Let $G$ be a locally compact group, let $\Gamma$ be a lattice in $G$, and let $\eta$ be a spread-out probability measure on $G$. A Furstenberg discretization of $\eta$ to $\Gamma$ is a probability measure $\mu$ on $\Gamma$ such that every $(G,\eta)$-stationary space is also $(\Gamma,\mu)$-stationary. For cocompact lattices $\Gamma$ and under mild regularity conditions on $\eta$, we construct Furstenberg discretizations satisfying finite moment conditions of arbitrary order with respect to arbitrary prescribed subadditive functions. Our approach relies on a rather general study of differential entropy and Lyapunov exponents in the setting of locally compact groups. We relate Lyapunov exponents to spectral radii in weighted $L^{1}$-algebras and prove vanishing results for weights coming from length functions. Additionally, we prove that if $G$ has the property of rapid decay or the Kunze--Stein property, this yields entropy formulas for stationary actions whose Koopman representation is weakly contained in the left-regular representation. As consequences, we obtain preservation results for Poisson boundaries under discretization and characterize Zimmer amenability of stationary actions in terms of weak containment and maximal Furstenberg entropy.
Let $(M^n,g)$ be a complete, connected, noncompact Riemannian manifold, and let $f\ge0$ be a proper $C^2$ weight. We assume polynomial bounds on $f$, $|\nabla f|$, and $\Delta f$ (Assumptions H1-H3), with growth exponent $\alpha>0$. We study the drift Laplacian $L_f=\Delta-\langle\nabla f,\nabla\cdot\rangle$ on $L^2(M,...
We show that if $G$ is a real semisimple Lie group and $\Gamma<G$ is a discrete subgroup with slow growth, in the sense that its growth indicator function is smaller than $\rho$, then $\Gamma$ acts totally dissipatively on the Furstenberg boundary of $G$. Moreover, for $G= SO(n,2)$ and $n\geq 3$, we construct infinite-...
S. Dey, Mikołaj Frączyk, Sebastian Hurtado· 2 citations· ⚡1
Let $g$ be a $C^\infty$ diffeomorphism in the isotopy class of a pseudo-Anosov homeomorphism $f$ such that $g$ and $f$ have the same topological entropy. In 1988, Handel proved that this implies the existence of a semiconjugacy $\pi$ from $g$ to $f$. He stated that, in general, there is at least one point $x$ such that...
Pablo D. Carrasco, F. Hertz, J. R. Hertz et al.· 0 citations
Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective...
Ren-Yu Chen, Song-Ying Li, Su-Juan Long et al.· 0 citations
Let $\Gamma$ be an infinite finitely generated group that is hyperbolic or of polynomial growth. For $f$ belonging to the reduced group $C^*$ algebra $C_r^*(\Gamma)$, we construct a rearrangement of its individual Fourier terms with a subsequence converging to $f$ in the operator norm. For every $2\le p<\infty$ and $f$...
Ono, Schneider and Wagner found a partition-theoretic analogue of Alladi's duality formula: for many sets $S\subseteq\N$, the series $-\sum\mu_{\mathcal P}(\lambda)q^{|\lambda|}$ over partitions into distinct parts with smallest part in $S$ tends to the density of $S$ as $q\to1$. We determine when this happens. Radiall...
Mohammad F. Marashdeh· 0 citations
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