# Invariant measure for composition on space of continuous functions

Let $$C_g:C(\mathbb{R}^d;\mathbb{R}^d)\rightarrow C(\mathbb{R}^d;\mathbb{R}^d)$$ be defined by $$C_g(f)\triangleq f\circ g$$ for some fixed $$g \in C(\mathbb{R}^d;\mathbb{R}^d)$$.

What are examples of Borel probability measures on $$C(\mathbb{R}^d;\mathbb{R}^d)$$ which are invariant with respect to $$C_g$$ and are non-atomic (preferably also locally positive: ie: positive value on non-empty open subsets).

Note: Here, $$C(\mathbb{R}^d;\mathbb{R}^d)$$ is equipped with the compact-open topology; ie: uniform convergence on compacts.

• I'm not sure this can have a meaningful answer in this generality. Of course it's trivial to give examples in special cases such as $g(x)=0$ or $g(x)=x+1$ (or anything that has a periodic orbit for that matter), but I suppose that's not what you wanted. – Christian Remling Nov 6 '19 at 17:46
• (Though I find the general case most interesting) and example which I'm also particularly curious about is the case where $g(x)=Ax+b$ and $A\in Mat_{d\times d},\, b \in \mathbb{R}^d$. – AIM_BLB Nov 6 '19 at 18:16
• By the way, the extra assumptions on the measures that you added don't really rule out the trivial examples from periodic orbits (for something like $g(x)=x+1$, say) since you can take averages of measures supported by periodic orbits. – Christian Remling Nov 6 '19 at 18:19
• if g is in the orthogonal group I suppose you can take any proba on R^d and average it by the compact group. – sanette Nov 8 '19 at 9:42
• I'd phrase it in a different way: the distribution of the Ornstein-Uhlenbeck process is shift-invariant. It also satisfies your hypothesis of locality (full support in continuous functions). – D. Thomine Nov 9 '19 at 10:37

Let us assume that $$g$$ is invertible. Then everything depends on the "size" of the closed subgroup $$K_g$$ generated by $$g$$. If $$K_g$$ is compact, then its Haar measure and all its right convolutions are invariant with respect to the left $$g$$-translations. If $$G$$ is non-compact, then there are no probability $$g$$-invariant measures. I presume that a similar dichotomy should hold in the non-invertible case as well.
• Isn't $\delta_c$ invariant for any constant function $c$? I think your argument only gives the non-existence of invariant probability measures on $Homeo (\mathbb{R}^d)$ (or similar groups)... – D. Thomine Nov 8 '19 at 21:21
• I said "Let us assume that $g$ is invertible" at the very beginning – R W Nov 8 '19 at 22:15
• Yes, but $f$ needs not be. – D. Thomine Nov 9 '19 at 10:29