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I'm starting to study geodesic currents and I have a question concerning uniform continuity.

Let's take $S$ a closed surface of genus $g$ and $C(S)$ the space of geodesic currents on $S$ (as it is defined by Bonahon, endowed with the weak star topology). Let's call $\mathcal{C}(S)$ the set of isotopy classes of closed curves on $S$ and $\mathcal{S}\subset\mathcal{C}(S)$ the subset corresponding to simple closed curves.

It is known that $\mathcal{C}(S)$ injects in $C(S)$ and that there is a continuous extension $i:C(S)\times C(S)\rightarrow \mathbb{R}$ of the intersection number on $\mathcal{C}(S)$ .

Writing $T(S)$ for the Teichmuller space of $S$, it is also known that there is an injection $T(S)\rightarrow C(S)$ which sends a hyperbolic metric $h$ to the Liouville current $L_h$. For every Liouville current $L_h$ and every $\alpha\in \mathcal{S}$ it is true $l_h(\alpha)=i(L_h,\alpha)$.

My question is: I know that, for every $h\in T(S)$ hyperbolic metric, that the function $i(L_h,-):\mathcal{S}\rightarrow \mathbb{R}^+$ is continuous, but is it also $uniformly$ continuous?

I'm starting to study geodesic currents and I have a question concerning uniform continuity.

Let's take $S$ a closed surface of genus $g$ and $GC(S)$ the space of geodesic currents on $S$ (as it is defined by Bonahon, endowed with the weak star topology). Let's call $\mathcal{C}(S)$ the set of isotopy classes of closed curves on $S$ and $\mathcal{S}\subset\mathcal{C}(S)$ the subset corresponding to simple closed curves.

It is known that $\mathcal{C}(S)$ injects in $GC(S)$ and that there is a continuous extension $i:GC(S)\times GC(S)\rightarrow \mathbb{R}$ of the intersection number on $\mathcal{C}(S)$ .

Writing $T(S)$ for the Teichmuller space of $S$, it is also known that there is an injection $T(S)\rightarrow GC(S)$ which sends a hyperbolic metric $h$ to the Liouville current $L_h$. For every Liouville current $L_h$ and every $\alpha\in \mathcal{S}$ it is true $l_h(\alpha)=i(L_h,\alpha)$.

My question is: I know that, for every $h\in T(S)$ hyperbolic metric, that the function $i(L_h,-):\mathcal{S}\rightarrow \mathbb{R}^+$ is continuous, but is it also uniformly continuous?

I'm starting to study geodesic currents and I have a question concerning uniform continuity.

Let's take $S$ a closed surface of genus $g$ and $C(S)$ the space of geodesic currents on $S$ (as it is defined by Bonahon, endowed with the weak star topology). Let's call $\mathcal{C}(S)$ the set of isotopy classes of closed curves on $S$ and $\mathcal{S}\subset\mathcal{C}(S)$ the subset corresponding to simple closed curves.

It is known that $\mathcal{C}(S)$ injects in $C(S)$ and that there is a continuous extension $i:C(S)\times C(S)\rightarrow \mathbb{R}$ of the intersection number on $\mathcal{C}(S)$ .

Writing $T(S)$ for the Teichmuller space of $S$, it is also known that there is an injection $T(S)\rightarrow C(S)$ which sends a hyperbolic metric $h$ to the Liouville current $L_h$. For every Liouville current $L_h$ and every $\alpha\in \mathcal{S}$ it is true $l_h(\alpha)=i(L_h,\alpha)$.

My question is: I know that, for every $h\in T(S)$ hyperbolic metric, that the function $i(L_h,-):\mathcal{S}\rightarrow \mathbb{R}^+$ is continuous, but is it also $uniformly$ continuous?

Edit: yes, as Misha observed of course to talk about uniform continuity we must have a metric. As Bruno Martelli said $C(S)$ is naturally a metric space (I should have mentioned it, my bad).

The metric of $C(S)$: consider $\mathcal{G}$ the space of geodesics on $\widetilde{S}$ (the universal cover of $S$), so that $C(S)$ is the space of radon measures on $\mathcal{G}$ invariant under the action of the covering group. Then we can find a countable family of continuous functions $\varphi_n:\mathcal{G}\rightarrow \mathbb{R}$ such that the pullback of the product topology on $\prod_{n\in\mathbb{N}}\varphi_n$ by the natural map $C(S)\rightarrow \prod_{n\in\mathbb{N}}\varphi_n$ coincides with the weak star topology on $C(S)$. $\prod_{n\in\mathbb{N}}\varphi_n$ is metrizable and so is $C(S)$. For more details and an explicit metric look at http://www-bcf.usc.edu/~fbonahon/Research/Preprints/Bouquin.pdf page 39

I'm starting to study geodesic currents and I have a question concerning uniform continuity.

Let's take $S$ a closed surface of genus $g$ and $C(S)$ the space of geodesic currents on $S$ (as it is defined by Bonahon, endowed with the weak star topology). Let's call $\mathcal{C}(S)$ the set of isotopy classes of closed curves on $S$ and $\mathcal{S}\subset\mathcal{C}(S)$ the subset corresponding to simple closed curves.

It is known that $\mathcal{C}(S)$ injects in $C(S)$ and that there is a continuous extension $i:C(S)\times C(S)\rightarrow \mathbb{R}$ of the intersection number on $\mathcal{C}(S)$ .

Writing $T(S)$ for the Teichmuller space of $S$, it is also known that there is an injection $T(S)\rightarrow C(S)$ which sends a hyperbolic metric $h$ to the Liouville current $L_h$. For every Liouville current $L_h$ and every $\alpha\in \mathcal{S}$ it is true $l_h(\alpha)=i(L_h,\alpha)$.

My question is: I know that, for every $h\in T(S)$ hyperbolic metric, that the function $i(L_h,-):\mathcal{S}\rightarrow \mathbb{R}^+$ is continuous, but is it also $uniformly$ continuous?

I'm starting to study geodesic currents and I have a question concerning uniform continuity.

Let's take $S$ a closed surface of genus $g$ and $C(S)$ the space of geodesic currents on $S$ (as it is defined by Bonahon, endowed with the weak star topology). Let's call $\mathcal{C}(S)$ the set of isotopy classes of closed curves on $S$ and $\mathcal{S}\subset\mathcal{C}(S)$ the subset corresponding to simple closed curves.

It is known that $\mathcal{C}(S)$ injects in $C(S)$ and that there is a continuous and bilinear extension $i:C(S)\times C(S)\rightarrow \mathbb{R}$ of the intersection number on $\mathcal{C}(S)$ .

Writing $T(S)$ for the Teichmuller space of $S$, it is also known that there is an injection $T(S)\rightarrow C(S)$ which sends a hyperbolic metric $h$ to the Liouville current $L_h$. For every Liouville current $L_h$ and every $\alpha\in \mathcal{S}$ it is true $l_h(\alpha)=i(L_h,\alpha)$.

My question is: I know that, for every $h\in T(S)$ hyperbolic metric, that the function $i(L_h,-):\mathcal{S}\rightarrow \mathbb{R}^+$ is continuous, but is it also $uniformly$ continuous?

Edit: yes, as Misha observed of course to talk about uniform continuity we must have a metric. As Bruno Martelli said $C(S)$ is naturally a metric space (I should have mentioned it, my bad).

The metric of $C(S)$: consider $\mathcal{G}$ the space of geodesics on $\widetilde{S}$ (the universal cover of $S$), so that $C(S)$ is the space of radon measures on $\mathcal{G}$ invariant under the action of the covering group. Then we can find a countable family of continuous functions $\varphi_n:\mathcal{G}\rightarrow \mathbb{R}$ such that the pullback of the product topology on $\prod_{n\in\mathbb{N}}\varphi_n$ by the natural map $C(S)\rightarrow \prod_{n\in\mathbb{N}}\varphi_n$ coincides with the weak star topology on $C(S)$. $\prod_{n\in\mathbb{N}}\varphi_n$ is metrizable and so is $C(S)$. For more details and an explicit metric look at http://www-bcf.usc.edu/~fbonahon/Research/Preprints/Bouquin.pdf page 39

I'm starting to study geodesic currents and I have a question concerning uniform continuity.

Let's take $S$ a closed surface of genus $g$ and $C(S)$ the space of geodesic currents on $S$ (as it is defined by Bonahon, endowed with the weak star topology). Let's call $\mathcal{C}(S)$ the set of isotopy classes of closed curves on $S$ and $\mathcal{S}\subset\mathcal{C}(S)$ the subset corresponding to simple closed curves.

It is known that $\mathcal{C}(S)$ injects in $C(S)$ and that there is a continuous extension $i:C(S)\times C(S)\rightarrow \mathbb{R}$ of the intersection number on $\mathcal{C}(S)$ .

Writing $T(S)$ for the Teichmuller space of $S$, it is also known that there is an injection $T(S)\rightarrow C(S)$ which sends a hyperbolic metric $h$ to the Liouville current $L_h$. For every Liouville current $L_h$ and every $\alpha\in \mathcal{S}$ it is true $l_h(\alpha)=i(L_h,\alpha)$.

My question is: I know that, for every $h\in T(S)$ hyperbolic metric, that the function $i(L_h,-):\mathcal{S}\rightarrow \mathbb{R}^+$ is continuous, but is it also $uniformly$ continuous?

Edit: yes, as Misha observed of course to talk about uniform continuity we must have a metric. As Bruno Martelli said $C(S)$ is naturally a metric space (I should have mentioned it, my bad).

The metric of $C(S)$: consider $\mathcal{G}$ the space of geodesics on $\widetilde{S}$ (the universal cover of $S$), so that $C(S)$ is the space of radon measures on $\mathcal{G}$ invariant under the action of the covering group. Then we can find a countable family of continuous functions $\varphi_n:\mathcal{G}\rightarrow \mathbb{R}$ such that the pullback of the product topology on $\prod_{n\in\mathbb{N}}\varphi_n$ by the natural map $C(S)\rightarrow \prod_{n\in\mathbb{N}}\varphi_n$ coincides with the weak star topology on $C(S)$. $\prod_{n\in\mathbb{N}}\varphi_n$ is metrizable and so is $C(S)$. For more details and an explicit metric look at http://www-bcf.usc.edu/~fbonahon/Research/Preprints/Bouquin.pdf page 39