arXiv subjects
Publications and source records attributed to Serge Troubetzkoy.
We compute the complexity of the billiard language of the regular Euclidean $N$-gons (and other families of rational lattice polygons), answering a question posed by Cassaigne-Hubert-Troubetzkoy. Our key technical result is a counting result for saddle connections on lattice surfaces, when we count by combinatorial length.
We consider the space $C_λ$ of all continuous interval maps preserving the Lebesgue measure $λ$. A continuous function $f\colon~[0,1]\to \mathbb R$ is called Besicovitch if it does not have any finite or infinite unilateral derivative. It is known that the set of Besicovitch functions in $C_λ$ is nonempty and meager. We prove that no Besicovitch function is invertible $λ$-almost everywhere. As a consequence, every Besicovitch function in $C_λ$ has positive measure-theoretic entropy with respect to $λ$. Furthermore, we show that Besicovitch functions are dense in $C_λ$ and, consequently, also dense in the class of interval maps with a dense set of periodic points.
We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or $1 - \id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In this article we show that the generic map has zero measure-theoretic entropy. This implies that there are dramatic differences in the topological versus measure-theoretic behavior both for injectivity as well as for the structure of the level sets of generic maps. As a consequence we get a surprising corollary for a family of planar attractors homeomorphic to the pseudo-arcs.
We consider the class of interval maps with dense set of periodic points CP and its closure Cl(CP) equipped with the metric of uniform convergence. Besides studying basic topological properties and density results in the spaces CP and Cl(CP) we prove that Cl(CP) is dynamically characterized as the set of interval maps for which every point is chain-recurrent. Furthermore, we prove that a strong topological expansion property called topological exactness (or leo property) is attained on the open dense set of maps in CP and on a residual set in Cl(CP). Moreover, we show that every second category set in CP and Cl(CP) is rich in a sense that it contains uncountably many conjugacy classes. An analogous conclusion also holds in the setting of interval maps preserving any fixed non-atomic probability measure with full support. Finally, we give a detailed description of the structure of periodic points of generic maps in CP and Cl(CP) and show that generic maps in CP and Cl(CP) satisfy the shadowing property.
We show that the complexity of the billiard in a typical polygon grows cubically and the number of saddle connections grows quadratically along certain subsequences. It is known that the set of points whose first n-bounces hits the same sequence of sides as the orbit of an aperiodic phase point z converges to z. We establishe a polynomial lower bound estimate on this convergence rate for almost every z. This yields an upper bound on the upper metric complexity and upper slow entropy of polygonal billiards. We also prove significant deviations from the expected convergence behavior. Finally we extend these results to higher dimensions as well as to arbitrary invariant measures.
We show that the topological entropy of the billiard map in a Bunimovich stadium is at most log(3.49066).
We study the set of lengths of the horizontal chords of a continuous function. We give a new proof of Hopf's characterization of this set, and show that it implies that no matter which function we choose, at least half of the possible lengths occur. We prove several results about functions for which all the possible lengths occur.
We survey the current state-of-the-art about the dynamical behavior of continuous Lebesgue measure-preserving maps on one-dimensional manifolds.
We construct ergodic probability measures with infinite metric entropy for typical continuous maps and homeomorphisms on compact manifolds. We also construct sequences of such measures that converge to a zero-entropy measure.
We give a simple proof based on symmetries that there are no geodesics from a vertex to itself in the cube, tetrahedron, octahedron, and icosahedron.
In this paper we provide a detailed topological and measure-theoretic study of Lebesgue measure-preserving circle maps that are rotated with inner and outer rotations which are independent of each other. In particular, we analyze the stability of the locally eventually onto and measure-theoretic mixing properties.
We consider piecewise monotone maps, we show that an ergodic measure for which the map is invertible almost everywhere can not be mixing. It follows that every ergodic measure for an interval translation mapping is not mixing. We also show that double rotations without periodic points have an ergodic but not weakly mixing invariant measure. This article is dedicated to the memory of Anatoly Mikhailovich Stepin.
We show that for the generic continuous maps of the interval and circle which preserve the Lebesgue measure it holds for each k $\ge$ 1 that the set of periodic points of period k is a Cantor set of Hausdorff dimension zero and of upper box dimension one. Furthermore, building on this result, we show that there is a dense collection of transitive Lebesgue measure preserving interval map whose periodic points have full Lebesgue measure and whose periodic points of period k have positive measure for each k $\ge$ 1. Finally, we show that the generic continuous maps of the interval which preserve the Lebesgue measure satisfy the shadowing and periodic shadowing property.
In this paper we show that generic continuous Lebesgue measure preserving circle maps have the s-limit shadowing property. In addition we obtain that s-limit shadowing is a generic property also for continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings as well.
We show the chess billiard map, which was introduced in [HM] in order to study a generalization of the $n$-Queens problem in chess, is a circle homeomorphism. We give a survey of some of the known results on circle homeomorphisms, and apply them to this map. We prove a number of new results which give answers to some of the open questions posed in [HM].
Let us denote $λ$ the Lebesgue measure on $[0,1]$, put$$ C(λ)=\{f\in C([0,1]);\ \forall~A\subset [0,1], A~\text{Borel}:\ λ(A)=λ(f^{-1}(A))\}.$$ We endow the set $C(λ)$ by the uniform metric $ρ$ and investigate dynamical properties of typical maps in the complete metric space $(C(λ),ρ)$.
In this short note we prove that a continuous map which is locally eventually onto and is expansive satisfies the periodic specification property. We also discuss the role of continuity as a key condition in the previous characterization. We include several examples to illustrate the relation between these concepts.
We consider infinite staircase translation surfaces with varying step sizes. For typical step sizes we show that the translation flow is uniquely ergodic in almost every direction. Our result also hold for typical configurations of the Ehrenfest wind-tree model endowed with the Hausdorff topology. In contrast, we show that the translation flow on a periodic translation surface can not be uniquely ergodic in any direction.