Homotopical theory of periodic points of periodic homeomorphisms on closed surfaces

In this work we show that the Wecken theorem for periodic points holds for periodic homeomorphisms on closed surfaces, which therefore completes the periodic point theory in such a special case. Using it we derive the set of homotopy minimal periods for such homeomorphisms. Moreover we show that the results hold for homotopically periodic self-maps of closed surfaces. This let us to re-formulate our results as a statement on properties of elements of finite order in the group of outer automorphisms of the fundamental group of a surface with non-positive Euler characteristic.     

Measurements of sensitivity to hydrostatic pressure and temperature in highly birefringent photonic crystal fibers

We report on experimental studies of polarimetric sensitivity to hydrostatic pressure and temperature in two highly birefringent index guided photonic crystal fibers, in which birefringence is induced by one row of the cladding holes with diameters smaller than the other cladding holes. The sensitivity measurements were carried out in the spectral range from 0.6 μm to 1.6 μm. Our results show that absolute value of the polarimetric sensitivity to hydrostatic pressure can reach 23 rad/MPa × m, which is almost one order of magnitude higher than in conventional fibers with elliptical core. Simultaneously, polarimetric sensitivity to temperature is at least two orders of magnitude lower than in conventional highly birefringent fibers. Moreover, we proved experimentally that one of the investigated fibers is completely insensitive to temperature at certain wavelength.     

A symmetry of sphere map implies its chaos*

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere Sⁿ, n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : Sⁿ → Sⁿ commutes with a free homeomorphism g : Sⁿ → Sⁿ of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points. *Research supported by KBN grant nr 2 P03A 045 22.     

Molecular dynamics and residual entropy in the soft crystal, SmE phase, of 4-butyl-4'-isothiocyano-1,1'-biphenyl

Molecular dynamics and resulting disorder in the soft crystal, smectic E (SmE) phase, were studied in detail for the title compound, 4-butyl-4'-isothiocyano-1,1'-biphenyl (4TCB), by (1)H NMR spectroscopy and adiabatic calorimetry. The ordered crystal phase of 4TCB was realized for the first time under ambient pressure after long two-step annealing and used as the reference state in the analysis of the experimental results. Four motional modes were identified in the SmE phase through the analysis of the (1)H NMR T(1). The residual entropy was determined as ca. 6 J K(-1) mol(-1). This magnitude implies that most of the disorder in the SmE phase at high temperatures is removed on cooling except for the head-to-tail disorder of the rod-shaped 4TCB molecule. Standard thermodynamic functions are tabulated below 375 K.     

The Unstable Equivariant Fixed Point Index and the Equivariant Degree

A correspondence between the equivariant degree introduced byIze, Massabó, and Vignoli and an unstable version ofthe equivariant fixed point index defined by Prieto and Ulrichis shown. With the help of conormal maps and properties of theunstable index, a sum decomposition formula is proved for theindex and consequently also for the degree. As an application,equivariant homotopy groups are decomposed as direct sums ofsmaller groups of fixed orbit types, and a geometric interpretationof each summand is given in terms of conormal maps.