Invariant Borel Measure Research Articles

Topological speedups of ℤd-actions

We study minimal -Cantor systems and the relationship between their speedups, their collections of invariant Borel measures, their associated unital dimension groups, and their orbit equivalence classes. In the particular case of minimal -odometers, we show that their bounded speedups must again be odometers but, contrary to the 1-dimensional case, they need not be conjugate, or even isomorphic, to the original. Furthermore, we give examples of speedups of -odometers which show the significant role played by a choice of ‘cone’ associated to the speedup.

On Kharazishvili type measures in infinite-dimensional Polish vector spaces

Abstract Some questions are discussed concerning the existence of nonzero σ-finite invariant Borel measures in infinite-dimensional Polish vector spaces.

Invariant measures on topological groups

In this paper, we prove that every locally K (see Definition 3.4) topological group has a nonzero outer regular invariant Borel measure when K is an admissible invariant family which is separated by NG. In this case, every open set and every member of S(K0) are K-inner regular. This extends the existence theorem of Haar measure on locally compact Hausdorff groups.

Existence and uniqueness of translation invariant measures in separable Banach spaces

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.

A universal measure for a pencil of conics and the Great Poncelet Theorem

Borel measures on conics which are invariant under the Poncelet map are investigated. For a pencil of conics the existence of a universal measure, which is invariant with respect to each conic in the pencil, is proved. Using this measure a new proof of the Great Poncelet Theorem is given. A full description of invariant Borel measures is also presented. Bibliography: 10 titles.

On the uniqueness property of non-separable extensions of invariant Borel measures and relative measurability of real-valued functions

Abstract. It is shown that the class of all non-separable extensions of a nonzero σ $\sigma $ -finite Borel measure in the topological vector space ℝ ℕ $\mathbb {R}^{\mathbb {N}}$ , which are invariant under some everywhere dense continual subgroup of ℝ ℕ $\mathbb {R}^{\mathbb {N}}$ and which possess the uniqueness property, has maximal cardinality 2 2 c $2^{2^{c}}$ . Some related questions concerning the measurability properties of real-valued functions with respect to the class of non-separable measures are also discussed.

Is Lebesgue measure the only σ-finite invariant Borel measure?

S. Saks and recently R.D. Mauldin asked if every translation invariant σ-finite Borel measure on R d is a constant multiple of Lebesgue measure. The aim of this paper is to investigate the versions of this question, since surprisingly the answer is “yes and no,” depending on what we mean by Borel measure and by constant. According to a folklore result, if the measure is only defined for Borel sets, then the answer is affirmative. We show that if the measure is defined on a σ-algebra containing the Borel sets, then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant σ-finite Borel measure (in the wider sense) on R d can be non- σ-finite when we restrict it to the Borel sets.

Borel sets which are null or non- [formula omitted]-finite for every translation invariant measure

We show that the set of Liouville numbers is either null or non- σ -finite with respect to every translation invariant Borel measure on R , in particular, with respect to every Hausdorff measure H g with gauge function g. This answers a question of R.D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch–Eggleston numbers, as well as all Borel subgroups of R that are not F σ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.

Carleson measures theorems for generalized Bergman spaces on the unit polydisk

We provide sufficient conditions on a positive finite rotation invariant Borel measure on {𝔻}^n which guarantee that the analogue of the Carleson measure theorem remains valid for Bergman spaces of holomorphic and n-harmonic functions on {𝔻}^n generated by the measure.

Measurable distal and topological distal systems

In this paper we prove that any ergodic measurably distal system can be realized as a minimal topologically distal system with an invariant Borel measure of full support. The proof depends upon a theorem stating that every measurable function from a measurable system with its base space being a compact metric space to a connected compact group is cohomologous to a continuous function.

Measure theoretic ergonic theory in quantum mechanics

We apply the measure theoretic ergonic theory to quantum systems. The time evolution is considered as a point transformation in the set of states of a C ∗-algebra. Invariant Borel measures on this set are shown to exist, so there is a measure theoretic dynamic system. Interrelations between this system and the corresponding quantum dynamical system are established.

Entropy-Expansive Maps

Let $f:X \to X$ be a uniformly continuous map of a metric space. f is called h-expansive if there is an $\varepsilon > 0$ so that the set ${\Phi _\varepsilon }(x) = \{ y:d({f^n}(x),{f^n}(y)) \leqq \varepsilon$ for all $n \geqq 0$} has zero topological entropy for each $x \in X$. For X compact, the topological entropy of such an f is equal to its estimate using $\varepsilon :h(f) = h(f,\varepsilon )$. If X is compact finite dimensional and $\mu$ an invariant Borel measure, then ${h_\mu }(f) = {h_\mu }(f,A)$ for any finite measurable partition A of X into sets of diameter at most $\varepsilon$. A number of examples are given. No diffeomorphism of a compact manifold is known to be not h-expansive.

Entropy-expansive maps

Let f : X → X f:X \to X be a uniformly continuous map of a metric space. f is called h-expansive if there is an ε > 0 \varepsilon > 0 so that the set Φ ε ( x ) = { y : d ( f n ( x ) , f n ( y ) ) ≦ ε {\Phi _\varepsilon }(x) = \{ y:d({f^n}(x),{f^n}(y)) \leqq \varepsilon for all n ≧ 0 n \geqq 0 } has zero topological entropy for each x ∈ X x \in X . For X compact, the topological entropy of such an f is equal to its estimate using ε : h ( f ) = h ( f , ε ) \varepsilon :h(f) = h(f,\varepsilon ) . If X is compact finite dimensional and μ \mu an invariant Borel measure, then h μ ( f ) = h μ ( f , A ) {h_\mu }(f) = {h_\mu }(f,A) for any finite measurable partition A of X into sets of diameter at most ε \varepsilon . A number of examples are given. No diffeomorphism of a compact manifold is known to be not h-expansive.

Symbolic dynamics and transformations of the unit interval

transformation. The main result includes a classical theorem of Poincare-Denjoy [2] on homeomorphisms of the circle onto itself. It also provides a partial answer to a question of Ulam's [3] concerning the possibility of piecewise linearising continuous transformations of the unit interval. This problem was also mentioned by Stein and Ulam in [4], together with the remark that necessary conditions can be given in terms of the trees of points, but that no meaningful sufficient conditions are known. In the same work a few special examples are examined. Our main theorem also has a bearing on certain transformations discussed by Renyi [5]. In ??2-4 we consider the shift transformation acting on a compact invariant subset of the space of one-way infinite sequences of symbols chosen from a finite set. The shift transformation on such a set is continuous but not necessarily open. If X, T are the compact invariant set and the shift transformation, respectively, we refer to (X, T) as a symbolic dynamical system [6]. For a symbolic dynamical system (X, T) we define a number called the absolute entropy(') which dominates the entropy of T with respect to each normalised T invariant Borel measure, and show that if T is regionally transitive then there is always one invariant measure with respect to which the entropy of T equals the absolute entropy of T. When T is open, (or equivalently, when (X, T) is an intrinsic Markov chain) this maximal measure is unique. A further theorem states that, under certain conditions, there exists a normalised Borel measure with respect to which T acts in a linear fashion. In ??5-6 we apply this latter theorem to certain transformations of the unit interval and obtain our main result, Theorem 5.

The Mean and Transitive Points of Homeomorphisms

A point p of a compact metric space Q is called a mean point of a homeomorphism T of Q onto itself if the sequence of arithmetic means F.(p) = 1/n J:= f(T'p) converges for every real valued continuous function f(x) on a. This notion of a mean point was introduced by Kryloff and Bougoliouboff who called such a point a quasi-regular point [cf. 5, p. 93].' The mean points derive their importance from the fact that, as was shown by Kryloff and Bougoliouboff, they determine completely the totality of all normalized invariant Borel measures in Q. Indeed, to each mean point there corresponds a unique normalized invariant Borel measure and every normalized invariant Borel measure is a limit of convex linear combinations of measures corresponding to mean points. In fact a more restricted class of points called transitive points is already sufficient to determine all normalized invariant Borel measures in Q. These points are the mean points to which correspond ergodic normalized invariant Borel measures. The question of whether there always exist mean or transitive points in Q2 was answered by Kryloff and Bougoliouboff who showed that the set of transitive (and hence also mean) points of Q is not empty. Moreover they have also shown that this set has maximal invariant measure. I.e., it has measure one for every normalized invariant measure. In this paper it is shown that there are always infinitely many transitive (and hence mean) points in Q provided I has infinitely many points (cf. Theorems 1 and 1' below). Moreover under the added restriction that Q is locally connected it is shown that there are c transitive (and hence mean) points in Q (cf. Theorems 2 and 2'). It is also shown that while the mean points are the general case as far as measure is concerned, they are by no means the general case as far as category is concerned. Indeed, under conditions which occur in many examples of dynamical systems, the set of mean points is a set of first category (cf. Theorem 3). That an added restriction on Q is needed in order to insure the existence of c mean points in Q is shown in the last paragraph by an example of a non locally connected compact metric space Q with a homeomorphism of it onto itself, for which there are only a countable infinity of mean points although the space Q has c points. The proofs of our main theorems are based partly on the theory of invariant measures in dynamical systems and partly on modifications of theorems due

Stepanoff flows on the torus

A statement in a recent paper by Saito [61 suggested the following question: Does there exist on the torus an analytic, area preserving, ergodic flow which has a stationary point? The purpose of this note is to describe an elementary example of such a flow, and to discuss some properties of a topological class of flows to which it belongs. Stepanoff [7) (cf. also [4, pp. 395-400, 506-507]) considered the flows defined by equations of the form (1) where Y= aX, a is an irrational number, and X is periodic, non-negative, continuous, satisfies a Lipschitz condition in x and y, and vanishes at one and only one point of the torus. He showed that these flows are metrically transitive with respect to the invariant Borel measure