Expansive dynamics on locally compact groups

2020 ◽  
pp. 1-12
Author(s):  
BRUCE P. KITCHENS
Keyword(s):  
Dynamical System ◽  
Topological Group ◽  
Finite Number ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Coset Space ◽  
Locally Compact ◽  
Countable State ◽  
Full Shift ◽  
Totally Disconnected

Abstract Let $\mathcal {G}$ be a second countable, Hausdorff topological group. If $\mathcal {G}$ is locally compact, totally disconnected and T is an expansive automorphism then it is shown that the dynamical system $(\mathcal {G}, T)$ is topologically conjugate to the product of a symbolic full-shift on a finite number of symbols, a totally wandering, countable-state Markov shift and a permutation of a countable coset space of $\mathcal {G}$ that fixes the defining subgroup. In particular if the automorphism is transitive then $\mathcal {G}$ is compact and $(\mathcal {G}, T)$ is topologically conjugate to a full-shift on a finite number of symbols.

Locally compact groups: maximal compact subgroups and N-groups

1988 ◽  
Vol 104 (1) ◽  
pp. 47-64
Author(s):  
R. W. Bagley ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Compact Set ◽  
Locally Compact ◽  
Group A ◽  
Totally Disconnected ◽  
Compactly Generated

AbstractIf G is a locally compact group such thatG/G0contains a uniform compactly generated nilpotent subgroup, thenGhas a maximal compact normal subgroupKsuch thatG/Gis a Lie group. A topological groupGis an N-group if, for each neighbourhoodUof the identity and each compact setC⊂G, there is a neighbourhoodVof the identity such thatfor eachg∈G. Several results on N-groups are obtained and it is shown that a related weaker condition is equivalent to local finiteness for certain totally disconnected groups.

Locally compact groups with totally disconnected space of subgroups

2019 ◽  
Vol 22 (1) ◽  
pp. 119-132
Author(s):  
Hatem Hamrouni ◽  
Bilel Kadri
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Compact Topology ◽  
Chabauty Topology ◽  
Locally Compact Topological Group ◽  
Totally Disconnected

Abstract The closed subsets {\mathcal{F}(X)} of any topological space can be given a topology, called the Chabauty–Fell topology, in which {\mathcal{F}(X)} is quasi-compact. If X is a locally compact space, then {\mathcal{F}(X)} is Hausdorff. If {X=G} is a locally compact group, then the subset {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} of closed subgroups is closed in {\mathcal{F}(G)} , and the set of closed subgroups inherits its own compact topology, called the Chabauty topology. We develop some of the important properties of this topology. More precisely, the results contained in this paper deal with the following question: Given a locally compact topological group, when is {\mathcal{F}(G)} (resp. {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} ) a totally disconnected space?

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

scholarly journals Homomorphisms into totally disconnected, locally compact groups with dense image

2019 ◽  
Vol 31 (3) ◽  
pp. 685-701 ◽  
Author(s):  
Colin D. Reid ◽  
Phillip R. Wesolek
Keyword(s):  
Normal Subgroup ◽  
Open Subgroup ◽  
Locally Compact Groups ◽  
Group Homomorphism ◽  
Compact Groups ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Dense Image ◽  
Profinite Completions ◽  
Totally Disconnected

Abstract Let {\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair {(G,\phi^{-1}(L))} , where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.

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