×
Zava, Nicolò

On a notion of entropy in coarse geometry. (English) Zbl 1484.54019

Topol. Algebra Appl. 7, 48-68 (2019).
Summary: The notion of entropy appears in many branches of mathematics. In each setting (e.g., probability spaces, sets, topological spaces) entropy is a non-negative real-valued function measuring the randomness and disorder that a self-morphism creates. In this paper we propose a notion of entropy, called coarse entropy, in coarse geometry, which is the study of large-scale properties of spaces. Coarse entropy is defined on every bornologous self-map of a locally finite quasi-coarse space (a recent generalisation of the notion of coarse space, introduced by Roe). In this paper we describe this new concept, providing basic properties, examples and comparisons with other entropies, in particular with the algebraic entropy of endomorphisms of monoids.

Cited in 4 Documents

MSC:

54C70 Entropy in general topology
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
37B40 Topological entropy
54A99 Generalities in topology
54E15 Uniform structures and generalizations
54E99 Topological spaces with richer structures

Keywords:

coarse space; quasi-coarse space; coarse entropy; algebraic entropy; growth of metric spaces; coarse equivalence
Cite Review PDF
Full Text: DOI
OA License

References:

[1] R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy, Trans. Amer. Math. 114 (1965) 309-319. · Zbl 0127.13102
[2] J. A. Álvarez López, A. Candel, Generic coarse geometry of leaves, Lecture Notes in Mathematics, Springer International Publishing (2018). · Zbl 1426.57001
[3] G. Bell, A. N. Dranishnikov, Asymptotic dimension. Topology Appl. 155 (2008), 1265-1296. · Zbl 1149.54017
[4] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971) 401-414. · Zbl 0212.29201
[5] T. Ceccherini-Silberstein, M. Coornaert, F. Krieger, An analogue of Fekete’s lemma for subadditive functions on cancellative amenable semigroups, J. Analyse Math. 124 (2014) 59-81. · Zbl 1308.43002
[6] D. Dikranjan, A. Fornasiero, A. Giordano Bruno, Algebraic entropy for amenable semigroup actions, preprint. · Zbl 1453.20074
[7] D. Dikranjan, A. Giordano Bruno, Functorial topologies, Topology Appl. 158 (17) (2011), 2261-2442. · Zbl 1237.54033
[8] D. Dikranjan, A. Giordano Bruno, Topological entropy and algebraic entropy for group endomorphisms, Proceedings ICTA2011 Islamabad, Pakistan July 4-10 2011 Cambridge Scientific Publishers 133-214. · Zbl 1300.54002
[9] D. Dikranjan, A. Giordano Bruno, The connection betwwen topological and algebraic entropy, Topology Appl. 159 (13) (2012) 2980-2989. · Zbl 1256.54061
[10] D. Dikranjan, A. Giordano Bruno, Entropy on abelian groups, Advances in Mathematics 298 (2016), 612-653. · Zbl 1368.37015
[11] D. Dikranjan, A. Giordano Bruno, Entropy on normed semigroups (Towards a unifying approach to entropy), Dissertationes Math. 542 (2019), 1-90. · Zbl 1429.16002
[12] D. Dikranjan, B. Goldsmith, L. Salce, P. Zanardo, Algebraic entropy for abelian groups, Trans. Amer. Math. Soc. 361 (2009), 3401-3434. · Zbl 1176.20057
[13] D. Dikranjan, N. Zava, Some categorical aspects of coarse spaces and balleans, Topology Appl., 225 (2017), 164-194. · Zbl 1377.54027
[14] D. Dikranjan, N. Zava, Categories of coarse groups: quasi-homomorphisms and functorial coarse structures, Topology Appl., to appear. · Zbl 1377.54027
[15] D. Dikranjan, N. Zava, The impact of Pontryagin and Bohr functors on large-scale properties of LCA groups, Topology Appl., to appear. · Zbl 1423.54053
[16] A. Dranishnikov, J. Smith, Asymptotic dimension of discrete groups. Fund. Math. 189 (1) (2006) 27-34. · Zbl 1100.20034
[17] J. Dydak, C. S. Hoffland, An alternative definition of coarse structures, Topology Appl. 155 (2008), no. 9, 1013-1021. · Zbl 1145.54032
[18] R. Engelking, Dimension Theory. North Holland Publishing Company, 1978. · Zbl 0401.54029
[19] M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, vol. 2 (Sussex, 1991), 1-295, London Math. Soc. Lecture Notes Ser., 182, Cambridge Univ. Press, Cambridge, 1993. · Zbl 0841.20039
[20] P. de la Harpe, Topics in geometric group theory. Chicago Lectures in Math., the University of Chicago Press, Chicago (2000). · Zbl 0965.20025
[21] K.H. Hofmann, L. Stoyanov, Topological entropy of group and semigroup actions, Adv. Math. 115 (1995) no. 1, 54-98. · Zbl 0865.22003
[22] B. M. Hood, Topological entropy and uniform spaces, J. London Math. Soc. 8 (2) (1974) 235-242. · Zbl 0291.54051
[23] J. R. Isbell, Uniform Spaces, American Mathematical Society (1964). · Zbl 0124.15601
[24] P. Fletcher, W. F. Lindgren, Quasi-Uniform Spaces, Lecture Notes Pure Appl. Math., 77, Dekker, New York (1982). · Zbl 0501.54018
[25] A. N. Kolmogorov, New metric invariants of transitive dynamical systems and automorphisms of Lebesgue spaces, Doklady Akad. Nauk. SSSR 119 (1958) 861-864. · Zbl 0083.10602
[26] H.-P. Kunzi, Nonsymmetric distances and their associated topologies: about the origins of basic ideals in the area of asymmetric topology, in Handbook of the history of general topology, Vol. 3, 853-968, Hist. Topol., 3, Kluwer Acad. Publ., Dordrecht, 2001. · Zbl 1002.54002
[27] A. Nicas, D. Rosenthal, Coarse structures on groups, Topology Appl. 159 (14) (2012) 3215-3228. · Zbl 1247.22003
[28] A. Nicas, D. Rosenthal, On the asymptotic dimension of the dual group of a locally compact abelian group, Topology Appl. 160 (2013), 682-684. · Zbl 1261.22004
[29] P. W. Nowak, G. Yu, Large Scale Geometry. European Mathematical Society, 2012. · Zbl 1264.53051
[30] D. Ornstein, B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987) 1-141. · Zbl 0637.28015
[31] J. Peters, Entropy on discrete abelian groups, Adv. Math. 33 (1979), 1-13. · Zbl 0421.28019
[32] J. Peters, Entropy of automorphisms on L.C.A. groups, Pacific J. Math. 96 (2) (1981) 475-488. · Zbl 0478.28010
[33] L. S. Pontryagin, Topological Groups. Gordon and Breach, Science Publishers, Inc., 1966.
[34] I. V. Protasov, Asymptotic proximities, Applied Gen. Top., 9 (2008), no. 2, 189-195. · Zbl 1179.54041
[35] I. V. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs, Math. Stud. Monogr. Ser., vol. 11, VNTL, Lviv, 2003. · Zbl 1147.05033
[36] I. V. Protasov, O. I. Protasova, Sketch of group balleans, Matematychni Studii, 22 (2004), no. 1, 10-20. · Zbl 1061.22002
[37] I. V. Protasov, M. Zarichnyi, General Asymptology, 2007 VNTL Publishers, Lviv, Ukraine. · Zbl 1172.54002
[38] J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, American Mathematical Society, Providence RI, 2003. · Zbl 1042.53027
[39] C. E. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal, 27 (3) (1948) 379-423. · Zbl 1154.94303
[40] Y. G. Sinai, On the concept of entropy of a dynamical system, Doklady Akad. Nauk. SSSR 124 (1959) 789-781 (in Russian).
[41] L. N. Stoyanov, Uniqueness of topological entropy for endomorphisms on compact groups, Boll. Un. Mat. Ital. 1 (3) (1987) 829-847. · Zbl 0648.22002
[42] S. Virili, Entropy for endomorphisms of LCA groups, Topology Appl. 159 (9) (2012), 2546-2556. · Zbl 1243.22007
[43] M. D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory 8 (3) (1974/75) 243-248. · Zbl 0298.28014
[44] N. Zava, Generalisations of coarse spaces, Topology Appl. 263 (2019), 230-256. · Zbl 1423.54054
[45] N. Zava, Cowellpoweredness and closure operators in categories of coarse spaces, Topology Appl. 268 (2019), 106899. · Zbl 1429.54036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.