Random mappings with a single absorbing center and combinatorics of discretizations of the logistic mapping.
Ratner's property for special flows over irrational rotations under functions of bounded variation. II
We consider special flows over the rotation on the circle by an irrational α under roof functions of bounded variation. The roof functions, in the Lebesgue decomposition, are assumed to have a continuous singular part coming from a quasi-similar Cantor set (including the devil's staircase case). Moreover, a finite number of discontinuities is allowed. Assuming that α has bounded partial quotients, we prove that all such flows are weakly mixing and enjoy the weak Ratner property. Moreover, we provide...
Real Koebe principle
We prove a version of the real Koebe principle for interval (or circle) maps with non-flat critical points.
Reconstructing the global dynamics of attractors via the Conley index
Given an unknown attractor 𝓐 in a continuous dynamical system, how can we discover the topology and dynamics of 𝓐? As a practical matter, how can we do so from only a finite amount of information? One way of doing so is to produce a semi-conjugacy from 𝓐 onto a model system 𝓜 whose topology and dynamics are known. The complexity of 𝓜 then provides a lower bound for the complexity of 𝓐. The Conley index can be used to construct a simplicial model and a surjective semi-conjugacy for a large...
Regularity of conjugacies between critical circle maps: an experimental study.
Renormalization of cocycles and linear ODE with almost-periodic coefficients.
Renormalization of exponential sums and matrix cocycles
In this paper, we present a new point of view on the renormalization of some exponential sums stemming from number theory. We generalize this renormalization procedure to study some matrix cocycles arising in spectral problems of quantum mechanics
Rescaling of Markov shifts.
Retracts, fixed point index and differential equations.
Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions.
Return time statistics for unimodal maps
We prove that a non-flat S-unimodal map satisfying a weak summability condition has exponential return time statistics on intervals around a.e. point. Moreover we prove that the convergence to the entropy in the Ornstein-Weiss formula enjoys normal fluctuations.
Reversibility in the diffeomorphism group of the real line.
Rigidity and inflexibility in conformal dynamics.
Rigidity and renormalization in one dimensional dynamical systems.
Rigidity of critical circle mappings I
We prove that two critical circle maps with the same rotation number in a special set are conjugate for some provided their successive renormalizations converge together at an exponential rate in the sense. The set has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of critical circle maps with the same rotation number that are not conjugate for any . The class of rotation numbers for which such examples exist contains...
Roots of continuous piecewise monotone maps of an interval.
Rosen fractions and Veech groups, an overly brief introduction
We give a very brief, but gentle, sketch of an introduction both to the Rosen continued fractions and to a geometric setting to which they are related, given in terms of Veech groups. We have kept the informal approach of the talk at the Numerations conference, aimed at an audience assumed to have heard of neither of the topics of the title.The Rosen continued fractions are a family of continued fraction algorithms, each gives expansions of real numbers in terms of elements of a corresponding algebraic...
Rotation Numbers of Products of Circle Homeomorphisms.
Rotation sets and monotone periodic orbits for annulus homeomorphisms.
Rotation sets for graph maps of degree 1
For a continuous map on a topological graph containing a loop it is possible to define the degree (with respect to the loop ) and, for a map of degree , rotation numbers. We study the rotation set of these maps and the periods of periodic points having a given rotation number. We show that, if the graph has a single loop then the set of rotation numbers of points in has some properties similar to the rotation set of a circle map; in particular it is a compact interval and for every rational...
Rotation sets for some non-continuous maps of degree one.