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We prove a ratio ergodic theorem for non-singular free ${\mathbb{Z}}^d$ and ${ℝ}^d$ actions, along balls in an arbitrary norm. Using a Chacon–Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in ${ℝ}^d$. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general group...

Let $\mu$ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_a\left(x\right)=ax\mathrm{𝚖𝚘𝚍}1$, and $0<\mathrm{𝚍𝚒𝚖}\mu <1$. Let $f$ be a local diffeomorphism on some open set. We show that if $E\subseteq ℝ$ and $\left(f\mu \right){\left|{}_E\sim \mu \right|}_E$, then $f^{\text{'}}\left(x\right)\in \left\{\pm a^r:r\in \mathbb{Q}\right\}$ at $\mu$-a.e. point $x\in f^{-1}E$. In particular, if $g$ is a piecewise-analytic map preserving $\mu$ then there is an open $g$-invariant set $U$ containing supp $\mu$ such that $g\left|{}_U\right.$ is piecewise-linear with slopes which are rational powers of $a$. In a similar vein, for $\mu$ as above, if $b$ is another integer and $a,b$ are not powers of a common integer, and if $\nu$ is a $T_b$-invariant...