7 corrected question
Great thanks for realized calculations.The initial problem led me to the next situation: there are two homeomorphisms of the line and a continuous linear positive functional $l$ on $C_b(\mathbb R)$ which is invariant with respect to these homeomorphisms. Also this functional is permanent: $l([C_0(\mathbb R)])=0$, so $l$ is "concentrate at infinity". After Stone-Čech compactification, the homeomorphism still will be a homeomorphism and I can show that it will transfer $\mathbb R$ to $\mathbb R$ and the remainder $\mathbb R^*$ to $\mathbb R^*$ ($\mathbb R^* = \beta\mathbb R\setminus\mathbb R$). By the Riesz representation theorem, for our linear functional (already on $\beta\mathbb R$ and still invariant) there is a unique regular countably additive Borel measure $\mu$ on $\beta\mathbb R$. I can show that this measure will be trivial zero at $\mathbb R$. I need to understand under which conditions on the homeomorphisms this measure will be trivial zero at $\mathbb R^*$. So, figuratively speaking, we need a affect on the "compactification of infinity" from finite intervals of the homeomorphisms. I will be very grateful for links on this problem. For me the Stone-Čech compactification is a very strange think because I obtained that sequence from $\mathbb R$ can not converge to the point in $\mathbb R^*$, but each point in $\mathbb R^*$ is a limit point.

6 fixed grammar

Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it means that a finite interval does not affect on the "compactification of infinity".

Update: Great thanks for realized calculations. The initial problem led me to the next situation: there are two homeomorphisms of the line and a continuous linear positive functional $l$ on $C_b(\mathbb R)$ which is invariant with respect to these homeomorphisms. Also this functional is permanent: $l([C_0(\mathbb R)])=0$, so $l$ is "concentrate at infinity". After Stone-Čech compactification, the homeomorphism still will be a homeomorphism and I can show that it will transfer $\mathbb R$ to $\mathbb R$ and the remainder $\mathbb R^*$ to $\mathbb R^*$ ($\mathbb R^* = \beta\mathbb R\setminus\mathbb R$). By the Riesz representation theorem, for our linear functional (already on $\beta\mathbb R$ and still invariant) there is a unique regular countably additive Borel measure $\mu$ on $\beta\mathbb R$. I can show that this measure will be trivial zero at $\mathbb R$. I need to understand under which conditions on the homeomorphisms this measure will be trivial zero at $\mathbb R^*$. So, figuratively speaking, we need a affect on the "compactification of infinity" from finite intervals of the homeomorphisms. I will be very grateful for links on this problem. For me the Stone-Čech compactification is a very strange think because I obtained that sequence from $\mathbb R$ can not converge to the point in $\mathbb R^*$, but each point in $\mathbb R^*$ is a limit point.

5 fixed grammar

Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it means that a finite interval does not affect on the "compactification of infinity".

Update: Great thanks for realized calculations. The initial problem led me to the next situation: there are two homeomorphisms of the line and a continuous linear positive functional $l$ on $C_b(\mathbb R)$ which is invariant with respect to these homeomorphisms. Also this functional is permanent: $l([C_0(\mathbb R)])=0$, so $l$ is "concentrate at infinity". After Stone-Čech compactification, the homeomorphism still will be a homeomorphism and I can show that it will transfer $\mathbb R$ to $\mathbb R$ and the remainder $\mathbb R^*$ to $\mathbb R^*$ ($\mathbb R^* = \beta\mathbb R\setminus\mathbb R$). By the Riesz representation theorem, for our linear functional (already on $\beta\mathbb R$ and still invariant) there is a unique regular countably additive Borel measure $\mu$ on $\beta\mathbb R$. I can show that this measure will be trivial zero at $\mathbb R$. I need to understand under which conditions on homeomorphisms this measure will be trivial zero at $\mathbb R^*$. So, figuratively speaking, we need a affect on the "compactification of infinity" from finite intervals of the homeomorphisms. I will be very grateful for links on this problem. For me the Stone-Čech compactification is a very strange think because I obtained that sequence from $\mathbb R$ can not converge to the point in $\mathbb R^*$, but each point in $\mathbb R^*$ is a limit point.

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