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Relative uniformly positive entropy of induced amenable group actions
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 2 / February 2024
- Published online by Cambridge University Press:
- 17 March 2023, pp. 569-593
- Print publication:
- February 2024
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- Article
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Let G be a countably infinite discrete amenable group. It should be noted that a G-system
$(X,G)$ naturally induces a G-system 
$(\mathcal {M}(X),G)$, where 
$\mathcal {M}(X)$ denotes the space of Borel probability measures on the compact metric space X endowed with the weak*-topology. A factor map 
$\pi : (X,G)\to (Y,G)$ between two G-systems induces a factor map 
$\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$. It turns out that 
$\widetilde {\pi }$ is open if and only if 
$\pi $ is open. When Y is fully supported, it is shown that 
$\pi $ has relative uniformly positive entropy if and only if 
$\widetilde {\pi }$ has relative uniformly positive entropy.
