arXiv subjects
Publications and source records attributed to S. Troubetzkoy.
This paper had a serious error. In fixing the error the emphasis of the paper has changed completely, thus meriting a new name: ``Periodic orbits in right triangles''. I have made a new submission to arXiv with this name.
We study the dynamics of a class of interval translation map on three intervals. We show that in this class the typical ITM is of finite type (reduce to an interval exchange transformation) and that the complement contains a Cantor set. We relate our maps to substitution subshifts. Results on Hausdorff dimension of the attractor and on unique ergodicity are obtained.
We prove two inequalities between the Lyapunov exponents of a diffeomorphism and its local recurrence properties. We give examples showing that each of the inequalities is optimal.
We consider an Axiom A diffeomorphism and the invariant set $Ω$ of orbits which never falls into a fixed hole. We study various aspects of the complexity of the symbolic representation of $Ω$. Our main result are that each topologically transitive component of $Ω$ is coded and that typically $Ω$ is of finite type.
We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the complexity has cubic asymptotic growth.
We show that the Poincaré return time of a typical cylinder is at least its length. For one dimensional maps we express the Lyapunov exponent and dimension via return times.