On the entropy of a product endomorphism in infinite measure spaces

Summary In this note, a relationship is established between the entropy, defined by Krengel for an endomorphism of a -finite measure space, and the notion of a spreading partition. This relationship is used to answer in the quasi-finite case a question raised by Krengel concerning the entropy of the product endomorphism on the direct product of a finite and -finite measure space.     

On non-singular transformations of a measure space. I

We consider a Lebesgue measure space (M, , m). By an automorphism of (M, , m) we mean a bi-measurable transformation of (M, , m) that together with its inverse is non-singular with respeot to m. We study an equivalence relation between these automorphisms that we call the weak equivalence. Two automorphisms S and T are weakly equivalent if there is an automorphism U such that for almost all x M U maps the S-orbit of x onto the T-orbit of U x. Ergodicity, the existence of a finite invariant measure, the existenoe of a -finite infinite invariant measure, and the non-existence of such measures are invariants of weak equivalenoe. In this paper and in its sequel we solve the problem of weak equivalenoe for a class of automorphisms that comprises all ergodic automorphisms that admit a -finite invariant measure, and also certain ergodic automorphisms that do not admit such a measure.     

Solenoidal automorphisms with specifications

The specification property for solenoidal automorphisms is discussed; a solenoidal automorphism satisfies specification iff is expansive, and satisfies weak specification but not specification iff is ergodic and central spin. These are problems set up byK. Sigmund for homeomorphisms with specification. The proofs for toral case are given byD. Lind. For solenoidal case, a key ingredient in our proofs is splitting theorems on solenoidal groups with respect to described in § 2. Moreover, the following is proved: (i) If obeys specification then satisfies weak specification and is densely periodic. But the converse is not necessarily true. (ii) Every solenoidal automorphism with specification admits a Markov partition. (iii) Every ergodic solenoidal automorphism without specification does not admit Markov partitions. (iv) There exists an expansive homeomorphism with specification which has not Markov partitions.     

Commuting point transformations

Summary The question considered is the following: If two invertible measure preserving point transformations commute, in what sense is one a function of the other? The main theorem is the following: If two invertible measure preserving transformations commute, and if the first admits of approximation by periodic transformations, then the second transformation is a piecewise power of the first, where we say that is a piecewise power of if there exists a sequence [j(n)] of positive integers such that for each measurable set A the limit of the measure of the symmetric difference of (A) and ^j(n) (A) tends to zero.Research supported in part by NSF grant GP-3752.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

A skew-product which is Bernoulli

We show that if is the shift on sequences of {0,1} and is the entropy zero transformation used by Ornstein in constructing a counter-example toPinsker's conjecture, then the skew-product transformationT defined byT(x,y)=(x, ^x0 y) is Bernoulli. ThisT is conditionally mixing with respect to the independent generator for , a partition with full entropy.This research was done while the first author was a visitor at Stanford, supported in part by NSF Grant MP-575-08324.     

Stationary regenerative phenomena

Summary Corresponding to anyp-functionp, a stationary version of the associated regenerative phenomenon is constructed for which the underlying probability measure may have infinite total mass (though it will always be -finite ifp is standard). As a trivial consequence,p is a positive-definite function. The construction is generalised to quasi-Markov chains.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

On mixing in infinite measure spaces

Summary Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote -algebra (=tail -field). A transformation T is called mixing if (T ^–n A) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures ₁,₂, in total variation norm.Research of this author is supported by the National Science Foundation (U.S.A.) under grant GP 7693.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

On the existence of regular conditional probabilities

Summary Let (X, A, P) be a measure space with P(X)=1 and a sub--algebra. It is well known (see e.g. Doob, p. 624) that even if A is separable, a regular conditional probability (r.c.p.) on A, given , does not always exist. All theorems assuring the existence of a r.c.p. known hitherto use in addition to the separability of A conditions of an essentially topological nature such as compact approximation (for references see e.g. Pfanzagl-Pierlo (1966), section 7).It is the purpose of this paper to explore the possibility of sufficient conditions of a purely measure-theoretic nature. Without loss of generality we may assume that (X, A, P) is complete and is complete with respect to P ¦A. For many applications it will be sufficient to have the r.c.p. defined on a sub--algebra A ₀ A with A ₀ ~ A(P|A) rather than on the whole -algebra A. It will be shown that a r.c.p. in this more restricted sense always exists if A is the completion of a separable -algebra and if it contains a P-null set of the power of the continuum. The P-null-set -condition cannot be omitted without any replacements. To be more specific: Even a r.c.p. in the restricted sense does not necessarily exist if A is the completion of a separable -algebra (see Example 1). Furthermore, there are instances, in which a r.c.p. in the restricted sense exists but a r.c.p. in the usual sense does not exist (see Example 2).Even if a r.c.p. exists, it is not necessarily proper. The deep study by Blackwell and Ryll-Nardzewski (1963) reveals that this negative statement even holds under rather restrictive conditions (X = complete separable metric space, A = Borel algebra of X). It therefore seems to be of some interest to state that the r.c.p. in the restricted sense can always be chosen such that it is proper with respect to some sub--algebra +B_o +(+B A ₀ with ₀₀(P¦A).     

Group automorphisms with finite entropy

A compact metrie abelian group X with the normalized Haar measure is a Lebesgue probability space. A group automorphism ofX is an invertible measure preserving transformation of the probability space. This paper is to show that if the entropy of is finite, then there exist totally disconnected subgroupsH andN, a finite-dimensional subgroupS and a subgroupT satisfying the conditions: (i)H, N, S andT are strictly -invariant, (ii)N=HST, (iii)h(| _N )=0, (iv) ifS/N is non-trivial then it is a finite-dimensional solenoidal group with condition (**) (see the definition in §1), (v) ifT/N is non-trivial then it is connected and locally connected, such thatX/N splits into a direct sumX/N=H/NS/NT/N. This result characterizes the structure of finite entropy automorphisms.     

Subgroups of the full linear group over a semilocal ring

Let be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal idealmC contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals such that G()HN(), whereN() is the normalizer of the D-net subgroup . A transparent classification of subgroups GL(n, ) normalizable by diagonal matrices is thus obtained. Further, the factor groupN()/G() is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 32–34, 1978.     

A characterization of subgroups of the orthogonal group

LetG be a subgroup of the general linear group GL_n(K), where charK 2. Put Kⁿ =V. AssumeG is generated by the setS of all elements inG for which dimV( – 1) = 1, and suppose ²=1_V for each inS. If {V(–1)¦S} contains a simplex, if – 1_V G, and if inG is a product of dim v(–1) elements inS wheneverV(–1) is not contained in the kernel of–1, thenG is a subgroup of an orthogonal group.This research was supported in part by NSERC Canada grant A7251.To Helmut Mäurer on his 60th birthday     

Self-intersections of 1-dimensional random walks

Summary Consider a random walk S _n on the integers, where the steps _i have mean 0 and variance ². Let T be the time of first self-intersection of the random walk. It is shown that, as , T grows at rate ^2/3. More precisely, T^2/3 has a non-degenerate limit distribution which can be described in terms of Brownian motion local time.Research supported by National Science Foundation Grant MCS80-02698.     

Une démonstration unifiée des théorèmes de sections de P. A. Meyer

Summary In this note we get a simplified and unified proof of Meyer's section theorems by introducing a convenient paving which generates the -field of well measurable sets and using only elementary results from the theory of analytic sets and capacity.     

Extending partial isomorphisms of graphs

Theorem Let X be a finite graph. Then there exists a finite graph Z containing X as an induced subgraphs, such that every isomorphism between induced subgraphs of X extends to an automorphism of Z.The graphZ may be required to be edge-transitive. The result implies that for anyn, there exists a notion of a genericn-tuple of automorphism of the Rado graphR (the random countable graph): for almost all automorphism ₁,..., _n and ₁ ofR (in the sense of Baire category), (R,₁,..., _n ), (R,₁,..., _n ). The problem arose in a recent paper of Hodges, Hodgkinson, Lascar and Shelah, where the theorem is used to prove the small index property forR.Work supported by a Sloan Fellowship and by NSF grant DMS-8903378.     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

Exact SOR convergence regions for a general class ofp-cyclic matrices

Linear systems whose associated block Jacobi iteration matrixB is weakly cyclic generated by the cyclic permutation = (₁,₂,..., _p ) in the spirit of Li and Varga are considered. Regions of convergence for the corresponding blockp-cyclic SOR method are derived and the exact convergence domains for real spectra, (B ^p ), of the same sign are obtained. Moreover, analytical expressions for two special cases forp = 5 are given and numerical results are presented confirming the theory developed. The tools used for this work are mainly from complex analysis and extensive use of (asteroidal) hypocycloids in the complex plane is made to produce our results.This work was supported in part by AFOSR grant F49620-92-J-0069 and NSF grant 9202536-CCR.