Shift dynamical systems and remotely almost periodic functions

Abstract The purpose of this work is to study remotely almost periodic functions defined on the semi-axis <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo>:=</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathbb{R}_{+}:=[0,+\infty)} (respectively, on the real axis <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathbb{R}:=(-\infty,+\infty)} ) with values in the Banach space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔅</m:mi> </m:math> {\mathfrak{B}} . We establish a relation between remotely almost periodic movements in the shift dynamical system <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>C</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝕋</m:mi> <m:mo>,</m:mo> <m:mi>𝔅</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>𝕋</m:mi> </m:mrow> </m:math> {(C(\mathbb{T},\mathfrak{B}),\mathbb{T}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>σ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {\sigma)} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝕋</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msub> </m:mrow> </m:math> {\mathbb{T}=\mathbb{R}_{+}} or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℝ</m:mi> </m:math> {\mathbb{R}} ) in the space of continuous functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝕋</m:mi> <m:mo>,</m:mo> <m:mi>𝔅</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> C(\mathbb{T},\mathfrak{B}) and remotely almost periodic functions. Using this connection, we prove some important analytical and topological properties of remotely almost periodic functions.