We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. We investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. We also present a proof due to Weiss for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy.
If $X$ is a sofic shift and $\varphi : X\rightarrow X$ is a homeomorphism such that ${\varphi }^{2} = {\text{id} }_{X} $ and $\varphi {\sigma }_{X} = { \sigma }_{X}^{- 1} \varphi $, the number of points in $X$ that are fixed by ${ \sigma }_{X}^{m} $ and ${ \sigma }_{X}^{n} \varphi , m= 1, 2, \ldots , n\in \mathbb{Z} $, is expressed in terms of a finite number of square matrices: the matrices are obtained from Krieger’s joint state chain of a sofic shift which is conjugate to $X$.
We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. On route, we investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. We also present a proof due to Weiss for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy.
A
$D_{\infty }$
-topological Markov chain is a topological Markov chain provided with an action of the infinite dihedral group
$D_{\infty }$
. It is defined by two zero-one square matrices A and J satisfying
$AJ=JA^{\textsf {T}}$
and
$J^2=I$
. A flip signature is obtained from symmetric bilinear forms with respect to J on the eventual kernel of A. We modify Williams’ decomposition theorem to prove the flip signature is a
$D_{\infty }$
-conjugacy invariant. We introduce natural
$D_{\infty }$
-actions on Ashley’s eight-by-eight and the full two-shift. The flip signatures show that Ashley’s eight-by-eight and the full two-shift equipped with the natural
$D_{\infty }$
-actions are not
$D_{\infty }$
-conjugate. We also discuss the notion of
$D_{\infty }$
-shift equivalence and the Lind zeta function.
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