Frequency Locking on the Boundary of the Barycentre Set

We consider the doubling map T : z Z2 of the circle. For each T-invariant probability measure μ we define its barycentre b(μ) = ∫S1 Z dμ(z), which describes its average weight around the circle. We study the set Ω of all such barycentres, a compact convex set with nonempty interior. Its boundary ∂Ω has a countable dense set of points of nondifferentiability, the worst possible regularity for the boundary of a convex set. We explain this behaviour in terms of the frequency locking of rotation numbers for a certain class of invariant measures, each supported on the closure of a Sturmian orbit.

[1]  Philip Boyland,et al.  Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals , 1986 .

[2]  P. Shiu.  An introduction to the theory of numbers (5th edition) , by I. Niven, H. S. Zuckerman and H. L. Montgomery. Pp 529. £14·50. 1991. ISBN 0-471-5460031 (Wiley) , 1991 .

[3]  J. Zukas.  Introduction to the Modern Theory of Dynamical Systems , 1998 .

[4]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[5]  G. A. Hedlund,et al.  Symbolic Dynamics II. Sturmian Trajectories , 1940 .

[6]  Peter Veerman,et al.  Symbolic dynamics of order-preserving orbits , 1987 .

[7]  Thierry Bousch,et al.  Le poisson n'a pas d'arêtes , 2000 .

[8]  Caroline Series,et al.  The geometry of markoff numbers , 1985 .

[9]  Peter Veerman,et al.  Symbolic dynamics and rotation numbers , 1986 .

[10]  Jean Berstel,et al.  Recent Results on Sturmian Words , 1995, Developments in Language Theory.

[11]  Floris Takens,et al.  Bifurcations and stability of families of diffeomorphisms , 1983 .

[12]  O. Jenkinson.  Conjugacy rigidity, cohomological triviality and barycentres of invariant measures , 1996 .

[13]  Oscar E. Lanford,et al.  Dynamique symbolique des rotations , 1984 .

[14]  Ott,et al.  Optimal periodic orbits of chaotic systems occur at low period. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  Tom C. Brown,et al.  Descriptions of the Characteristic Sequence of an Irrational , 1993, Canadian Mathematical Bulletin.

[16]  Shaun Bullett,et al.  Ordered orbits of the shift, square roots, and the devil's staircase , 1994, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.