Self-dual codes with automorphism of order 3 having 8 cycles

All optimal binary self-dual codes which have an automorphism of order 3 with 8 independent cycles are obtained up to equivalence.     

New 3-designs and 2-designs having U(3,3) as an automorphism group

In this paper we classify $2$-designs and $3$-designs with $28$ or $36$ points admitting a transitive action of the unitary group $U(3,3)$ on points and blocks. We also construct $2$-designs and $3$-designs with $56$ or $63$ points and strongly regular graphs on 36, 63 or 126 vertices having $U(3,3)$ as a transitive automorphism group. Further, we show that this completes the classification of $3$-designs admitting a transitive action of the group $U(3,3)$, in terms of parameters. A number of the $3$-designs and $2$-designs obtained in this paper have not been known before up to our best knowledge.     

On an infinite class of non-bipartite and non-cayley graphs having 2-arc transitive automorphism groups

Letq be a prime of the formq = 40x + 13,q = 40x + 27,q = 40x + 37, orq = 40x + 43. Then a connected, undirected, 4-valent, non-bipartite graph on whichPSL ₂ (q) acts 2-arc transitively is non-Cayley.     

On nilpotent groups having an almost regular automorphism of prime order

Translated from Sibirskii Matematicheskii, Vol. 35, No. 3, pp. 630–632, May–June, 1994.     

Dimension and measure for semi-hyperbolic rational maps of degree 2

We prove that almost every non-hyperbolic rational map of degree 2 has at least one recurrent critical point. This estimate is optimal because the set of rational maps with all critical points non-recurrent is of full Hausdorff dimension. To cite this article: M. Aspenberg, J. Graczyk, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Attractors for graph critical rational maps

We call a rational map graph critical if any critical point either belongs to an invariant finite graph , or has minimal limit set, or is non-recurrent and has limit set disjoint from . We prove that, for any conformal measure, either for almost every point of the Julia set its limit set coincides with , or for almost every point of its limit set coincides with the limit set of a critical point of .

     

Mean value conjectures for rational maps

Let p be a polynomial in one complex variable. Smale's mean value conjecture estimates |p′(z)| in terms of the gradient of a chord from (z,?p(z)) to some stationary point on the graph of p. The conjecture does not immediately generalize to rational maps since its formulation is invariant under the group of affine maps, not the full Möbius group. Here we give two possible generalizations to rational maps, both of which are Möbius invariant. In both cases we prove a version with a weaker constant, in parallel to the situation for Smale's mean value conjecture. Finally, we discuss some candidate extremal rational maps, namely rational maps all of whose critical points are fixed points.     

Continuous rational maps into the unit 2-sphere

Investigated are continuous rational maps from a compact nonsingular real algebraic set into unit spheres. Special attention is devoted to such maps with values in the unit 2-sphere.     

On the Lyapunov spectrum for rational maps

We study the dimension spectrum for Lyapunov exponents for rational maps on the Riemann sphere.     

The least number of edges for graphs having dihedral automorphism group

The least (and greatest) number of edges realizable by a graph having n vertices and automorphism group isomorphic to D_2m, the dihedral group of order 2m, is determined for all admissible n.     

Conjugacies between rational maps and extremal quasiconformal maps

We show that two rational maps which are -quasiconformally combinatorially equivalent are -quasiconformally conjugate. We also study the relationship between the boundary dilatation of a combinatorial equivalence and the dilatation of a conjugacy.

     

Solvable groups having only three rational classes of 2-elements

The main result of this paper is that a solvable group that has only three or fewer rational classes consisting of 2-elements has 2-length at most 1.     

Exceptional curves on rational surfaces having K2⩾0

We characterize the rational surfaces X which have a finite number of (?1)-curves under the assumption that ?K_X is nef, where K_X is a canonical divisor on X, and has self-intersection zero. We prove also that if ?K_X is not nef and has self-intersection zero, then X has a finite number of (?1)-curves. To cite this article: M. Lahyane, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Hyperbolic-parabolic deformations of rational maps

We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.     

Foliations invariant by rational maps

We give a classification of pairs ${(\mathcal{F}, \phi)}$ where ${\mathcal{F}}$ is a holomorphic foliation on a projective surface and ${\phi}$ is a non-invertible dominant rational map preserving ${\mathcal{F}}$ .