The behavior of Bernoulli shifts relative to their factors

1999 ◽  
Vol 19 (5) ◽  
pp. 1255-1280 ◽  
Author(s):  
CHRISTOPHER HOFFMAN
Keyword(s):  
Bernoulli Shift ◽  
Bernoulli Shifts ◽  
Ergodic Transformations ◽  
Zero Entropy ◽  
Almost All

We present numerous examples of ways that a Bernoulli shift can behave relative to a family of factors. This shows the similarities between the properties which collections of ergodic transformations can have and the behavior of a Bernoulli shift relative to a collection of its factors. For example, we construct a family of factors of a Bernoulli shift which have the same entropy, and any extension of one of these factors has more entropy, yet no two of these factors sit the same. This is the relative analog of Ornstein and Shields uncountable collection of nonisomorphic $K$ transformations of the same entropy. We are able to construct relative analogs of almost all the zero entropy counter-examples constructed by Rudolph (1979), as well as the $K$ counterexamples constructed by Hoffman (1997). This paper provides a solution to a problem posed by Ornstein (1975).

Entropy for extensions of Bernoulli shifts

1996 ◽  
Vol 16 (6) ◽  
pp. 1197-1206 ◽  
Author(s):  
Marie Choda
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebras ◽  
Bernoulli Shift ◽  
Product Formula ◽  
Von Neumann ◽  
Bernoulli Shifts ◽  
Outer Automorphisms ◽  
Finite Von Neumann Algebras

AbstractWe give a condition for automorphisms α and β on finite von Neumann algebras which induces the tensor product formula for entropies: H(α ⊗ β) = H(α) + H(β). As an application, the Bernoulli shift (1/n, 1/n, …, 1/n) has extensions to ergodic outer automorphisms {αk; k = 1,2, …} on the hyperfinite II1 factor R with the entropies H(αk) = (1/2)kn log n.

scholarly journals Entropy of systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1118-1126
Author(s):  
RADU B. MUNTEANU
Keyword(s):  
Positive Integer ◽  
Probability Space ◽  
Bernoulli Shift ◽  
Ergodic Measure ◽  
Measure Preserving Transformation ◽  
Zero Entropy ◽  
Standard Probability

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is $\text{AT}(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not $\text{AT}(n)$. We also give an example of a transformation which has zero entropy but does not have property $\text{AT}(n)$ for any integer $n\geq 1$.

scholarly journals Periodic Points of Asymmetric Bernoulli Shifts

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yong-Guo Shi ◽  
Kai Chen ◽  
Wei Liao
Keyword(s):  
Bernoulli Shift ◽  
Piecewise Linear ◽  
Chaotic Map ◽  
Periodic Points ◽  
Complete Structure ◽  
Bounded Interval ◽  
Bernoulli Shifts ◽  
Piecewise Linear Chaotic Map

It is well-known that Sharkovskii’s theorem gives a complete structure of periodic order for a continuous self-map on a closed bounded interval. As a further study, a natural problem is how to determine the location and number of periodic points for a specific map. This paper considers the periodic points of asymmetric Bernoulli shift, which is a piecewise linear chaotic map.

On Finite Coding Factors of a Class of Random Markov Chains

1994 ◽  
Vol 37 (3) ◽  
pp. 399-407 ◽  
Author(s):  
M. Rahe
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Direct Product ◽  
Finite Length ◽  
Bernoulli Shift ◽  
Finite Rotation ◽  
Step Size ◽  
Zero Entropy

AbstractFor k-step Markov chains, factors generated by finite length codes split off with Bernoulli complement when maximal in entropy. Those not maximal are relatively finite in another factor which generates or splits off.These results extend to random Markov chains with finite expected step size, implying that random Markov chains with finite expected step size can have only finitely many ergodic components, each of which is isomorphic to a finite rotation, a Bernoulli shift, or a direct product of a Bernoulli shift with a finite rotation. This result limits the type of zero entropy factors which occur in random Markov chains with finite expected step size, providing a counterpoint to the work of Kalikow, Katznelson, and Weiss, who have shown that each zero entropy process can be embedded in some random Markov chain.Extending Rudolph and Schwarz, random Markov chains with finite expected step size are limits in of their canonical Markov approximants. The -closure of the class is the Bernoulli cross Generalized Von Neuman processes.Finitary isomorphism of aperiodic ergodic random Markov chains with finite expected step size is considered.Applications are made to a class of generalized baker's transformations.

scholarly journals The zero-type property and mixing of Bernoulli shifts

2012 ◽  
Vol 33 (2) ◽  
pp. 549-559 ◽  
Author(s):  
ZEMER KOSLOFF
Keyword(s):  
Bernoulli Shift ◽  
Type Iii ◽  
Bernoulli Shifts ◽  
Type Property ◽  
Weakly Mixing

AbstractWe prove that every non-singular Bernoulli shift is either zero-type or there is an equivalent invariant stationary product probability. We also give examples of a type III1Bernoulli shift and a Markovian flow which are power weakly mixing and zero-type.

The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains

1991 ◽  
Vol 11 (1) ◽  
pp. 129-180 ◽  
Author(s):  
Brian Marcus ◽  
Selim Tuncel
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Periodic Orbits ◽  
Bernoulli Shift ◽  
Beta Function ◽  
Bernoulli Shifts ◽  
Canonical Extensions

AbstractWe study Markov chains via invariants constructed from periodic orbits. Canonical extensions, based on these invariants, are used to establish a constraint on the degree of finite-to-one block homomorphisms from one Markov chain to another. We construct a polytope from the normalized weights of periodic orbits. Using this polytope, we find canonically-defined induced Markov chains inside the original Markov chain. Each of the invariants associated with these Markov chains gives rise to a scaffold of invariants for the original Markov chain. This is used to obtain counterexamples to the finite equivalence conjecture and to a conjecture regarding finitary isomorphism with finite expected coding time. Also included are results related to the problem of minimality (with respect to block homomorphism) of Bernoulli shifts in the class of Markov chains with beta function equal to the beta function of the Bernoulli shift.

scholarly journals Automorphisms of blowups of threefolds being Fano or having Picard number 1

2016 ◽  
Vol 37 (7) ◽  
pp. 2255-2275
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Main Idea ◽  
Chern Classes ◽  
Picard Number ◽  
Zero Entropy ◽  
Almost All ◽  
Dynamical Degrees ◽  
Holomorphic Automorphisms

Let $X_{0}$ be a smooth projective threefold which is Fano or which has Picard number 1. Let $\unicode[STIX]{x1D70B}:X\rightarrow X_{0}$ be a finite composition of blowups along smooth centers. We show that for ‘almost all’ of such $X$, if $f\in \text{Aut}(X)$, then its first and second dynamical degrees are the same. We also construct many examples of blowups $X\rightarrow X_{0}$, on which any automorphism is of zero entropy. The main idea is that, because of the log-concavity of dynamical degrees and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology classes. We will also discuss a possible application of these results to a threefold constructed by Kenji Ueno.

Two facts concerning the transformations which satisfy the weak Pinsker property

2008 ◽  
Vol 28 (2) ◽  
pp. 689-695 ◽  
Author(s):  
J.-P. THOUVENOT
Keyword(s):  
Bernoulli Shift ◽  
Block Structure ◽  
Bernoulli Shifts ◽  
Finite Generator ◽  
The One

AbstractWe show that every ergodic, finite entropy transformation which satisfies the weak Pinsker property possesses a finite generator whose two-sided tail field is exactly the Pinsker algebra. This is proved by exhibiting a generator endowed with a block structure quite analogous to the one appearing in the construction of the Ornstein–Shields examples of non Bernoulli K-automorphisms. We also show that, given two transformations T1 and T2 in the previous class (i.e. satisfying the weak Pinsker property), and a Bernoulli shift B, if T1×B is isomorphic to T2×B, then T1 is isomorphic to T2. That is, one can ‘factor out’ Bernoulli shifts.

scholarly journals Subsystems, Perron numbers, and continuous homomorphisms of Bernoulli shifts

1989 ◽  
Vol 9 (3) ◽  
pp. 561-570 ◽  
Author(s):  
Selim Tuncel
Keyword(s):  
Finite Type ◽  
Bernoulli Shift ◽  
Block Code ◽  
Block Codes ◽  
Probability Vector ◽  
Finiteness Conditions ◽  
Subshift Of Finite Type ◽  
Bernoulli Shifts ◽  
Subshifts Of Finite Type ◽  
Markov Measures

AbstractLet S, T be subshifts of finite type, with Markov measures p, q on them, and let φ: (S, p) → (T, q) be a block code. Let Ip, Iq denote the information cocycles of p, q. For a subshift of finite type ⊂T, the pressure of equals that of . Applying this to Bernoulli shifts and using finiteness conditions on Perron numbers, we have the following. If the probability vector p = (p1…, pk+1) is such that the distinct transcendental elements of {p1/pk+1…pk/pk+1) are algebraically independent then the Bernoulli shift B(p) has finitely many Bernoulli images by block codes.

scholarly journals Approximately transitive flows and ITPFI factors

1985 ◽  
Vol 5 (2) ◽  
pp. 203-236 ◽  
Author(s):  
A. Connes ◽  
E. J. Woods
Keyword(s):  
Lebesgue Measure ◽  
Group Action ◽  
Measure Space ◽  
Finite Measure ◽  
Type Iii ◽  
Ergodic Transformations ◽  
Zero Entropy ◽  
Approximate Transitivity

AbstractWe define a new property of a Borel group action on a Lebesgue measure space, which we call approximate transitivity. Our main results are (i) a type III0 hyperfinite factor is ITPFI if and only if its flow of weights is approximately transitive, and (ii) for ergodic transformations preserving a finite measure, approximate transitivity implies zero entropy.

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