scholarly journals Loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli

2021 ◽  
pp. 1-57
Author(s):  
MARLIES GERBER ◽  
PHILIPP KUNDE
Keyword(s):  
Structure Theorem ◽  
Classification Problem ◽  
Global Structure ◽  
Classification Result ◽  
Rank One ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Circular System ◽  
Factor Maps

Abstract Foreman and Weiss [Measure preserving diffeomorphisms of the torus are unclassifiable. Preprint, 2020, arXiv:1705.04414] obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor $\mathcal {F}$ (see [Foreman and Weiss, From odometers to circular systems: a global structure theorem. J. Mod. Dyn.15 (2019), 345–423]) mapping odometer-based systems, $\mathcal {OB}$ , to circular systems, $\mathcal {CB}$ . This functor transfers the classification problem from $\mathcal {OB}$ to $\mathcal {CB}$ , and it preserves weakly mixing extensions, compact extensions, factor maps, the rank-one property, and certain types of isomorphisms. Thus it is natural to ask whether $\mathcal {F}$ preserves other dynamical properties. We show that $\mathcal {F}$ does not preserve the loosely Bernoulli property by providing positive and zero-entropy examples of loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli. We also construct a loosely Bernoulli circular system whose corresponding odometer-based system has zero entropy and is not loosely Bernoulli.

scholarly journals A topological lens for a measure-preserving system

2010 ◽  
Vol 31 (1) ◽  
pp. 49-75 ◽  
Author(s):  
E. GLASNER ◽  
M. LEMAŃCZYK ◽  
B. WEISS
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Positive Entropy ◽  
Topological Properties ◽  
Topologically Transitive ◽  
Topological System ◽  
Zero Entropy ◽  
Weakly Mixing ◽  
Dynamical Properties

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

scholarly journals Dynamical properties of some adic systems with arbitrary orderings

2016 ◽  
Vol 37 (7) ◽  
pp. 2131-2162 ◽  
Author(s):  
SARAH FRICK ◽  
KARL PETERSEN ◽  
SANDI SHIELDS
Keyword(s):  
Probability Measure ◽  
Fixed Points ◽  
Invariant Probability Measure ◽  
Random Order ◽  
Periodic Points ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Necessary And Sufficient

We consider arbitrary orderings of the edges entering each vertex of the (downward directed) Pascal graph. Each ordering determines an adic (Bratteli–Vershik) system, with a transformation that is defined on most of the space of infinite paths that begin at the root. We prove that for every ordering the coding of orbits according to the partition of the path space determined by the first three edges is essentially faithful, meaning that it is one-to-one on a set of paths that has full measure for every fully supported invariant probability measure. We also show that for every$k$the subshift that arises from coding orbits according to the first$k$edges is topologically weakly mixing. We give a necessary and sufficient condition for any adic system to be topologically conjugate to an odometer and use this condition to determine the probability that a random order on a fixed diagram, or a diagram constructed at random in some way, is topologically conjugate to an odometer. We also show that the closure of the union over all orderings of the subshifts arising from codings of the Pascal adic by the first edge has superpolynomial complexity, is not topologically transitive, and has no periodic points besides the two fixed points, while the intersection over all orderings consists of just four orbits.

scholarly journals Some Chaotic Properties of Discrete Fuzzy Dynamical Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yaoyao Lan ◽  
Qingguo Li ◽  
Chunlai Mu ◽  
Hua Huang
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Continuous Map ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Fuzzy Dynamical Systems ◽  
Compact Subsets

Letting(X,d)be a metric space,f:X→Xa continuous map, and(ℱ(X),D)the space of nonempty fuzzy compact subsets ofXwith the Hausdorff metric, one may study the dynamical properties of the Zadeh's extensionf̂:ℱ(X)→ℱ(X):u↦f̂u. In this paper, we present, as a response to the question proposed by Román-Flores and Chalco-Cano 2008, some chaotic relations betweenfandf̂. More specifically, we study the transitivity, weakly mixing, periodic density in system(X,f), and its connections with the same ones in its fuzzified system.

scholarly journals On rationally ergodic and rationally weakly mixing rank-one transformations

2014 ◽  
Vol 35 (4) ◽  
pp. 1141-1164 ◽  
Author(s):  
IRVING DAI ◽  
XAVIER GARCIA ◽  
TUDOR PĂDURARIU ◽  
CESAR E. SILVA
Keyword(s):  
Weak Mixing ◽  
Infinite Measure ◽  
Markov Shifts ◽  
Rank One ◽  
Measure Spaces ◽  
Metric Invariant ◽  
Weakly Mixing

AbstractWe study the notions of weak rational ergodicity and rational weak mixing as defined by J. Aaronson [Rational ergodicity and a metric invariant for Markov shifts.Israel J. Math. 27(2) (1977), 93–123; Rational weak mixing in infinite measure spaces.Ergod. Th. & Dynam. Sys.2012, to appear.http://arxiv.org/abs/1105.3541]. We prove that various families of infinite measure-preserving rank-one transformations possess or do not posses these properties, and consider their relation to other notions of mixing in infinite measure.

scholarly journals Symbolic dynamics on amenable groups: the entropy of generic shifts

2016 ◽  
Vol 37 (4) ◽  
pp. 1187-1210 ◽  
Author(s):  
JOSHUA FRISCH ◽  
OMER TAMUZ
Keyword(s):  
Symbolic Dynamics ◽  
Amenable Group ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Strongly Irreducible ◽  
Isolated Points ◽  
Zero Entropy ◽  
Weakly Mixing

Let$G$be a finitely generated amenable group. We study the space of shifts on$G$over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that, more generally, the shifts of entropy$c$are generic in the space of shifts with entropy at least $c$. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that, for every entropy value$c\in [0,\log |A|]$, there is a weakly mixing subshift of$A^{G}$with entropy $c$. We also show that the set of strongly irreducible shifts does not form a$G_{\unicode[STIX]{x1D6FF}}$in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.

Rank one and loosely Bernoulli actions in ${\Bbb Z}^{\bi d}$

1998 ◽  
Vol 18 (5) ◽  
pp. 1159-1172 ◽  
Author(s):  
AIMEE S. A. JOHNSON ◽  
AYŞE A. ŞAHİN
Keyword(s):  
One Dimensional ◽  
Rank One ◽  
Zero Entropy

We define rank one for ${\Bbb Z}^d$ actions and show that those rank one actions with a certain tower shape are loosely Bernoulli for $d\ge 1$. We also construct a zero entropy ${\Bbb Z}^2$ loosely Bernoulli action with a zero entropy, ergodic, non-loosely Bernoulli one-dimensional subaction.

scholarly journals Rank-one power weakly mixing non-singular transformations

2001 ◽  
Vol 21 (05) ◽  
Author(s):  
TERRENCE ADAMS ◽  
NATHANIEL FRIEDMAN ◽  
CESAR E. SILVA
Keyword(s):  
Rank One ◽  
Weakly Mixing

scholarly journals From odometers to circular systems: A global structure theorem

2019 ◽  
Vol 15 (0) ◽  
pp. 345-423
Author(s):  
Matthew Foreman ◽  
◽  
Benjamin Weiss ◽  
Keyword(s):  
Structure Theorem ◽  
Global Structure

scholarly journals Fuzzy classifiers in cardiovascular disease diagnostics: Review

2020 ◽  
Vol 35 (4) ◽  
pp. 22-31
Author(s):  
I. A. Hodashinsky
Keyword(s):  
Artificial Intelligence ◽  
Medical Diagnosis ◽  
Classification Problem ◽  
Medical Diagnostics ◽  
Medical Systems ◽  
Classification Result ◽  
Biological Objects ◽  
Disease Diagnostics ◽  
Fuzzy Classifiers ◽  
Human Thinking

The complexity of biological objects makes the development of computerized medical systems a difficult algorithmic decision due to the natural uncertainty inherent in these objects. Human thinking is based on vague and approximate data that can be analyzed to form clear decisions. An exact mathematical model of biological objects may not exist in practice, or such a model may be too complex to implement. In this case, fuzzy logic is a suitable tool for solving the specified problem. The problem of medical diagnosis can be viewed as a classification problem. The article presents a literature review of the use of fuzzy classifiers in diagnostics of cardiovascular diseases. The main advantage of fuzzy classifiers in comparison with other artificial intelligence methods is the ability to interpret the resulting classification result. The review aims to expand the knowledge of various researchers working in the field of medical diagnostics.

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