surface diffeomorphisms
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Author(s):  
Vaughn Climenhaga ◽  
Stefano Luzzatto ◽  
Yakov Pesin

2021 ◽  
Vol 17 (3) ◽  
pp. 321-334
Author(s):  
M. K. Barinova ◽  
◽  
E. Y. Gogulina ◽  
O. V. Pochinka ◽  
◽  
...  

The present paper gives a partial answer to Smale’s question which diagrams can correspond to $(A,B)$-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by “Smale surgery” are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class $G$ of $(A,B)$-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class $G$ realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from $G$ with isomorphic labeled Smale diagrams which are not ambiently $\Omega$-conjugated are constructed. Moreover, a subset $G_{*}\subset G$ of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient $\Omega$-conjugacy is singled out.


2020 ◽  
pp. 1-26
Author(s):  
SNIR BEN OVADIA

Abstract The papers [O. M. Sarig. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc.26(2) (2013), 341–426] and [S. Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn.13 (2018), 43–113] constructed symbolic dynamics for the restriction of $C^r$ diffeomorphisms to a set $M'$ with full measure for all sufficiently hyperbolic ergodic invariant probability measures, but the set $M'$ was not identified there. We improve the construction in a way that enables $M'$ to be identified explicitly. One application is the coding of infinite conservative measures on the homoclinic classes of Rodriguez-Hertz et al. [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys.306(1) (2011), 35–49].


Author(s):  
Carlos Matheus ◽  
Jacob Palis ◽  
Jean-Christophe Yoccoz

We show that the stable and unstable sets of non-uniformly hyperbolic horseshoes arising in some heteroclinic bifurcations of surface diffeomorphisms have the value conjectured in a previous work by the second and third authors of the present paper. Our results apply to first heteroclinic bifurcations associated with horseshoes with Hausdorff dimension ${<}22/21$ of conservative surface diffeomorphisms.


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