Optimization and majorization of invariant measures

Jenkinson, Oliver¹

¹ School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK

Electronic research announcements of the American Mathematical Society, Tome 13 (2007), pp. 1-12

Résumé

The set of $\times 2$-invariant measures can be equipped with the partial order of majorization, describing relative dispersion. The minimal elements for this order are precisely the Sturmian measures of Morse and Hedlund. This yields new characterisations of Sturmian measures, and has applications to the ergodic optimization of convex functions.

Détail

DOI : 10.1090/S1079-6762-07-00170-9

Bibliographie

[1] Allouche, Jean-Paul, Shallit, Jeffrey Automatic sequences 2003

[2] Bawa, Vijay S. Stochastic dominance: a research bibliography Management Sci. 1982 698 712

[3] Beardon, Alan F. Iteration of rational functions 1991

[4] Bernhardt, C. Rotation intervals of a class of endomorphisms of the circle Proc. London Math. Soc. (3) 1982 258 280

[5] Blackwell, David Comparison of experiments 1951 93 102

[6] Blackwell, David Equivalent comparisons of experiments Ann. Math. Statistics 1953 265 272

[7] Bousch, Thierry Le poisson n’a pas d’arêtes Ann. Inst. H. Poincaré Probab. Statist. 2000 489 508

[8] Bousch, Thierry La condition de Walters Ann. Sci. École Norm. Sup. (4) 2001 287 311

[9] Bousch, Thierry, Jenkinson, Oliver Cohomology classes of dynamically non-negative 𝐶^{𝑘} functions Invent. Math. 2002 207 217

[10] Bowen, Rufus Equilibrium states and the ergodic theory of Anosov diffeomorphisms 1975

[11] Bullett, Shaun, Sentenac, Pierrette Ordered orbits of the shift, square roots, and the devil’s staircase Math. Proc. Cambridge Philos. Soc. 1994 451 481

[12] Cartier, Pierre, Fell, J. M. G., Meyer, Paul-André Comparaison des mesures portées par un ensemble convexe compact Bull. Soc. Math. France 1964 435 445

[13] Choquet, Gustave Le théorème de représentation intégrale dans les ensembles convexes compacts Ann. Inst. Fourier (Grenoble) 1960 333 344

[14] Chothi, V., Everest, G., Ward, T. 𝑆-integer dynamical systems: periodic points J. Reine Angew. Math. 1997 99 132

[15] Contreras, G., Lopes, A. O., Thieullen, Ph. Lyapunov minimizing measures for expanding maps of the circle Ergodic Theory Dynam. Systems 2001 1379 1409

[16] Douady, A., Hubbard, J. H. Étude dynamique des polynômes complexes. Partie I 1984 75

[17] Elton, J., Hill, T. P. Fusions of a probability distribution Ann. Probab. 1992 421 454

[18] Gambaudo, Jean-Marc, Lanford, Oscar, Iii, Tresser, Charles Dynamique symbolique des rotations C. R. Acad. Sci. Paris Sér. I Math. 1984 823 826

[19] Hardy, G. H., Littlewood, J. E., Pólya, G. Inequalities 1952

[20] Jenkinson, Oliver Frequency locking on the boundary of the barycentre set Experiment. Math. 2000 309 317

[21] Jenkinson, Oliver Ergodic optimization Discrete Contin. Dyn. Syst. 2006 197 224

[22] Karlin, Samuel, Novikoff, Albert Generalized convex inequalities Pacific J. Math. 1963 1251 1279

[23] Katok, Anatole, Hasselblatt, Boris Introduction to the modern theory of dynamical systems 1995

[24] Keane, Michael Strongly mixing 𝑔-measures Invent. Math. 1972 309 324

[25] Keller, Karsten Invariant factors, Julia equivalences and the (abstract) Mandelbrot set 2000

[26] Kertz, Robert P., Rösler, Uwe Stochastic and convex orders and lattices of probability measures, with a martingale interpretation Israel J. Math. 1992 129 164

[27] Lindvall, Torgny Lectures on the coupling method 1992

[28] Lothaire, M. Algebraic combinatorics on words 2002

[29] Marshall, Albert W., Olkin, Ingram Inequalities: theory of majorization and its applications 1979

[30] Meilijson, Isaac Stochastic orders and stopping times in Brownian motion 2003 207 219

[31] Milnor, John Dynamics in one complex variable 1999

[32] Morse, Marston, Hedlund, Gustav A. Symbolic dynamics II. Sturmian trajectories Amer. J. Math. 1940 1 42

[33] Ott, Edward Chaos in dynamical systems 2002

[34] Phelps, Robert R. Lectures on Choquet’s theorem 1966

[35] Fogg, N. Pytheas Substitutions in dynamics, arithmetics and combinatorics 2002

[36] Rothschild, Michael, Stiglitz, Joseph E. Increasing risk. I. A definition J. Econom. Theory 1970 225 243

[37] Schmidt, Klaus Dynamical systems of algebraic origin 1995

[38] Schur, Issai Gesammelte Abhandlungen. Band I 1973

[39] Schuster, Heinz Georg Deterministic chaos 1995

[40] Series, Caroline The geometry of Markoff numbers Math. Intelligencer 1985 20 29

[41] Strassen, V. The existence of probability measures with given marginals Ann. Math. Statist. 1965 423 439

[42] Tong, Yung Liang Probability inequalities in multivariate distributions 1980

[43] Van Der Vecht, D. P. Inequalities for stopped Brownian motion 1986

[44] Veerman, Peter Symbolic dynamics of order-preserving orbits Phys. D 1987 191 201