Geodesic
Coverings of solenoids and automorphisms of semigroup $C^*$-algebras
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 2, pp. 275-286

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The paper deals with finite-sheeted covering mappings onto the $P$-adic solenoids and limit endomorphisms of semigroup $C^*$-algebras. The aim of our exposition is two-fold: firstly, to present the results concerning the above-mentioned mappings and endomorphisms; secondly, to demonstrate proofs for some of the results. It has been shown that every covering mapping onto a solenoid is isomorphic to a power mapping. We have considered dynamical properties of the covering mappings. A power mapping for the $P$-adic solenoid is topologically transitive. A criterion for the covering mapping to be chaotic has been given. The classical Euler–Fermat theorem may be used in its proof. We have studied limit endomorphisms of $C^*$-algebras generated by isometric representations for semigroups of rational numbers. We formulate criteria for limit endomorphisms to be automorphisms in number-theoretic, algebraic, and functional terms. The necessity of such a criterion has been given from the category-theoretic viewpoint.
Keywords: automorphism of $C^*$-algebras, chaotic, inductive sequence of Toeplitz algebras associated with sequence of prime numbers, inverse limit and sequence, finite-sheeted covering mapping, semigroup $C^*$-algebra, Toeplitz algebra, topologically transitive.
Mots-clés : solenoid, $*$-homomorphism 
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     author = {R. N. Gumerov},
     title = {Coverings of solenoids and automorphisms of semigroup $C^*$-algebras},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {275--286},
     publisher = {mathdoc},
     volume = {160},
     number = {2},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2018_160_2_a7/}
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R. N. Gumerov. Coverings of solenoids and automorphisms of semigroup $C^*$-algebras. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 2, pp. 275-286. http://geodesic.mathdoc.fr/item/UZKU_2018_160_2_a7/