Abstract
The Pinsker subgroup of an abelian group with respect to an endomorphism was introduced in the context of algebraic entropy. Motivated by the nice properties and characterizations of the Pinsker subgroup, we generalize its construction in two directions. Indeed, we introduce the concept of entropy function h of an abelian category, and we define the Pinsker radical with respect to h, so that the class of all objects with trivial Pinsker radical is the torsion-free class of a torsion theory.
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References
Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)
Aoki, N.: Topological entropy and measure-theoretic entropy for automorphisms on compact groups. Math. Syst. Theory 5, 4–7 (1971)
Blanchard, F., Lacroix, Y.: Zero entropy factors of topological flows. Proc. Am. Math. Soc. 119(3), 985–992 (1993)
Borceux, F., Clementino, M.M.: Topological semi-abelian algebras. Adv. Math. 130, 425–453 (2005)
Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)
Clementino, M.M., Dikranjan, D., Tholen, W.: Torsion theories and radicals in normal categories. J. Algebra 305, 98–129 (2006)
Dickson, S.E.: A torsion theory for Abelian categories. Trans. Am. Math. Soc. 121, 223–235 (1966)
Dikranjan, D.: A uniform approach to chaos. Algebra meets Topology: Advances and Applications (Abstracts). UPC - Barcelona Tech. Barcelona, Spain, 19–23 July 2010. http://atlas-conferences.com/cgi-bin/abstract/cbah-54
Dikranjan, D., Giordano Bruno, A.: Entropy on abelian groups (preprint)
Dikranjan, D., Giordano Bruno, A.: The Pinsker subgroup of an algebraic flow. J. Pure Appl. Algebra (to appear)
Dikranjan, D., Giordano Bruno, A., Virili, S.: On the Pinsker subgroup of the i-entropy (work in progress)
Dikranjan, D., Giuli, E.: Factorizations, injectivity and compactness in categories of modules. Commun. Algebra 19(1), 45–83 (1991)
D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy of endomorphisms of abelian groups, Trans. Amer. Math. Soc. 361 (2009), 3401–3434.
Dikranjan, D., Gong, K., Zanardo, P.: Endomorphisms of abelian groups with small algebraic entropy. (submitted)
Dikranjan, D., Sanchis, M., Virili, S.: New and old facts about entropy in uniform spaces and topological groups. Topol. its Appl. (to appear)
Dikranjan, D., Tholen, W.: Categorical Structure of Closure Operators with Applications to Topology, Algebra and Discrete Mathematics. Mathematics and its Applications, vol. 346. Kluwer Academic Publishers, Dordrecht-Boston-London (1995)
Dikranjan, D., Virili, S.: A general approach to chaos (work in progress)
Everest, G., Ward, T.: Heights of polynomials and entropy in algebraic dynamics. Universitext, Springer London Ltd., London (1999)
Giordano Bruno, A., Virili, S.: Algebraic Yuzvinski Formula (preprint)
Golan, J.S.: Torsion theories. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 29. Longman Scientific & Technical, Harlow. Wiley, New York (1986)
Halmos, P.: On automorphisms of compact groups. Bull. Am. Math. Soc. 49, 619–624 (1943)
Hironaka, E.: What is...Lehmer’s number?. Not. Am. Math. Soc. 56(3), 374–375 (2009)
Janelidze, G., Márki, L., Tholen, W.: Semi-abelian categories. J. Pure Appl. Algebra 168(2–3), 367–386 (2002)
Janelidze, G., Tholen, W.: Characterization of torsion theories in general categories. Contemp. Math. 431, 249–256 (2007)
Kerr, D., Li, H.: Dynamical entropy in Banach spaces. Invent. Math 162(3), 649–686 (2005)
Lind, D., Ward, T.: Automorphisms of solenoids and p-adic entropy. Ergodic Theory Dynam. Systems 83, 411–419 (1988)
Northcott, D.G., Reufel, M.: A generalization of the concept of length. Q. J. Math. (Oxford) (2) 16, 297–321 (1965)
Peters, J.: Entropy on discrete Abelian groups. Adv. Math. 33, 1–13 (1979)
Poguntke, D.: Epimorphisms of compact groups are onto. Proc. Am. Math. Soc. 26, 503–504 (1970)
Salce, L., Vamos, P., Virili, S.: Extending length functions to polynomial rings via algebraic entropy. Forum Math. (to appear)
Salce, L., Zanardo, P.: A general notion of algebraic entropy and the rank entropy. Forum Math. 21(4), 579–599 (2009)
Stojanov, L.N.: Uniqueness of topological entropy for endomorphisms on compact groups. Boll. Unione Mat. Ital., B (7) 1(3), 829–847 (1987)
Vámos, P.: Additive functions and duality over Noetherian rings. Q. J. Math. (Oxford) (2) 19, 43–55 (1968)
Virili, S.: Algebraic i-entropies. Master Sci. thesis, University of Padova, Padova (2010)
Virili, S.: Entropy for endomorphisms of LCA groups. Topol. its Appl. (to appear)
Walters, P.: An Introduction to Ergodic Theory. Springer, New-York (1982)
Ward, T.: Entropy of compact group automorphisms. Online Lecture Notes. http://www.mth.uea.ac.uk/~h720/lecturenotes/
Weiss, M.D.: Algebraic and other entropies of group endomorphisms. Math. Syst. Theory 8(3), 243–248 (1974/75)
Yuzvinski, S.: Metric properties of endomorphisms of compact groups. Izv. Acad. Nauk SSSR, Ser. Mat. 29, 1295–1328 (1965) (in Russian); Am. Math. Soc. Transl. (2) 66, 63–98 (1968) (English translation)
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Dedicated to the memory of Maria Silvia Lucido.
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Dikranjan, D., Bruno, A.G. Entropy in a Category. Appl Categor Struct 21, 67–101 (2013). https://doi.org/10.1007/s10485-011-9256-1
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DOI: https://doi.org/10.1007/s10485-011-9256-1