On Ergodic Extensions of Stationary Measures with Minimal Support

1983 ◽  
Vol 26 (1) ◽  
pp. 20-25
Author(s):  
William B. Krebs ◽  
James B. Robertson
Keyword(s):  
Positive Integer ◽  
Discrete Spectrum ◽  
Ergodic Measure ◽  
Finite Partition ◽  
Minimal Support ◽  
Stationary Measures ◽  
Root Of Unity ◽  
Measure Preserving Transformation

AbstractLet T be an ergodic measure preserving transformation with the following property: there exists a positive integer n and a finite partition α such that the number of atom of is one more than that of , and the probability of at least one of the atoms is irrational. Then there exists a unique (up to conjugacy) transformation S such that there is a partition β with S restricted to isomorphic to T restricted to and the number of atoms in is one more than the number of atoms in for all m ≥ n. Moreover this transformation has discrete spectrum with at most two generators. If there are two generators, one of them must be a root of unity.

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 $n\geq 1$.

Pairs of Consecutive Power Residues or Non-Residues

1964 ◽  
Vol 16 ◽  
pp. 310-314 ◽  
Author(s):  
J. H. Jordan
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Proper Subgroup ◽  
Multiplicative Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Root Of Unity ◽  
Necessary And Sufficient

For a positive integer k and a prime p ≡ 1 (mod k), there is a proper subgroup, R, of the multiplicative group (mod p) consisting of the kth power residues (mod p). A necessary and sufficient condition that an integer t be an element of R is that the congruence xk ≡ t (mod p) be solvable. The cosets, not R, formed with respect to R are called classes of kth power nonresidues, and form with R a cyclic group of order k. Let ρ be a primitive kth root of unity and let S be a class of non-residues that is a generator of this cyclic group. There is a kth power character X (mod p) such that

Mixing on Sequences

1983 ◽  
Vol 35 (2) ◽  
pp. 339-352 ◽  
Author(s):  
Nathaniel A. Friedman
Keyword(s):  
Ergodic Measure ◽  
General Definition ◽  
Weak Mixing ◽  
Independent Sequence ◽  
Density Zero ◽  
Measure Preserving Transformation ◽  
Upper Density ◽  
Mixing Sequences ◽  
Definition Of ◽  
Single Sequence

Our aim is to study the mixing sequences of a weak mixing transformation. An ergodic measure preserving transformation is weak mixing if and only if for each pair of sets there exists a sequence of density one on which the transformation mixes the sets [9]. An unpublished result of S. Kakutani implies there actually exists a single sequence of density one on which the transformation is mixing for all sets (see Section 3). This result motivated the general définition of a transformation being mixing on a sequence, as well as mixing of higher order on a sequence. Given a weak mixing transformation, there exist sequences along which it is mixing of all degrees. In particular, this is the case for an eventually independent sequence [7].In Section 3 it will be shown that if T is weak mixing but not mixing, then a sequence on which T is two-mixing must have upper density zero.

scholarly journals PRIME SPECTRA OF AMBISKEW POLYNOMIAL RINGS

2018 ◽  
Vol 61 (1) ◽  
pp. 49-68
Author(s):  
CHRISTOPHER D. FISH ◽  
DAVID A. JORDAN
Keyword(s):  
Positive Integer ◽  
Polynomial Algebra ◽  
Central Element ◽  
Universal Enveloping Algebra ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Closed Field ◽  
Prime Spectrum ◽  
Enveloping Algebra ◽  
Root Of Unity

AbstractWe determine sufficient criteria for the prime spectrum of an ambiskew polynomial algebra R over an algebraically closed field 𝕂 to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra U(sl2) (in characteristic 0) and its quantization Uq(sl2) (when q is not a root of unity). More precisely, we determine sufficient criteria for the prime spectrum of R to consist of 0, the ideals (z − λ)R for some central element z of R and all λ ∈ 𝕂, and, for some positive integer d and each positive integer m, d height two prime ideals P for which R/P has Goldie rank m.

scholarly journals Infinite partitions and Rokhlin towers

2011 ◽  
Vol 32 (2) ◽  
pp. 707-738 ◽  
Author(s):  
STEVEN KALIKOW
Keyword(s):  
Positive Integer ◽  
Lebesgue Space ◽  
Complete Analysis ◽  
Positive Entropy ◽  
Periodic Measure ◽  
Measure Preserving Transformation ◽  
Distributed Process ◽  
Independent Identically Distributed

AbstractWe find a countable partition P on a Lebesgue space, labeled {1,2,3,…}, for any non-periodic measure-preserving transformation T such that P generates T and, for the T,P process, if you see an n on time −1 then you only have to look at times −n,1−n,…−1 to know the positive integer i to put at time 0 . We alter that proof to extend every non-periodic T to a uniform martingale (i.e. continuous g function) on an infinite alphabet. If T has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining questions on uniform martingales. In the process of proving the uniform martingale result we make a complete analysis of Rokhlin towers which is of interest in and of itself. We also give an example that looks something like an independent identically distributed process on ℤ2 when you read from right to left but where each column determines the next if you read left to right.

scholarly journals Power partitions and a generalized eta transformation property

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Don Zagier
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
General Type ◽  
Circle Method ◽  
Exact Formula ◽  
Transformation Property ◽  
Roots Of Unity ◽  
Root Of Unity ◽  
Famous Paper ◽  
The One

In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number $p_s(n)$ of partitions of a positive integer~$n$ into $s$-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1-q^{n^s}\bigr)^{-1}$ near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case $s=1$. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for $p_s(n)$ of the same general type of the one that they gave for~$s=1$. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of $p(200)$ the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for $p_2(100000)$ these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying sizes

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