Let $K$ be a compact set of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(K,\mathbb{R}^n)$ be a speed field on $K$ such that for any initial condition $x_0\in K$, the following dynamical system

\begin{equation} \begin{cases} x(0)=x_0 \\ \dot{x}(t)=v\big(x(t)\big),\>t\geq 0 \end{cases} \end{equation}

has a solution $x$ defined on $\mathbb{R}_+$. Hence, there exists a probability measure on $K$, denoted $\mu_{|x_0}$, which is invariant for this dynamical system, and the pointwise ergodic theorem states that for any $f\in\mathcal{C}^0(K,\mathbb{R})$

\begin{equation} \lim\limits_{T\rightarrow+\infty}\frac{1}{T}\int_0^T f\big(x(t)\big)\,\mathrm{d}t=\int_K f\,\mathrm{d}\mu_{|x_0}. \end{equation}

Given $f$, if $v$ is nowhere-vanishing, is this convergence uniform toward the choice of $x_0\in K$?

Edit: is it easier if the dynamical system is assumed to be uniquely ergodic?

Let $K$ be a compact set of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(K,\mathbb{R}^n)$ be a speed field on $K$ such that for any initial condition $x_0\in K$, the following dynamical system

\begin{equation} \begin{cases} x(0)=x_0 \\ \dot{x}(t)=v\big(x(t)\big),\>t\geq 0 \end{cases} \end{equation}

has a solution $x$ defined on $\mathbb{R}_+$. Hence, there exists a probability measure on $K$, denoted $\mu_{|x_0}$, which is invariant for this dynamical system, and the pointwise ergodic theorem states that for any $f\in\mathcal{C}^0(K,\mathbb{R})$

\begin{equation} \lim\limits_{T\rightarrow+\infty}\frac{1}{T}\int_0^T f\big(x(t)\big)\,\mathrm{d}t=\int_K f\,\mathrm{d}\mu_{|x_0}. \end{equation}

Given $f$, if $v$ is nowhere-vanishing, is this convergence uniform toward the choice of $x_0\in K$?

Let $K$ be a compact set of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(K,\mathbb{R}^n)$ be a speed field on $K$ such that for any initial condition $x_0\in K$, the following dynamical system

\begin{equation} \begin{cases} x(0)=x_0 \\ \dot{x}(t)=v\big(x(t)\big),\>t\geq 0 \end{cases} \end{equation}

has a solution $x$ defined on $\mathbb{R}_+$. Hence, there exists probability measure on $K$, denoted $\mu_{|x_0}$, which is invariant for this dynamical system, and the pointwise ergodic theorem states that for any $f\in\mathcal{C}^0(K,\mathbb{R})$

\begin{equation} \lim\limits_{T\rightarrow+\infty}\frac{1}{T}\int_0^T f\big(x(t)\big)\,\mathrm{d}t=\int_K f\,\mathrm{d}\mu_{|x_0}. \end{equation}

Given $f$, if $v$ is nowhere-vanishing, is this convergence uniform toward the choice of $x_0\in K$?

Let $K$ be a compact set of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(K,\mathbb{R}^n)$ be a speed field on $K$ such that for any initial condition $x_0\in K$, the following dynamical system

\begin{equation} \begin{cases} x(0)=x_0 \\ \dot{x}(t)=v\big(x(t)\big),\>t\geq 0 \end{cases} \end{equation}

has a solution $x$ defined on $\mathbb{R}_+$. Hence, there exists a probability measure on $K$, denoted $\mu_{|x_0}$, which is invariant for this dynamical system, and the pointwise ergodic theorem states that for any $f\in\mathcal{C}^0(K,\mathbb{R})$

\begin{equation} \lim\limits_{T\rightarrow+\infty}\frac{1}{T}\int_0^T f\big(x(t)\big)\,\mathrm{d}t=\int_K f\,\mathrm{d}\mu_{|x_0}. \end{equation}

Given $f$, if $v$ is nowhere-vanishing, is this convergence uniform toward the choice of $x_0\in K$?

Edit: is it easier if the dynamical system is assumed to be uniquely ergodic?