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**11 - 15**of**15**### Measures of ǫ-complexity

, 2008

"... We study some measures which are related to the notion of the ǫ-complexity. We prove that measure of ǫ-complexity defined on the base of the notion of ǫ-separability is equivalent to the dual measure that is defined through ǫ-nets. ..."

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We study some measures which are related to the notion of the ǫ-complexity. We prove that measure of ǫ-complexity defined on the base of the notion of ǫ-separability is equivalent to the dual measure that is defined through ǫ-nets.

### In search of a Lebesgue density theorem for R ∞

, 810

"... M.Sc. program is a joint program with Carleton University, administered by the Ottawa- ..."

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M.Sc. program is a joint program with Carleton University, administered by the Ottawa-

### 1 1 Measures related to (ǫ, n)-complexity functions

, 705

"... The (ǫ, n)-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance ǫ during the time interval n. Behavior of the (ǫ, n)-complexity functions as n → ∞ is reflected in the proper ..."

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The (ǫ, n)-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance ǫ during the time interval n. Behavior of the (ǫ, n)-complexity functions as n → ∞ is reflected in the properties of special measures. These measures are constructed as limits of atomic measures supported at points of (ǫ, n)-separated sets. We study such measures. In particular, we prove that they are invariant if the (ǫ, n)-complexity function grows subexponentially.

### ON THE ACTION OF THE GROUP OF ISOMETRIES ON A LOCALLY COMPACT METRIC SPACE: CLOSED-OPEN PARTITIONS AND CLOSED ORBITS

, 902

"... Abstract. In the present work we study the dynamic behavior of the orbits of the natural action of the group G of isometries on a locally compact metric space X using suitable closed-open subsets of X. Precisely, we study the dynamic behavior of an orbit even in cases where G is not locally compact ..."

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Abstract. In the present work we study the dynamic behavior of the orbits of the natural action of the group G of isometries on a locally compact metric space X using suitable closed-open subsets of X. Precisely, we study the dynamic behavior of an orbit even in cases where G is not locally compact with respect to the compactopen topology. In case G is locally compact we decompose the space X into closed-open invariant disjoint sets that are related to various limit behaviors of the orbits. We also provide a simple example of a locally compact separable and complete metric space X with discrete group of isometries G such that the natural action of G on X has closed and non-closed orbits. 1.

### Approach Merotopies and Associated Near Sets

"... This article introduces associated near sets of a collection of sets. The proposed approach introduces a means of defining as well as describing anε-approach merotopy in terms of the members of associated sets of collections that are sufficiently near. A characterization for continuous functions is ..."

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This article introduces associated near sets of a collection of sets. The proposed approach introduces a means of defining as well as describing anε-approach merotopy in terms of the members of associated sets of collections that are sufficiently near. A characterization for continuous functions is established using associated near sets. This article also introduces p-containment considered in the context of near sets. An application of the proposed approach is given in terms of digital image classification.