Jointly ergodic measure-preserving transformations

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.

Few, if any, of the properties enjoyed by ergodic measure-preserving transformations defined on a finite measure space generalize in a natural way to those defined on an infinite measure space. Concrete examples of ergodic transformations which preserve a finite measure and ones which preserve an infinite measure exist in the literature, see [1]. It is not difficult to see that ergodic transformations never admit wandering sets of positive measure. In [2 ] it was shown that a basic difference exists between ergodic transformations which preserve a finite measure and those which preserve an infinite measure; namely, an ergodic measure-preserving transformation defined on an infinite measure space always admits weakly wandering sets of positive measure (Theorem 2 of [2]). Unlike wandering sets, it is not true in general that the union of two weakly wandering sets is again a weakly wandering set even if we require that a class of mutually disjoint images of one weakly wandering set does not intersect a class of mutually disjoint images of the other. One may ask then if there are any ergodic measure-preserving transformations defined on an infinite measure space which admit only weakly wandering sets of finite measure. In this paper we show that this is not the case. We prove that there always exist weakly wandering sets of infinite measure for any ergodic measure-preserving transformation defined on an infinite measure space (Theorem 3). In [2] the existence of a weakly wandering set of positive measure for an ergodic measure-preserving transformation defined on an infinite measure space was discovered while studying the necessary and sufficient conditions for the existence of a finite, invariant, and equivalent measure for a given measurable and nonsingular transformation. In this paper we construct the weakly wandering sets in a different way. Using the pointwise ergodic theorem we prove a simple and yet a useful fact about ergodic measure-preserving transformations defined on an infinite measure space (Theorem 2). It states that given two sets A and B both of finite measure, it is possible to find an image of A under some power of the transformation T which has small intersection with the set B. This fact is basic in proving Lemma 1 which shows the existence of a

Given an increasing sequence of positive integers { m n } \left \{ {{m_n}} \right \} , a non-decreasing sequence of positive integers { b n } \left \{ {{b_n}} \right \} , and a measurable, measure-preserving ergodic transformation τ \tau on a probability space ( Ω , F , μ ) \left ( {\Omega ,\mathcal {F},\mu } \right ) , the a.s. convergence of the moving averages T n ( f ) = b n − 1 ∑ k = m n + 1 m n + b n f ( τ k ) {T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )} is considered, for f ∈ L p ( Ω ) f \in {L_p}\left ( \Omega \right ) . A counterexample is constructed in the case of polynomial-like { m n } \left \{ {{m_n}} \right \} .

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.

Let x be a real separable Banach space and (ω,A,μ) a probability space. Also let be a measurable, measure preserving transformation on (ω,A,μ) and a semigroup of measurable measure preserving transformations on (ω,A,μ) so that the induced semigroup of isometries on L1 is strongly continous on [0,∞]Given a measurable multifunction with we prove the almost sure convergence in the mosco sense of the following ergodic averages of the multifunction F. and identifications of the corresponding limits are given.

Let $(X,\mathcal{B},m,\tau)$ be a dynamical system with $(X,\mathcal{B},m)$ a probability space and $\tau$ a measurable, invertible, measure preserving transformation. The present thesis deals with the almost everywhere convergence in $\mbox{L}^1(X)$ of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are convolution products of members of a sequence of probability measures $\{\nu_i\}$ on $\mathbb{Z}$. In the last section, we also prove a variation inequality for this type of sequence operators.

FRANK HAHNO) AND YITZHAK KATZNELSON(2) 0. Introduction.Ergodic theory involves itself with the study of transformations of a measure space.Topological dynamics is involved with homeomorphisms of a topological space.The entropy of a measure preserving transformation is a gauge of its randomness.There are certain qualitative properties in topological dynamics which seem to play a similar role.For example, equicontinuity, distality, and minimality restrict the "randomness" of a transformation group.The question arises : If a measure preserving transformation is also a homeomorphism is there a strict relation between these notions ?For instance it is known that equicontinuity and distality imply zero entropy.It was not known whether minimality implied zero entropy.W. Parry raised the following question : Let X be a compact metric space and Ta continuous map of Xinto itself.Suppose there is a unique Tinvariant probability measure and suppose (X, T) is minimal (every orbit is dense).Can (X, T) have positive entropy ?The difficulty in answering such a question is the scarcity of a wide class of minimal, uniquely ergodic transformation groups (X, T).We give here a method of constructing minimal uniquely ergodic transformation groups (X, T).Furthermore our method is designed to show that there exist transformation groups (X, T), X a compact metric space, which are minimal and uniquely ergodic but have arbitrarily large finite entropy.Thus minimality and unique ergodicity do not bound the entropy of (X, T) and the answer to Parry's question is no.We carry out our program by constructing closed shift invariant subspaces of a sequence space.In 1 we establish some notation and terminology.In 2 and 3 we either modify or translate some known theorems into combinatorial terms which are to be used in the succeeding sections.In 4 we define the sequences which will be used to build the uniquely ergodic minimal pairs (X, T) of large but finite entropy. 5, 6, 7, 8 are used to carry out the detailed examination of the properties of our sequence.9 indicates the modification necessary in 4 to show that the same construction can be done on bisequerices and thus Tmay also be a homeomorphism.The results are summarized in Corollaries 8.14 and 9.7.The principal idea which makes our result possible is the weak law of large numbers.Lemmas 1.1 and 1.2 are merely forms of this law.The remainder is

The Oldroyd 8-constant framework for continuum constitutive theory contains a rich diversity of popular special cases for polymeric liquids. In this paper, we focus on the normal stress difference responses to large-amplitude oscillatory shear (LAOS) flow. The nonlinearity of the polymeric liquids, triggered by LAOS, causes these responses at even multiples of the test frequency. We call responses at a frequency higher than twice the test frequency higher harmonics. The normal stress difference responses for the Oldroyd 8-constant framework has recently yielded to the exact analytical solution. However, in its closed form, Bessel functions appear 24 times, each within summations to infinity. In this paper, to simplify the exact solution, we expand it in a Taylor series. We truncate the series after its 17th power of the shear rate amplitude. Our main result reduces to the well-known expression for the special cases of the corotational Jeffreys and corotational Maxwell fluids. Whereas these special cases yielded to the Goddard integral expansion (GIE), the more general Oldroyd 8-constant framework does not. We use Ewoldt grids to show our main result to be highly accurate for the corotational Jeffreys and corotational Maxwell fluids. For these two special cases, our solutions agree closely with the exact solutions as long as Wi/De&amp;lt;3310. We compare our main result, for the special case of the Johnson–Segalman fluid, with measurements on dissolved polyisobutylene in the isobutylene oligomer. For this, we use the Spriggs relations to generalize our main result to multimode, which then agrees closely with the measurements.

The central quantity in the theory of transport for Hamiltonian systems, and in particular the area-preserving twist maps, is the action of rotational periodic orbits. Usually this is a complicated discontinuous function of two arguments: some perturbation parameter k and a rational rotation number , denoted by . We applied the idea of modular smoothing to this complicated fractal. Our main result is that all the information contained in the fractal can be retrieved from a set of continuous or smooth functions of one variable.

Karlsson and Margulis (Commun. Math. Phys. 208:107–123, 1999) proved in the setting of uniformly convex geodesic spaces, which additionally satisfy a nonpositive curvature condition, an ergodic theorem that focuses on the asymptotic behavior of integrable cocycles of nonexpansive mappings over an ergodic measure-preserving transformation. In this note we show that this result holds true when assuming a weaker notion of uniform convexity.

Kingman’s subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformationTof a measure space$(X, \mu )$. We use a theorem of Silva and Thieullen to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that$\mu $and$\mu \circ T$have the same null sets. Using this, we are able to produce versions of the Furstenberg–Kesten theorem and the Oseledeč ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.

The formula for the entropy of the direct product of T 1 and T 2 holds both for measure-theoretic entropy, when T 1 and T 2 are measure-preserving transformations on probability spaces, and for topological entropy, when T 1 and T 2 are continuous transformations on compact topological spaces. Examples are given to show that the formula does not extend to sequence entropy in either case. Moreover, in the topological case, the formula does not hold, in general, when the spaces are non-compact.

We 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. &amp; 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.

This thesis falls naturally into two distinct parts. Both come under the general heading of the theory of conformal mapping though the later part incorporates work in potential theory. We first study the growth of means of the logarithmic derivative of a univalent function in the disc. Here results have been obtained by Hayman and by Baernstein and Brown. Hayman has shown that an elementary upper bound for the growth of these means is best possible in general. Later on, Baernstein and Brown showed that the means of certain classes of monotone slit mappings, including support points of the class S of normalised univalent functions, grow no faster than those of the Koebe function up to a multiplicative constant. The question remained open for unrestricted monotone slit mappings. We settle this question by constructing a monotone slit mapping the mean of whose logarithmic derivative grows faster than that of the Koebe function. Following this, we discuss some recent work by Burdzy on the boundary behaviour of positive harmonic functions in Lipschitz domains and applications of this work to the angular derivative problem. Burdzy obtains his results on the angular derivative by probabilistic methods. Rodin and Warschawski later gave a classical proof of part of Burdzy’s main result and related his criteria for the existence of an angular derivative to criteria which they had used previously. They were, however, unable to obtain a non-probabilistic proof of the full theorem. Using a new non-probabilistic method, we prove a theorem on the growth of positive harmonic functions vanishing near a boundary point of a Lipschitz domain. The plane case of this result and some special cases in space were proved by Burdzy in a series of articles. He went on to prove the full result in space in a later paper with R. J. Williams. Our result enables us to give an elementary proof of the remainder of Burdzy’s theorem on the angular derivative and so complements Rodin and Warschawski’s work. We complete our study of this problem by proving two further related results on the boundary behaviour of positive harmonic functions in Lipschitz domains. It is likely that the methods used will be helpful in problems of a similar nature.

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemerédi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter n n to those integers that have an even number of distinct prime factors or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are then analyzed using a recent structural result for bounded multiplicative functions proved by the authors.