scholarly journals Basic Sets for Unipotent Blocks of Finite Reductive Groups in Bad Characteristic

2020 ◽  
Author(s):  
Reda Chaneb
Keyword(s):  
Classical Type ◽  
Reductive Groups ◽  
Basic Set ◽  
Brauer Characters ◽  
Basic Sets

Abstract We show the existence of a unitriangular basic set for unipotent blocks of simple reductive groups of classical type in bad characteristic with some exceptions. Then, we introduce an algorithm to count unipotent irreducible Brauer characters and we verify that this algorithm is effective for exceptional adjoint groups.

The ambient structure of basic sets

1995 ◽  
Vol 15 (6) ◽  
pp. 1183-1188
Author(s):  
A. Löffler
Keyword(s):  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Product Structure ◽  
Axiom A ◽  
Basic Set ◽  
Local Product ◽  
Basic Sets

AbstractLet Λ be a basic set of an Axiom A diffeomorphism of a compact Riemannian manifold M without boundary. If ε is small enough one can find by local product structure that for x ε Λ there is a neighborhood V(x) in M such that V ∩ Λ is homeomorphic to . The author proves that this homeomorphism can be extended to a homeomorphism of V onto .

scholarly journals The templates of non-singular Smale flows on three manifolds

2011 ◽  
Vol 32 (3) ◽  
pp. 1137-1155 ◽  
Author(s):  
BIN YU
Keyword(s):  
Positive Integer ◽  
Symbolic Dynamics ◽  
Basic Set ◽  
Smale Flows ◽  
Basic Sets

AbstractIn this paper, we first discuss some connections between template theory and the description of basic sets of Smale flows on 3-manifolds due to F. Béguin and C. Bonatti. The main tools we use are symbolic dynamics, template moves and some combinatorial surgeries. Secondly, we obtain some relationship between the surgeries and the number of S1×S2 factors of M for a non-singular Smale flow on a given closed orientable 3-manifold M. We also prove that any template T can model a basic set Λ of a non-singular Smale flow on nS1×S2 for some positive integer n.

Dynamical behavior of endomorphisms on certain invariant sets

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Diana Putan ◽  
Diana Stan
Keyword(s):  
Hausdorff Dimension ◽  
Dynamical Behavior ◽  
Invariant Sets ◽  
Hyperbolic Polynomial ◽  
Stable Manifolds ◽  
Basic Set ◽  
Polynomial Endomorphisms ◽  
Complex Projective ◽  
Local Stable Manifolds ◽  
Basic Sets

AbstractWe study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space ℙ2. We consider the perturbation (z 2 +ɛz +bɛw 2, w 2) of (z 2, w 2) and we prove that, for b sufficiently small, it is injective on its basic set Λɛ close to Λ:= {0} × S 1. Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λɛ, in the case of these maps.

scholarly journals Cantor Type Basic Sets of Surface $A$-endomorphisms

2021 ◽  
Vol 17 (3) ◽  
pp. 335-345
Author(s):  
V. Z. Grines ◽  
◽  
E. V. Zhuzhoma ◽  
Keyword(s):  
Closed Surface ◽  
Orientable Surface ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Closed Orientable Surface ◽  
One Dimensional ◽  
Basic Set ◽  
Basic Sets ◽  
Nonwandering Set

The paper is devoted to an investigation of the genus of an orientable closed surface $M^{2}$ which admits $A$-endomorphisms whose nonwandering set contains a one-dimensional strictly invariant contracting repeller $\Lambda_{r}$ with a uniquely defined unstable bundle and with an admissible boundary of finite type. First, we prove that, if $M^{2}$ is a torus or a sphere, then $M^{2}$ admits such an endomorphism. We also show that, if $\Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^{2}\to M^{2}$ of a closed orientable surface $M^{2}$ and $f$ is not a diffeomorphism, then $\Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^{2}\to M^{2}$ is an $A$-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type $\Omega_{r}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_{r}$ is regular, then $M^{2}$ is a two-dimensional torus $\mathbb{T}^{2}$ or a two-dimensional sphere $\mathbb{S}^{2}$.

scholarly journals Minimal cohesive basic sets

1974 ◽  
Vol 19 (1) ◽  
pp. 73-75
Author(s):  
Donald L. Goldsmith
Keyword(s):  
Basic Sequence ◽  
Basic Set ◽  
Positive Integers ◽  
Basic Sets

A basic set (formerly basic sequence) ℬ is a set of pairs (a, b) of positive integers satisfying(1) if (a, b) ∈ ℬ, then (b, a) ∈ ℬ,(2) (a, bc) ∈ ℬ if and only if (a, b) ∈ ℬ and (a, c) ∈ ℬ,(3) (1, k ∈ ℬ, k = 1, 2, ….Some familiar examples of basic sets are, where Sk = {(1, k), (k, 1)}, = {(a, b)| a and b are relatively prime positive integers},ℒ = {(a, b)| a and b are any positive integers}.

On a class of stable conditional measures

2010 ◽  
Vol 31 (5) ◽  
pp. 1499-1515 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Invariant Measures ◽  
Reversible Systems ◽  
Absolutely Continuous ◽  
Stable Manifolds ◽  
Equilibrium Measures ◽  
Basic Set ◽  
Stable Dimension ◽  
Saddle Type ◽  
Local Stable Manifolds ◽  
Basic Sets

AbstractThe dynamics of endomorphisms (smooth non-invertible maps) presents many differences from that of diffeomorphisms or that of expanding maps; most methods from those cases do not work if the map has a basic set of saddle type with self-intersections. In this paper we study the conditional measures of a certain class of equilibrium measures, corresponding to a measurable partition subordinated to local stable manifolds. We show that these conditional measures are geometric probabilities on the local stable manifolds, thus answering in particular the questions related to the stable pointwise Hausdorff and box dimensions. These stable conditional measures are shown to be absolutely continuous if and only if the respective basic set is a non-invertible repeller. We find also invariant measures of maximal stable dimension, on folded basic sets. Examples are given, too, for such non-reversible systems.

scholarly journals An obstruction to the existence of certain dynamics in surface diffeomorphisms

1981 ◽  
Vol 1 (3) ◽  
pp. 255-260 ◽  
Author(s):  
Paul Blanchard ◽  
John Franks
Keyword(s):  
Zeta Function ◽  
Compact Manifold ◽  
Zeta Functions ◽  
Two Dimensional ◽  
Surface Diffeomorphisms ◽  
Basic Set ◽  
Subshifts Of Finite Type ◽  
Chain Recurrent Set ◽  
Basic Sets ◽  
Recurrent Set

AbstractLet M be a two-dimensional, compact manifold and g:Μ→ΜM be a diffeomorphism with a hyperbolic chain recurrent set. We find restrictions on the reduced zeta function p(t) of anyzero-dimensional basic set of g. If deg (p(t)) is odd, then p(1) = 0 (in ). Since there are infinitely many subshifts of finite type whose reduced zeta functions do not satisfy these restrictions, there are infinitely many subshifts which cannot be basic sets for any diffeomorphism of any surface.

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