Entropy of analytic endomorphisms of the riemannian sphere

1982 ◽  
Vol 15 (4) ◽  
pp. 300-302 ◽  
Author(s):  
M. Yu. Lyubich
Keyword(s):  
Riemannian Sphere

Local behavior of mappings of metric spaces with branching

2020 ◽  
Vol 17 (4) ◽  
pp. 574-593
Author(s):  
Serhii Skvortsov
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Local Behavior ◽  
Compact Closure ◽  
Fixed Domain ◽  
Riemannian Sphere ◽  
Main Inequality

The local behavior of mappings with the inverse Poletsky inequality between metric spaces is studied. The case where one of the spaces satisfies the condition of weak sphericalization, is similar to the Riemannian sphere (extended Euclidean space), and is locally linearly connected under a mapping is considered. It is proved that the equicontinuity of the corresponding families of mappings of two domains, one of which is a domain with a weakly flat boundary, and another one is a fixed domain with a compact closure, the corresponding weight in the main inequality being supposed to be integrable.

scholarly journals Modified Action and Differential Operators on the 3-D Sub-Riemannian Sphere

2010 ◽  
Vol 14 (4) ◽  
pp. 439-474
Author(s):  
Der-Chen Chang ◽  
Irina Markina ◽  
Alexander Vasil'ev
Keyword(s):  
Differential Operators ◽  
Riemannian Sphere

scholarly journals Sub-Riemannian sphere in Martinet flat case

1997 ◽  
Vol 2 ◽  
pp. 377-448 ◽  
Author(s):  
A. Agrachev ◽  
B. Bonnard ◽  
M. Chyba ◽  
I. Kupka
Keyword(s):  
Riemannian Sphere

scholarly journals On local structure of one-dimensional basic sets of non-reversible A-endomorphisms of surfaces

2020 ◽  
Vol 22 (4) ◽  
pp. 424-433
Author(s):  
Vyacheslav Z. Grines ◽  
Evgenii V. Zhuzhoma
Keyword(s):  
Local Structure ◽  
Complex Dynamics ◽  
Invariant Sets ◽  
Dimensional Torus ◽  
Remarkable Property ◽  
Closed Set ◽  
One Dimensional ◽  
Riemannian Sphere ◽  
Orientable Surfaces ◽  
Basic Sets

Recently the authors of the article discovered a meaningful class of non-reversible endomorphisms on a two-dimensional torus. A remarkable property of these endomorphisms is that their non-wandering sets contain nontrivial one-dimensional strictly invariant hyperbolic basic sets (in the terminology of S. Smale and F. Pshetitsky) which have the uniqueness of an unstable one-dimensional bundle. It was proved that nontrivial (other than periodic isolated orbits) invariant sets can only be repellers. Note that this is not the case for reversible endomorphisms (diffeomorphisms). In the present paper, it is proved that one-dimensional expanding uniquely hyperbolic and strictly invariant one-dimensional expanding attractors and one-dimensional contracting repellers of non-reversible A-endomorphisms of closed orientable surfaces have the local structure of the product of an interval by a zero-dimensional closed set (finite or Cantor). This result contrasts with the existence of one-dimensional fractal repellers arising in complex dynamics on the Riemannian sphere and not possessing the properties of the existence of a single one-dimensional unstable bundle.

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