Invariant idempotent measures
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 1
Abstract
The idempotent mathematics is a part of mathematics in which arithmetic operations in the reals are replaced by idempotent operations. In the idempotent mathematics, the notion of idempotent measure (Maslov measure) is a counterpart of the notion of probability measure. The idempotent measures found numerous applications in mathematics and related areas, in particular, the optimization theory, mathematical morphology, and game theory. In this note we introduce the notion of invariant idempotent measure for an iterated function system in a complete metric space. This is an idempotent counterpart of the notion of invariant probability measure defined by Hutchinson. Remark that the notion of invariant idempotent measure was previously considered by the authors for the class of ultrametric spaces. One of the main results is the existence and uniqueness theorem for the invariant idempotent measures in complete metric spaces. Unlikely to the corresponding Hutchinson's result for invariant probability measures, our proof does not rely on metrization of the space of idempotent measures. An analogous result can be also proved for the so-called in-homogeneous idempotent measures in complete metric spaces. Also, our considerations can be extended to the case of the max-min measures in complete metric spaces.
Authors and Affiliations
N. Mazurenko, M. Zarichnyi
Keywords
Related Articles
- EP ID EP532845
- DOI 10.15330/cmp.10.1.172-178
- Views 52
- Downloads 0
Optimal control problem for systems governed by nonlinear parabolic equations without initial conditions
An optimal control problem for systems described by Fourier problem for nonlinear parabolic equations is studied. Control functions occur in the coefficients of the state equations. The existence of the optimal control i...
Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$
It is known that the so-called elementary symmetric polynomials $R_n(x) = \int_{[0,1]}(x(t))^n\,dt$ form an algebraic basis in the algebra of all symmetric continuous polynomials on the complex Banach space $L_\infty,$...
Invariant idempotent measures
The idempotent mathematics is a part of mathematics in which arithmetic operations in the reals are replaced by idempotent operations. In the idempotent mathematics, the notion of idempotent measure (Maslov measure) is a...
Some analytic properties of the Weyl function of a closed linear relation
Let L and L0, where L is an expansion of L0, be closed linear relations (multivalued operators) in a Hilbert space H. In terms of abstract boundary operators (i.e. in the form which in the case of differential operators...
Separating polynomials and uniform analytical and separating functions
We present basic results of the theory of separating polynomials and uniformly analytic and separating functions on separable real Banach spaces. We consider basic properties of separating polynomials and uniformly analy...