Totally dissipative measures for the shift and conformal σ-finite measures for the stable holonomies

In this paper we investigate some results of ergodic theory with infinite measures for a subshift of finite type. We give an explicit way to construct σ-finite measures which are quasi-invariant by the stable holonomy and equivalent to the conditional measures of some σ-invariant measure. These σ-invariant measures are totally dissipative, σ-finite but satisfy a Birkhoff Ergodic-like Theorem. The constructions are done for the symbolic case, but can be extended for uniformly hyperbolic flows or diffeomorphisms.     

Tail invariant measures of the Dyck shift

We show that the one-sided Dyck shift has a unique tail invariant topologically σ-finite measure (up to scaling). This invariant measure of the one sided Dyck turns out to be a shift-invariant probability. Furthermore, it is one of the two ergodic probabilities obtaining maximal entropy. For the two sided Dyck shift we show that there are exactly three ergodic double-tail invariant probabilities. We show that the two sided Dyck has a double-tail invariant probability, which is also shift invariant, with entropy strictly less than the topological entropy. This article is a part of the author’s M.Sc. Thesis, written under the supervision of J. Aaronson, Tel-Aviv University.     

First return map and invariant measures

We give sufficient conditions for the existence of absolutely continuous invariant measures, finite or σ-finite, for maps on the interval. We givea priori bound for the number of different ergodic measures. The results are obtained via the first return map.     

An ergodic theorem of a parabolic Anderson model driven by L��vy noise

In this paper, we study an ergodic theorem of a parabolic Andersen model driven by Lévy noise. Under the assumption that A = (a(i, j)) _i,j∈S is symmetric with respect to a σ-finite measure gp, we obtain the long-time convergence to an invariant probability measure ν _h starting from a bounded nonnegative A-harmonic function h based on self-duality property. Furthermore, under some mild conditions, we obtain the one to one correspondence between the bounded nonnegative A-harmonic functions and the extremal invariant probability measures with finite second moment of the nonnegative solution of the parabolic Anderson model driven by Lévy noise, which is an extension of the result of Y. Liu and F. X. Yang.     

Onσ-finite invariant measures for Markov processes

Sufficient conditions are given for the existence ofσ-finite invariant measure for conservative and ergodic Markov processes. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem uuder the supervision of Professor S. R. Foguel. The author wishes to thank him for his helpful advice and kind encouragement.     

On the existence ofσ-finite invariant measures for operators

Several necessary and sufficient conditions are given for the existence of aσ-finite invariant measure for a positive operator onL _∞. They are ofσ-type: the entire space is an increasing union of setsX _k each of which is well-behaved. To the Memory of Shlomo Horowitz Research in part supported by the National Science Foundation (U.S.A.).     

Invariant measures for the horocycle flow on periodic hyperbolic surfaces

We classify the ergodic invariant Radon measures for the horocycle flow on geometrically infinite regular covers of compact hyperbolic surfaces. The method is to establish a bijection between these measures and the positive minimal eigenfunctions of the Laplacian of the surface. Two consequences arise: if the group of deck transformations G is of polynomial growth, then these measures are classified by the homomorphisms from G ₀ to ℝ where G ₀ ≤ G is a nilpotent subgroup of finite index; if the group is of exponential growth, then there may be more than one Radon measure which is invariant under the geodesic flow and the horocycle flow. We also treat regular covers of finite volume surfaces. The first author was supported by NSF grant 0500630. The second author was supported by NSF grant 0400687.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Examples of expandingC 1 maps having no σ-finite invariant measure equivalent to Lebesgue

In this paper we construct aC ¹ expanding circle map with the property that it has no σ-finite invariant measure equivalent to Lebesgue measure. We extend the construction to interval maps and maps on higher dimensional tori and the Riemann sphere. We also discuss recurrence of Lebesgue measure for the family of tent maps. Supported by the Deutsche Forschungsgemeinschaft (DFG). The research was carried out while HB was employed at the University of Erlangen-Nürnberg, Germany. Partially supported by NSF grant DMS # 9203489.     

Ergodic behaviour of stochastic parabolic equations

The ergodic behaviour of homogeneous strong Feller irreducible Markov processes in Banach spaces is studied; in particular, existence and uniqueness of finite and -finite invariant measures are considered. The results obtained are applied to solutions of stochastic parabolic equations.     

Entropy of conservative transformations

Summary The definition of entropy of a measure-preserving transformation (called: endomorphism) of a finite measure space into itself makes no sense for -finite measure spaces. Using induced transformations (introduced by Kakutani [1]) we give a definition which applies to conservative endomorphisms in -finite measure spaces. (This covers all cases of interest, since dissipative endomorphisms have a rather simple structure.) A theorem of Abramov [2] implies that for finite measure spaces the new definition is equivalent to the old one. Entropy as a metric invariant of conservative transformations has many, but not all of the properties discovered by Kolmogorov, Sinai, Rokhlin and others in the finite case. Major differences between the finite and the -finite case occur in the investigation of transformations with entropy 0.After giving the basic definitions in section 1 we first prove a theorem on antiperiodic transformations, which will be needed in all other sections, unless the reader is willing to assume that all transformations are ergodic. In section 3 we define entropy and prove a theorem which permits its computation. As an example the entropy of the Markov shift for null-recurrent Markov chains is computed in section 4. We then investigate simple properties such as h(T ⁿ )=nh(T) (section 5) and give the ergodic decomposition of h(T) in section 6. Section 7 is devoted to the investigation of transformations with entropy zero, especially an example is given which shows that a known necessary and sufficient condition for a transformation with finite invariant measure to have entropy zero is not sufficient for transformations with a -finite invariant measure unless they satisfy an additional assumption. Finally section 8 is devoted to the proof of category statements about the set of conservative transformations and the subset of those among them which have entropy zero.Prepared with the partial support of the National Science Foundation, Grant. No. GP-2593.Die übersetzung der vorliegenden Arbeit ins Deutsche wurde von der Naturwissenschaftlichen FakultÄt der Friedrich-Alexander-UniversitÄt Erlangen-Nürnberg im WS 1966/67 als Habilitationsschrift angenommen.I would like to thank Mr. H. Scheller for providing me with a copy of his unpublished paper [9]. My thanks are also due to Professor K. Jacobs, whose lectures made me familiar with the theory generalized in this paper and who kept me informed about some recent results.     

Kernel estimation for stationary density of Markov chains with general state space

Let {X _n } _n ≥0 be a Markov chain with stationary distributionf(x)ν(dx), ν being a σ-finite measure onE⊂R ^d . Under strict stationarity and mixing conditions we obtain the consistency and asymptotic normality for a general class of kernel estimates off(·). When the assumption of stationarity is dropped these results are extended to geometrically ergodic chains. Partially supported by CAPES. Partially supported by CNPq, PROCAD/CAPES, PRONEX/FAPDF and FINATEC/UnB.     

A Note on the characterization of Shannon-Wiener’s measure of information

Summary In [1] the author treated a characterization problem of the ShannonWiener measure of information for continuous probability distributions defined over an abstract measure space (R, S, m), wherem is a σ-finite measure over a σ-field S of subsets ofR, whose rangeM(S) is such thatM(S)=[0, ∞]. This condition on the range of the basic measure, however, can slightly be altered such thatM(S)=[0,1], and this modification is useful for characterization of the Kullback-Leibler mean information. In the present paper, it is shown that the characterization procedure of [1] can be applicable to continuous probability distributions defined on a finite measure space.     

Global random attractors are uniquely determined by attracting deterministic compact sets

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic. Entrata in Redazione il 10 marzo 1997.     

Absolutely continuous variational measures of Mawhin’s type

In this paper we study absolutely continuous and σ-finite variational measures corresponding to Mawhin, F- and BV -integrals. We obtain characterization of these σ-finite variational measures similar to those obtained in the case of standard variational measures. We also give a new proof of the Radon-Nikodym theorem for these measures.     

Fusion: a general framework for hierarchical tilings of \mathbb{R }^d

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli–Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing.     

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.     

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

   Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions), being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.     

On ordinary and standard products of infinite family of <Emphasis Type="Italic">σ</Emphasis>-finite measures and some of their applications

We introduce notions of ordinary and standard products of σ-finite measures and prove their existence. This approach allows us to construct invariant extensions of ordinary and standard products of Haar measures. In particular, we construct translation-invariant extensions of ordinary and standard Lebesgue measures on ℝ^∞ and Rogers-Fremlin measures on ℓ ^∞, respectively, such that topological weights of quasi-metric spaces associated with these measures are maximal (i.e., 2 ^c ). We also solve some Fremlin problems concerned with an existence of uniform measures in Banach spaces.