Abstract
We discuss the finiteness of the topological entropy of continuous endomorphims for some classes of locally compact groups. Firstly, we focus on the abelian case, imposing the condition of being compactly generated, and note an interesting behaviour of slender groups. Secondly, we remove the condition of being abelian and consider nilpotent periodic locally compact p-groups (p prime), reducing the computations to the case of Sylow p-subgroups. Finally, we investigate locally compact Heisenberg p-groups \(\mathbb{H}_n (\mathbb{Q}_ p )\) on the field \(\mathbb{Q}_ p \) of the p-adic rationals with n arbitrary positive integer.
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References
R. L. Adler, A. G. KonheimandM. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309–319.
M. Bonatto and D. Dikranjan, Generalized Heisenberg groups and self-duality, Quest. Answ. Gen. Topol., 37 (2019), 89–108.
R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401–414.
A. Caldas and M. Patrão, Entropy and its variational principle for locally compact metrizable systems, Ergodic Theory Dynam. Systems, 38 (2018), 540–565.
D. van Dantzig, Studien over Topologische Algebra, Dissertation (Amsterdam, 1931).
G. De Marco and A. Orsatti, Complete linear topologies on abelian groups, Sympos. Math., 13 (1974),153–161.
D. Dikranjan and M. Sanchis, Dimension and entropy in compact topological groups, J. Math. Anal. Appl., 476 (2019), 337–366.
D. Dikranjan, A. Giordano Bruno and F. G. Russo, Finiteness of topological entropy for locally compact abelian groups, Glasgow Math. J., 361 (2020), 403–442.
L. Fuchs, Abelian Groups, Springer (Cham, 2015).
A. Giordano Bruno and S. Virili, Topological entropy in totally disconnected locally compact groups, Ergod. Theory Dynam. Systems, 37 (2017), 2163–2186.
T. Grundhöfer and M. Stroppel, Automorphisms of Verardi groups: small upper triangular matrices over rings, Beitr. Algebra Geom., 49 (2008), 1–31.
W. Herfort, K. H. Hofmann and F. G. Russo, Periodic Locally Compact Groups, de Gruyter (Berlin, 2019).
K. H. Hofmann and S. A. Morris, The Structure of Compact Groups, de Gruyter (Berlin, 2020).
B. M. Hood, Topological entropy and uniform spaces, J. London Math. Soc., 8 (1974), 633–641.
C. R. Leedham-Green and S. McKay, The Structure of Groups of Prime Power Order, Oxford University Press (Oxford, 2002).
D. A. Lind and T. Ward, Automorphisms of solenoids and p-adic entropy, Ergod. Theory Dynam. Systems, 8 (1988), 411–419.
M. Patrão, The topological entropy of endomorphisms of Lie groups, Israel J. Math., 234 (2019), 55–80.
J. Peters, Entropy of automorphisms on locally compact abelian groups, Pacific J. Math., 96 (1981), 475–488.
L. Ribes and P. Zalesskii, Profinite Groups, Springer (Berlin, 2000).
D. J. S. Robinson, A Course in the Theory of Groups, Springer (New York, 1996).
K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser (Basel, 1995).
S. Virili, Entropy for endomorphisms of locally compact abelian groups, Topology Appl., 159 (2012), 2546–2556.
P. Walters, An Introduction to Ergodic Theory, Springer (Berlin, 1969).
S. Yuzvinski, Metric properties of endomorphisms of compact groups, Izv. Acad. Nauk SSSR, Ser. Mat., 29 (1965), 1295–1328 (in Russian); translation in Amer. Math. Soc. Transl., 66 (1968), 63–98.
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This paper is dedicated to the memory of Bernhard Banaschewski, whose inspiring seminars influenced the scientific works of the research group in "Topology, Algebra and Dynamical Systems" at the University of Cape Town for decades
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The authors thank the National Research Foundation of South Africa (South Africa) for the grant ISARP with Ref. No. ITAL22051410615 and for the project "Topology for Tomorrow", but also Gruppo Nazionale per la Fisica Matematica of INdAM (Italy).
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Russo, F.G., Waka, O. On locally compact groups of small topological entropy. Acta Math. Hungar. 172, 62–79 (2024). https://doi.org/10.1007/s10474-024-01405-2
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DOI: https://doi.org/10.1007/s10474-024-01405-2
Keywords and phrases
- topological entropy
- locally compact group
- dynamical system
- Sylow p-subgroup
- slender group
- Heisenberg p-group