Invariant measures on products and on the space of linear orders

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

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Extending Quasi-Invariant Measures by Using Subgroups of a Given Group

Georgian Mathematical Journal ◽

10.1515/gmj.2003.247 ◽

2003 ◽

Vol 10 (2) ◽

pp. 247-255

Author(s):

A. Kharazishvili

Keyword(s):

Invariant Measures ◽

Uncountable Group

Abstract A method of extending σ-finite quasi-invariant measures given on an uncountable group, by using a certain family of its subgroups, is investigated.

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Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽

10.3390/axioms10020080 ◽

2021 ◽

Vol 10 (2) ◽

pp. 80

Author(s):

Sergey Kryzhevich ◽

Viktor Avrutin ◽

Nikita Begun ◽

Dmitrii Rachinskii ◽

Khosro Tajbakhsh

Keyword(s):

Risk Management ◽

Invariant Measure ◽

Invariant Measures ◽

Interval Exchange ◽

Management Model ◽

Closing Lemma ◽

Symbolic Model ◽

Metric Properties ◽

Ergodic Measures ◽

Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

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