Nonwandering sets of maps on the circle.
Obtuse triangular billiards. I: Near the triangle.
Off-center reflections: caustics and chaos.
On a one-dimensional analogue of the Smale horseshoe
We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have , where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then for Lebesgue-a.e. x.
On boundaries of parallelizable regions of flows of free mappings.
On circle map coupled map lattice.
On closed subgroups of the group of homeomorphisms of a manifold
Let be a triangulable compact manifold. We prove that, among closed subgroups of (the identity component of the group of homeomorphisms of ), the subgroup consisting of volume preserving elements is maximal.
On common fixed points, periodic points, and recurrent points of continuous functions.
On commuting circle mappings and simultaneous Diophantine approximations.
On conjugacy equation in dimension one
In this paper, recent results on the existence and uniqueness of (continuous and homeomorphic) solutions φ of the equation φ ∘ f = g ∘ φ (f and g are given self-maps of an interval or the circle) are surveyed. Some applications of these results as well as the outcomes concerning systems of such equations are also presented.
On cusps and flat tops
Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to . The critical points are not required to verify a non-flatness condition, so the results are applicable to maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.
On disjointness properties of some smooth flows
Special flows over some locally rigid automorphisms and under L² ceiling functions satisfying a local L² Denjoy-Koksma type inequality are considered. Such flows are proved to be disjoint (in the sense of Furstenberg) from mixing flows and (under some stronger assumption) from weakly mixing flows for which the weak closure of the set of all instances consists of indecomposable Markov operators. As applications we prove that ∙ special flows built over ergodic interval exchange...
On fixed points of holomorphic type
We study a linearization of a real-analytic plane map in the neighborhood of its fixed point of holomorphic type. We prove a generalization of the classical Koenig theorem. To do that, we use the well known results concerning the local dynamics of holomorphic mappings in ℂ².
On fractional iterates of a homeomorphism of the plane
We find all continuous iterative roots of nth order of a Sperner homeomorphism of the plane onto itself.
On generic chaos of shifts in function spaces
On Kurzweil's 0-1 law in inhomogeneous Diophantine approximation
We give a necessary and sufficient condition such that, for almost all s ∈ ℝ, ||nθ - s|| < ψ(n) for infinitely many n ∈ ℕ, where θ is fixed and ψ(n) is a positive, non-increasing sequence. This can be seen as a dual result to classical theorems of Khintchine and Szüsz which dealt with the situation where s is fixed and θ is random. Moreover, our result contains several earlier ones as special cases: two old theorems of Kurzweil, a theorem of Tseng and a recent...
On local aspects of topological weak mixing in dimension one and beyond
We introduce the concept of weakly mixing sets of order n and show that, in contrast to weak mixing of maps, a weakly mixing set of order n does not have to be weakly mixing of order n + 1. Strictly speaking, we construct a minimal invertible dynamical system which contains a non-trivial weakly mixing set of order 2, whereas it does not contain any non-trivial weakly mixing set of order 3. In dimension one this difference is not that much visible, since we prove that every continuous...
On maximal transitive sets of generic diffeomorphisms
On Pawlak's problem concerning entropy of almost continuous functions
We prove that if f: → is Darboux and has a point of prime period different from , i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.
On some properties of orientation-preserving surjections on the circle