Logarithmic expansion, entropy, and dimension for set-valued maps

D. Carrasco-Olivera ; R. Metzger ; C. Morales

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Optimal control, Tome 178 (2020), pp. 31-40.

Résumé

We obtain a lower bound for the entropy of a (not necessarily invariant) Borel probability measure with respect to an upper semicontinuous set-valued map as the product of the lower dimension of the measure and the logarithmic expansion rate. This is a generalization of the well-known measure-preserving single-valued case.

Détail

Keywords: logarithm expansion, metric entropy
Mots-clés : dimension.

Bibliographie

[1] Dinaburg E. I., “Sootnoshenie mezhdu topologicheskoi entropiei i metricheskoi entropiei”, Dokl. AN SSSR., 190:1 (1970), 19–22 | Zbl

[2] Kolmogorov A. N., “Novyi metricheskii invariant tranzitivnykh dinamicheskikh sistem i avtomorfizmov prostranstv Lebega”, Tr. Mat. in-ta im. V. A. Steklova., 169 (1985), 94–98

[3] Sinai Ya. G., “O ponyatii entropii dinamicheskoi sistemy”, Dokl. AN SSSR., 124 (1959), 768–771 | Zbl

[4] Khinchin A. Ya., “Ob osnovnykh teoremakh teorii informatsii”, Usp. mat nauk., 11:1 (67) (1956), 17–75 | MR | Zbl

[5] Adler R. L., Konheim A. G., McAndrew M. H., “Topological entropy”, Trans. Am. Math. Soc., 114 (1965), 309–319 | DOI | MR | Zbl

[6] Akin E., The General Topology of Dynamical Systems, Am. Math. Soc., Providence, Rhode Island, 1993 | Zbl

[7] Aubin J.-P., Cellina A., Differential Inclusions. Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984 | MR | Zbl

[8] Aubin J.-P., Frankowska H., Set-Valued Analysis, Birkhäuser, Boston, Massachusetts, 2009 | MR | Zbl

[9] Aubin J.-P., Frankowska H., Lasota A. , “Poincaré's recurrence theorem for set-valued dynamical systems”, Ann. Polon. Math., 54:1 (1991), 85–91 | DOI | MR | Zbl

[10] Barreira L., Dimension and Recurrence in Hyperbolic Dynamics, Birkhäuser, Basel, 2008 | MR | Zbl

[11] Barreira L., Pesin Y., Schmeling J., “Dimension and product structure of hyperbolic measures”, Ann. Math. (2), 149:3 (1999), 755–783 | DOI | MR | Zbl

[12] Blumenthal L. M., “A new concept in distance geometry with applications to spherical subsets”, Bull. Am. Math. Soc., 47 (1941), 435–443 | DOI | MR

[13] Boltzmann L. , “Über die beziehung dem zweiten Haubtsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht”, Wiener Berichte., 76 (1877), 373–435

[14] Bowen R., “Entropy for group endomorphisms and homogeneous spaces”, Trans. Am. Math. Soc., 153 (1971), 401–414 | DOI | MR | Zbl

[15] Burachik R. S., Iusem A. N., Set-Valued Mappings and Enlargements of Monotone Operators, Springer-Verlag, New York, 2008 | MR

[16] Carrasco-Olivera D., Metzger R., Morales C. A., “Topological entropy for set-valued maps”, Discr. Contin. Dyn. Syst. Ser. B, 20:10 (2015), 3461–3474 | MR | Zbl

[17] Cheng W.-C., Zhao Y., Cao Y., “Pressures for asymptotically sub-additive potentials under a mistake function”, Discr. Contin. Dyn. Syst., 32:2 (2012), 487–497 | DOI | MR | Zbl

[18] Clausius R., “Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie”, Ann. Phys., 125, 353–400

[19] Clausius R., The Mechanical Theory of Heat with Its Applications to the Steam Engine and to Physical Properties of Bodies, John van Voorst, London, 1867

[20] Dai X., Zhou Z., Geng X., “Some relations between Hausdorff dimensions and entropies”, Sci. China Ser. A., 41:10 (1998), 1068–1075 | DOI | MR | Zbl

[21] Downarowicz T., Entropy in Dynamical Systems, Cambridge Univ. Press, Cambridge, 2011 | MR | Zbl

[22] Fathi A., “Expansiveness, hyperbolicity and Hausdorff dimension”, Commun. Math. Phys., 126:2 (1989), 249–262 | DOI | MR | Zbl

[23] Gillam B. E., “A new set of postulates for Euclidean geometry”, Revista Ci. Lima., 1940, 869–899 | MR | Zbl

[24] Hirsch M., “Asymptotic phase, shadowing and reaction-diffusion systems”, Lect. Notes Pure Appl. Math., 152 (1994), 87–99 | MR | Zbl

[25] Katok A., “Fifty years of entropy in dynamics: 1958–2007”, J. Mod. Dyn., 1:4 (2007), 545–596 | DOI | MR | Zbl

[26] Katok A., “Lyapunov exponents, entropy and periodic orbits for diffeomorphisms”, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137–173 | DOI | MR | Zbl

[27] Maschler M., Peleg B., “Stable sets and stable points of set-valued dynamic systems with applications to game theory”, SIAM J. Control Optim., 14:6 (1976), 985–995 | DOI | MR | Zbl

[28] Mañé R., “Expansive homeomorphisms and topological dimension”, Trans. Am. Math. Soc., 252 (1979), 313–319 | DOI | MR | Zbl

[29] McMillan B., “The basic theorems of information theory”, Ann. Math. Stat., 24 (1953), 196–219 | DOI | MR | Zbl

[30] Reddy W. L., “Expanding maps on compact metric spaces”, Topol. Appl., 13:3 (1982), 327–334 | DOI | MR | Zbl

[31] Rockafellar R. T., Convex Analysis, Princeton Univ. Press, Princeton, New Jersey, 1997 | MR | Zbl

[32] Schreiber S., “Expansion rates and Lyapunov exponents”, Discr. Contin. Dynam. Syst., 3:3 (1997), 433–438 | DOI | MR | Zbl

[33] Shannon C. E., “A mathematical theory of communication”, Bell System Tech. J., 27 (1948), 379–423, 623–656 | DOI | MR | Zbl

[34] Sinai Y., “Kolmogorov–Sinai entropy”, Scholarpedia., 4:3 (2009), 2034 | DOI

[35] Tarafdar E., Watson P., Yuan X.-Z., “Poincare's recurrence theorems for set-valued dynamical systems”, Appl. Math. Lett., 10:6 (1997), 37–44 | DOI | MR | Zbl

[36] Tarafdar E., Yuan X.-Z. , “The set-valued dynamic system and its applications to Pareto optima”, Acta Appl. Math., 46:1 (1997), 93–106 | DOI | MR | Zbl

[37] von Neumann J., Mathematische Grundlagen der Quantenmechanik, Springer-Verlag, Berlin–New York, 1968 (in German) | MR | Zbl

[38] Walters P., An Introduction to Ergodic Theory, Springer-Verlag, New York–Berlin, 1982 | MR | Zbl

[39] Williams R. K., “A note on expansive mappings”, Proc. Am. Math. Soc., 22 (1969), 145–147 | DOI | MR | Zbl