Comptes Rendus
no. 9-10
Mathematical Analysis
A Billingsley type theorem for Bowen entropy
[Un théorème de type Billingsley pour l'entropie de Bowen]

Présenté par : Jean-Pierre Kahane

Ji-Hua Ma1 ; Zhi-Ying Wen2
1 Department of Mathematics, Wuhan University, Wuhan 430072, PR China
2 Department of Mathematics, Tsinghua University, Beijing 100081, PR China
Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 503-507.

Pour les sous-ensembles d'un espace métrique muni d'une application continue, Bowen avait introduit une notion d'entropie. Dans cette Note nous démontrons que l'entropie de Bowen peut être déterminée par les entropies locales de mesures. Ce résultat est un analogue du théorème de Billingsley pour la dimension de Hausdorff.

For subsets of a metric space with a continuous map, Bowen introduced a notion of entropy. In this Note we show that the Bowen entropy can be determined via the local entropies of measures. This result can be considered as an analogue of Billingsley's Theorem for the Hausdorff dimension.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.03.010
Ji-Hua Ma 1 ; Zhi-Ying Wen 2

1 Department of Mathematics, Wuhan University, Wuhan 430072, PR China
2 Department of Mathematics, Tsinghua University, Beijing 100081, PR China 
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Ji-Hua Ma; Zhi-Ying Wen. A Billingsley type theorem for Bowen entropy. Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 503-507. doi : 10.1016/j.crma.2008.03.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.03.010/

[1] P. Billingsley Ergodic Theory and Information, John Wiley and Sons Inc., New York, 1965

[2] R. Bowen Topological entropy for noncompact sets, Trans. Amer. Math. Soc., Volume 184 (1973), pp. 125-136

[3] M. Brin; A. Katok On local entropy, Geometric Dynamics, Lecture Notes in Mathematics, vol. 1007, Springer, Berlin, 1983, pp. 30-38

[4] H. Federer Geometric Measure Theory, Springer-Verlag, New York, 1969

[5] P. Mattilla Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, Cambridge, 1995

[6] Y. Pesin Dimension Theory in Dynamical Systems: Contemporary Views and Applications, The University of Chicago Press, Chicago and London, 1997

[7] C.E. Pfister; W.G. Sullivan On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, Volume 27 (2007), pp. 929-956

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