On the entropy of uniquely ergodic transformations

We show that a certain type of quasifinite, conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic, measure preserving transformation wh

Strong singularity, disjointness, and strong finite additivity of finitely additive measures

In this paper we continue the study of Renyi entropies of measure-preserving transformations started in [22]. We have established there that for ergodic transformations with positive entropy, the Renyi entropies of orderq, q ∈ ℝ, are equal to either plus infinity (q 1 are equal to the essential infimum of the measure-theoretic entropies of measures forming the decomposition into ergodic components. Thus, it is possible that the Renyi entropies of orderq > 1 are strictly smaller than the measure-theoretic entropy, which is the average value of entropies of ergodic components.

Let T be an ergodic measure-preserving transformation of a Lebesgue measure space with entropy h(T).We prove that T has a generator of size k where 1. Introduction.In this paper we are concerned with ergodic invertible measurepreserving transformations of a Lebesgue measure space {E, 93, p).By a partition {An : n e 0} of E we shall mean a finite or countably infinite collection of disjoint sets An e 33 of positive measure such that E = U An. nee We call a partition {An : n e 0} a generator of an i.m.p.t.T of {E, 93, p) if 93 is generated by {T'An:ned}.i= -00 For the theory of entropy and generators of i.m.p.t.we refer to [1], [4], [5] and [6].It was proved by V. A. Rohlin that every aperiodic i.m.p.t. with finite entropy has a generator with finite entropy [6, 10.7].We shall prove in 2 that every ergodic i.m.p.t. with finite entropy has a finite generator, thereby solving a problem that was posed by V. A. Rohlin [6, p. 30].Throughout most of this paper we shall be given a finite or countably infinite state space Q.For finite Q we shall prove in 3 an approximation theorem for probability measures on Q.z that are invariant under the shift S, {Sx)i = xi+1, ieZ, x = (x,)," _"" e Q.z.This theorem will enable us to derive in 4 from the work of A. H. Zaslavskil [7] a formula for the minimal number of elements that a generator of an ergodic i.m.p.t.can contain.Denote this number by A(r).If the entropy h{T) of T is infinite then ACT) is also infinite, if h{T) < oo, then A(T) ^ eh(T\ Our result is A(D e(r)-r-l.

Investors often need to look for an optimal portfolio acting under ambiguity, as they may not be able to single out a unique real-world probability measure. In this paper a discrete-time dynamic portfolio selection problem is studied, referring to an epsilon -contaminated binomial market model and assuming investors’ preferences are consistent with the Choquet expected utility theory. We formulate the portfolio selection problem for a CRRA utility function in terms of the terminal wealth, and provide a characterization of the optimal solution in the case stock price returns are uniformly distributed. In this case, we further investigate the effect of the contamination parameter epsilon on the optimal portfolio.

A remark on Piasecki's Bayes formula for fuzzy probability measures

Throughout the paper n denotes a fixed positive integer unless otherwise specified. Let B = Bn denote the open unit ball of ℂn and let S = Sn denote its boundary, the unit sphere. The unique rotation-invariant probability measure on 5 will be denoted by σ = σn. For n = l, we use more customary notations D = B1, T = S1 and dσ1= dθ/2π. The Hardy space on B, denoted by H2(B), is then the space of functions f holomorphic on B for which

Maximum density for the Sierpinski carpet

The well known Kochen-Specker’s theorem (Kochen and Specker J. Math. Mech. 17:59–87, 1967) is devoted to the problem of hidden variables in quantum mechanics. In the paper we present a geometric proof for an indefinite analogy of Kochen-Specker’s theorem. On the real three-dimensional Krein space there exists unique two-valued probability measure.

Global solutions to an equation for mutation process in bacteria and its preservation of positive supports

We consider a finite-dimensional deterministic dynamical system with the global attractor 𝒜 which supports a unique ergodic probability measure P. The measure P can be considered as the uniform long-term mean of the trajectories staying in a bounded domain D containing 𝒜. We perturb the dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ϵ > 0 and solve the asymptotic first exit time and location problem from D in the limit of ϵ↘0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of ϵ, just as in the case when 𝒜 is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Lévy noise.

For a Polish Sample Space with a Borel σ-field with a surjective measurable transformation, we define an equivalence relation on sample points according to their ergodic limiting averages. We show that this equivalence relation partitions the subset of sample points on measurable invariant subsets, where each limiting distribution is the unique ergodic probability measure defined on each set. The results obtained suggest some natural objects for the model of a probabilistic time-invariant phenomenon are uniquely ergodic probability spaces. As a consequence of the results gained in this paper, we propose a notion of randomness that is weaker than recent approaches to Schnorr randomness.

For a continuous $\mathbb {N}^d$ or $\mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $\mathbb {Z}^d$ with positive entropy under the condition of existence of summable homoclinic points.

For let denote the class of generalized typically real functions i.e. the class of functions of the a formwhere , and is a the unique probability measure on the interval . For the same range of parameters, let the generalized Chebyshev polynomials of the second kind be defined as followsWe see thatThe main purpose this paper is the solution of coefficients problems in . A problem related to the well-known Zalcman conjecture is presented.

In eternally inflating cosmology, infinitely many pocket universes are seeded. Attempts to show that universes like our observable universe are probable amongst them have failed, since no unique probability measure is recoverable. This lack of definite probabilities is taken to reveal a complete predictive failure. Inductive inference over the pocket universes, it would seem, is impossible. I argue that this conclusion of impossibility mistakes the nature of the problem. It confuses the case in which no inductive inference is possible, with another in which a weaker inductive logic applies. The alternative, applicable inductive logic is determined by background conditions and is the same, non-probabilistic logic as applies to an infinite lottery. This inductive logic does not preclude all predictions, but does affirm that predictions useful to deciding for or against eternal inflation are precluded.