Jaroslaw Kwapisz

Poincaré rotation number for maps of the real line with almost periodic displacement

In generalizing the classical theory of circle maps, we study the rotation set for maps of the real line xf(x) with almost periodic displacement f(x) - x. Such maps are in one-to-one correspondence with maps of compact Abelian topological groups with the displacement taking values in a dense one-parameter subgroup. For homeomorphisms, we show the existence of the analogue of the Poincar? rotation number, which is the common rotation number of all orbits besides possibly those that have rotation zero. (The coexistence of zero and non-zero rotation numbers is the main new phenomenon compared with the classical circle case.) For non-invertible maps, we prove results concerning the realization of points of the rotation interval as the rotation numbers of orbits and ergodic measures. We also address the issue of practical computation of the rotation number.

A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps

Diffeomorphisms of the two torus that are isotopic to the identity have rotation sets that are convex compact subsets of the plane. We show that certain line segments (including all rationally sloped segments with no rational points) cannot be realized as a rotation set.

Combinatorics of torus diffeomorphisms

We construct dynamical partitions of the torus for a diffeomorphism that is isotopic to the identity. The existence and the combinatorics of the partitions is solely determined by the rotation set of the diffeomorphism. When the rotation set consists of a single non-resonant vector, there is a whole hierarchy of partitions analogous to the partitions of the circle into the closest return intervals under an irrational circle rotation. In particular, all such torus maps are infinitely renormalizable in a natural sense.

Rigidity and mapping class group for abstract tiling spaces

We study abstract self-affine tiling actions, which are an intrinsically defined class of minimal expansive actions of ℝ d on a compact space. They include the translation actions on the compact spaces associated to aperiodic repetitive tilings or Delone sets in ℝ d . In the self-similar case, we show that the existence of a homeomorphism between tiling spaces implies conjugacy of the actions up to a linear rescaling. We also introduce the general linear group of an (abstract) tiling, prove its discreteness, and show that it is naturally isomorphic with the (pointed) mapping class group of the tiling space. To illustrate our theory, we compute the mapping class group for a five-fold symmetric Penrose tiling.

Transfer operator, topological entropy and maximal measure for cocyclic subshifts

Cocyclic subshifts arise as the supports of matrix cocycles over a full shift and generalize topological Markov chains and sofic systems. We compute the topological entropy of a cocyclic subshift as the logarithm of the spectral radius of an appropriate transfer operator and give a concrete description of the measure of maximal entropy in terms of the eigenvectors. Unlike in the Markov or sofic case, the operator is infinite-dimensional and the entropy may be a logarithm of a transcendental number.

Stokes Space Representation of Modal Dispersion

Polarization-mode dispersion in single-mode fibers can be viewed as a special case of modal dispersion in multimode and multicore optical fibers. Exploiting the similarity between these two transmission effects, modal dispersion can be modeled in a way analogous to that of polarization-mode dispersion by modifying the conventional Jones–Stokes formalism. In this paper, we review the geometrical representation of modal dispersion in the generalized Stokes space by means of the modal dispersion vector. We summarize and unify the fundamental equations that encapsulate the properties of the modal dispersion vector. We prove that the modal dispersion vector can be expressed as a linear superposition of the Stokes vectors representing the principal modes. The coefficients of this expansion are the corresponding differential mode group delays. This concise and elegant expression can be considered as a simplified definition of the modal dispersion vector and can be used to facilitate analytical calculations.

Cocyclic subshifts from Diophantine equations

We give a method of constructing cocyclic subshifts from Diophantine equations. As an application, we produce examples of such subshifts that are not sofic and prove that the problem of equality of two cocyclic subshifts is algorithmically undecidable. This result also follows from the 1970 work by Paterson on mortality of matrices.

Diophantine and minimal but not uniquely ergodic (almost)

We demonstrate that minimal non-uniquely ergodic behaviour can be generated by slowing down a simple harmonic oscillator with diophantine frequency, in contrast with the known examples where the frequency is well approximable by the rationals. The slowing is effected by a singular time change that brings one phase point to rest. The time one-map of the flow has uncountably many invariant measures yet every orbit is dense, with the minor exception of the rest point.

A toral flow with a pointwise rotation set that is not closed

We give an example of a C1 flow on the two-dimensional torus for which the pointwise rotation set is not closed.