric and natural partial action of a free group. Since the latter can be realized as a transverse space of a foliated space with leaves Riemannian manifolds, this embedding allows us to obtain a suspension of such Bernoulli shift. By a similar argument, we show that the space of pointed trees is universal for compactly generated expansive pseudogroups of transformations. [ABSTRACT FROM AUTHOR]

In this paper, the chaotic behavior of a set-valued mapping F : X → 2X, where X is a compact space, is investigated. The existence of the generalized shadowing property in the hyperspace 2X is proved. Based on the generalized shadowing property of the set-valued mappings F and the assumption of the existence of an unstable chain recurrent point of the mapping F, it is shown that the Bernoulli system of bi-directional shifts is embedded in the sense of semiconjugacy into the image of mapping F, i.e. Smale's chaos in the set-valued system F is thereby proved. [ABSTRACT FROM AUTHOR]

Abstract: We first study the problem of chaotic behaviour in time-perturbed discontinuous systems whose unperturbed part has a piecewise homoclinic solution transversally crossing the discontinuity manifolds. We show that if a certain Melnikov function has a simple zero at some point, then the system has solutions that behave chaotically. The Melnikov function is explicitly constructed for two-dimensional systems and extends the usual Melnikov function for the smooth case. In the second part, we extend these results to sliding homoclinic bifurcations. We also mention some possibilities for further research. [Copyright &y& Elsevier]

From the viewpoint of symbolic dynamics, the complex dynamical behaviors of rule 35 in cellular automata are investigated in this paper. Rule 35 which is Bernoulli σT-shift rule and is member of Wolfram's class II, is said to be simple as periodic before. It is worthwhile studying dynamical behaviors of four rules, whether they possess chaotic attractors or not We find that rule 35 possess positive topological entropy, and is topologically mixing on its attractors. Therefore, dynamical behaviors of rule 35 are chaotic in the sense of both Li-York and Devaney. Then, we prove that four rules belonging to global equivalence Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. are topologically conjugate. Diagrams is used to explain the attractors of rule 35, where characteristic function is used to describe that some points fall into Bernoulli-shift map after several times iterations. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a group and be a contracting homomorphism from a subgroup of finite index. V. Nekrashevych (2005) associated with the pair the limit dynamical system and the limit G-space together with the covering by the tile . We develop the theory of self-similar measures on these limit spaces. It is shown that is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles for . We present applications to the invariant measures for the rational functions on the Riemann sphere and to the evaluation of the Lebesgue measure of integral self-affine tiles. [ABSTRACT FROM AUTHOR]

Recently the strings and the string number of self-maps were used in the computation of the algebraic entropy of specific abelian group endomorphisms. We introduce two special kinds of strings, and their relative string numbers. We show that a dichotomy holds for all these three string numbers; in fact, they admit only zero and infinity as values on abelian group endomorphisms. [ABSTRACT FROM AUTHOR]

new possible geometry of an attractor of a dynamical system, a bony attractor, is described. A bony attractor is the union of two parts. The first part is the graph of a continuous function defined on a subset of ∑, the set of bi-infinite sequences of integers m in the range 0 ≤ m < k. The second part is the union of uncountably many intervals contained in the closure of the graph. An open set of skew products over the Bernoulli shift (σω) = ω with fiber [0,1] is constructed such that each system in this set has a bony attractor. [ABSTRACT FROM AUTHOR]

Abstract: We study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function in such a way that if has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given. [Copyright &y& Elsevier]

This paper gives a short historical survey of basic events which had happend during the developmentb of chaos theory. [ABSTRACT FROM AUTHOR]

Abstract: We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo-type equation , previously considered by Grindrod and Sleeman [P. Grindrod, B.D. Sleeman, A model of a myelinated nerve axon: threshold behaviour and propagation, J. Math. Biol. 23 (1985) 119–135. ] and by Chen and Bell [P.-L. Chen, J. Bell, Spine-density dependence of the qualitative behavior of a model of a nerve fiber with excitable spines, J. Math. Anal. Appl. 187 (1994) 384–410.] in the study of nerve fiber models. In some recent works we discussed the case of nonexistence of nontrivial solutions as well as the case in which many positive periodic solutions may arise, the different situations depending upon threshold parameters related to the weight function . Here we show that for a step function (or for small perturbations of it) it is possible to obtain infinitely many periodic solutions and chaotic dynamics, due to the presence of a topological horseshoe (according to Kennedy and Yorke). [Copyright &y& Elsevier]

We develop and study change point detection and estimation procedures for the covariance kernel of functional data based on the norms of a generally weighted process of partial sample estimates. It is shown under mild weak dependence and moment conditions on the data that in the absence of a change point a detector based on integrating such a process over the partial sample parameter is asymptotically distributed as the norm of a Gaussian process, which furnishes a consistent change point detection procedure. We further derive consistency and local asymptotic results for this detector in the presence of a change in the covariance function. The corresponding change point estimator based on such a process is also shown to be rate optimal for estimating an existing change point, and further is asymptotically distributed as the argument maximum of a Gaussian process under a local asymptotic framework. We study the detector and change point estimator in a small simulation study to detect changes in the covariance of functional autoregressive and generalized conditionally heteroscedastic processes, which demonstrate that the use of the weighted CUSUM statistics in this context generally improves performance over existing methods. These new statistics are demonstrated in an application to detecting changes in the volatility of high resolution intraday asset price curves derived from oil futures prices. [ABSTRACT FROM AUTHOR]

Our scientific odyssey through the theory of 1-D cellular automata is enriched by the definition of quasi-ergodicity, a new empirical property discovered by analyzing the time-1 return maps of local rules. Quasi-ergodicity plays a key role in the classification of rules into six groups: in fact, it is an exclusive characteristic of complex and hyper Bernoulli-shift rules. Besides introducing quasi-ergodicity, this paper answers several questions posed in the previous chapters of our quest. To start with, we offer a rigorous explanation of the fractal behavior of the time-1 characteristic functions, finding the equations that describe this phenomenon. Then, we propose a classification of rules according to the presence of Isles of Eden, and prove that only 28 local rules out of 256 do not have any of them; this result sheds light on the importance of Isles of Eden. A section of this paper is devoted to the characterization of Bernoulli basin-tree diagrams through modular arithmetic; the formulas obtained allow us to shorten drastically the number of cases to take into consideration during numerical simulations. Last but not least, we present some theorems about additive rules, including an analytical explanation of their scale-free property. [ABSTRACT FROM AUTHOR]

Abstract: This paper shows that blocks of different entries and different lengths can have the same probability of occurrence in the representation of any continued fraction which its orbit distributed according to Gauss measure. More precisely there are blocks of even lengths and 3-blocks which their probability of occurrence equals to the probability of occurrence of a unit block, and this property is not satisfied by the Bernoulli shift. [Copyright &y& Elsevier]

This paper presents the basin tree diagrams of all hyper Bernoulli στ-shift rules for string lengths L = 3, 4, ..., 8. These diagrams have revealed many global and time-asymptotic properties that we have subsequently proved to be true for all L < ∞. In particular, we have proved that local rule $\kern.5pt{\fboxsep2pt\fbox{60}}\kern.5pt\xspace$ has no Isles of Eden for all L, and that local rules $\kern.5pt{\fboxsep2pt\fbox{154}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{45}}\kern.5pt\xspace$ are inhabited by a dense set (continuum) of Isles of Eden if, and only if, L is an odd integer. A novel and powerful graph-theoretic tool, called Isles-of-Eden digraph, has been developed and can be used to test the existence of dense Isles of Eden of any local rule which satisfies certain constraints, such as rules $\kern.5pt{\fboxsep2pt\fbox{154}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{45}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$, as well as all invariant local rules, such as rules $\kern.5pt{\fboxsep2pt\fbox{170}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{240}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{15}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{85}}\kern.5pt\xspace$, subject to no constraints. [ABSTRACT FROM AUTHOR]

This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulliστ-shift rules and eight hyper Bernoulliστ-shift rules, the latter including such famous rules $\kern.5pt{\fboxsep2pt\fbox{30}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{110}}\kern.5pt\xspace$. All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞. Basin tree diagrams of all ten complex Bernoulli στ-shift rules are exhibited for lengths L = 3, 4, ..., 8. Superficial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. Two such properties form the main results of this paper; namely, Explicit global state transition formulas are given for local rules $\kern.5pt{\fboxsep2pt\fbox{90}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$. Such formulas led to the rigorous proof of several surprising periodicity constraints for rule $\kern.5pt{\fboxsep2pt\fbox{90}}\kern.5pt\xspace$, and to the discovery of a new global, quasi-equivalence class, defined via an alternating transformation. In particular, local rules $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$are globally quasi-equivalent where corresponding space-time patterns can be derived from each other by simply complementing every other row. Another important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules $\kern.5pt{\fboxsep2pt\fbox{90}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$. In particular, the period "T" of all attractors of rules $\kern.5pt{\fboxsep2pt\fbox{90}}\kern.5pt\xspace$, $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$, as well as of all isles of Eden of rules $\kern.5pt{\fboxsep2pt\fbox{105}}\kern.5pt\xspace$ and $\kern.5pt{\fboxsep2pt\fbox{150}}\kern.5pt\xspace$, increases linearly with unit slope, in logarithmic scale, with the length L. [ABSTRACT FROM AUTHOR]

This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin ruleN†. In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible. The remaining 86 CA rules are time-irreversible in the sense that N and N† mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time. A simple "time-reversal test" is given for testing whether an attractor of a CA rule is time-reversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N†, and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter–antimatter "annihilation" or "pair-production" phenomenon from high-energy physics, as well as to mimic the "contraction" or "expansion" scenarios associated with the Big Bang from cosmology. Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors. Using an analytical approach, the time-τ return map of each Bernoulli στ-shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,. . or its inverse map. These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes. Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach à la Wolfram to an attractor-based analytical theory of cellular automata. [ABSTRACT FROM AUTHOR]

This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unified global representation for all 256 one-dimensional Cellular Automata local rules. Except for eight rather special local rules whose global dynamics are described by an affine (mod 1) function of only one binary cell state variable, all characteristic functions exhibit a fractal geometry where self-similar two-dimensional substructures manifest themselves, ad infinitum, as the number of cells (I + 1) → ∞. In addition to a complete gallery of time-1 characteristic functions for all 256 local rules, an accompanying table of explicit formulas is given for generating these characteristic functions directly from binary bit-strings, as in a digital-to-analog converter. To illustrate the potential applications of these fundamental formulas, we prove rigorously that the "right-copycat" local rule ${\fbox{170}}$ is equivalent globally to the classic "left-shift" Bernoulli map. Similarly, we prove the "left-copycat" local rule ${\fbox{240}}$ is equivalent globally to the "right-shift" inverse Bernoulli map. Various geometrical and analytical properties have been identified from each characteristic function and explained rigorously. In particular, two-level stratified subpatterns found in most characteristic functions are shown to emerge if, and only if, b1 ≠ 0, where b1 is the "synaptic coefficient" associated with the cell differential equation developed in Part I. Gardens of Eden are derived from the decimal range of the characteristic function of each local rule and tabulated. Each of these binary strings has no predecessors (pre-image) and has therefore no past, but only the present and the future. Even more fascinating, many local rules are endowed with binary configurations which not only have no predecessors, but are also fixed points of the characteristic functions. To dramatize that such points have no past, and no future, they are henceforth christened "Isles of Eden". They too have been identified and tabulated. [ABSTRACT FROM AUTHOR]

By exploiting the new concepts of CA characteristic functions and their associated attractor time-τ maps, a complete characterization of the long-term time-asymptotic behaviors of all 256 one-dimensional CA rules are achieved via a single "probing" random input signal. In particular, the graphs of the time-1 maps of the 256 CA rules represent, in some sense, the generalized Green's functions for Cellular Automata. The asymptotic dynamical evolution on any CA attractor, or invariant orbit, of 206 (out of 256) CA rules can be predicted precisely, by inspection. In particular, a total of 112 CA rules are shown to obey a generalized Bernoulli στ-shift rule, which involves the shifting of any binary string on an attractor, or invariant orbit, either to the left, or to the right, by up to 3 pixels, and followed possibly by a complementation of the resulting bit string. The most intriguing result reported in this paper is the discovery that the four Turing-universal rules $\fbox{110}$, $\fbox{124}$, $\fbox{137}$, and $\fbox{193}$, and only these rules, exhibit a 1/f power spectrum. [ABSTRACT FROM AUTHOR]

In this paper, the problem of making a nonlinear system chaotic by using state-feedback control is studied. The feedback controller uses a simple sine function of the system state, but only one component in each dimension. It is proved, by using the anti-integrable limit method, that the designed control system generates chaos in the sense of Devaney. In fact, the controlled system so designed is a perturbation of the original system, which turns out to be a simple Bernoulli shift. [ABSTRACT FROM AUTHOR]

We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if G contains a copy of F 2 , then there exist uncountably many, non-cocycle conjugate strongly outer actions of G on any tracial, unital, separable C*-algebra that absorbs tensorially the Jiang-Su algebra. Similar conclusions hold for outer actions on McDuff II 1 factors. We moreover show that G is amenable if and only if the Bernoulli shift on any finite strongly self-absorbing C*-algebra absorbs the trivial action on the Jiang-Su algebra. Our methods are inspired by Jones' work [27] , and consist in a careful study of weak containment for the Koopman representations of certain generalized Bernoulli actions. [ABSTRACT FROM AUTHOR]

For a topological flow (V , ϕ) - i.e., V is a linearly compact vector space and ϕ a continuous endomorphism of V - we gain a deep understanding of the relationship between (V , ϕ) and the Bernoulli shift: a topological flow (V , ϕ) is essentially a product of one-dimensional left Bernoulli shifts as many as ent ⁎ (V , ϕ) counts. This novel comprehension brings us to introduce a notion of corank for topological flows designed for coinciding with the value of the topological entropy of (V , ϕ). As an application, we provide an alternative proof of the so-called Bridge Theorem for locally linearly compact vector spaces connecting the topological entropy to the algebraic entropy by means of Lefschetz duality. [ABSTRACT FROM AUTHOR]

The article presents a study which explored the Rochlin's property in the mixing transformations. The study considered measure-preserving mixing transformations (mixings)'s space through an introduced metric. Results indicate that the contiguity class of positive entropy's arbitrary mixing transformation is dense everywhere in mixing space.