scholarly journals Exponential mixing and smooth classification of commuting expanding maps

2017 ◽  
Vol 11 (1) ◽  
pp. 263-312
Author(s):  
Ralf Spatzier ◽  
◽  
Lei Yang ◽  
Keyword(s):  
Expanding Maps ◽  
Exponential Mixing ◽  
Smooth Classification

Classifying C1+ structures on dynamical fractals: 2. Embedded trees

1995 ◽  
Vol 15 (5) ◽  
pp. 969-992 ◽  
Author(s):  
A. A. Pinto ◽  
D. A. Rand
Keyword(s):  
Moduli Spaces ◽  
Binary Tree ◽  
Conjugacy Classes ◽  
Expanding Maps ◽  
Bounded Geometry ◽  
Arbitrary Topology ◽  
Contact Points ◽  
Train Tracks ◽  
Smooth Conjugacy

AbstractWe classify the C1+α structures on embedded trees. This extends the results of Sullivan on embeddings of the binary tree to trees with arbitrary topology and to embeddings without bounded geometry and with contact points. We used these results in an earlier paper to describe the moduli spaces of smooth conjugacy classes of expanding maps and Markov maps on train tracks. In later papers we will use those results to do the same for pseudo-Anosov diffeomorphisms of surfaces. These results are also used in the classification of renormalisation limits of C1+α diffeomorphisms of the circle.

scholarly journals Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification

2021 ◽  
pp. 1-19
Author(s):  
D. Crowley ◽  
A. Skopenkov
Keyword(s):  
Connected Sum ◽  
Smooth Category ◽  
Simply Connected ◽  
Torsion Free ◽  
Smooth Classification

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$ , where $H_q := H_q(N; {\mathbb Z} )$ . Our main result is a readily calculable classification of embeddings $N \to {\mathbb R}^7$ up to isotopy, with an indeterminacy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9. The group of knots $S^4\to {\mathbb R}^7$ acts on the set of embeddings $N\to {\mathbb R}^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1 \ne 0$ , with an indeterminacy. Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008. For $N=S^1\times S^3$ we give a geometrically defined 1–1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set ${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$ .

The smooth classification of elliptic surfaces with $b^+>1$

1994 ◽  
Vol 75 (1) ◽  
pp. 1-50 ◽  
Author(s):  
Andr�s Stipsicz ◽  
Zolt�n Szab�
Keyword(s):  
Elliptic Surfaces ◽  
Smooth Classification

ON CONJUGATING EQUIVALENCE OF 0-RESONANT DIFFEOMORPHISMS ON ℝ3

2013 ◽  
Vol 23 (06) ◽  
pp. 1350100
Author(s):  
ZHIHUA REN ◽  
JIAZHONG YANG ◽  
FUWANG YU
Keyword(s):  
Smooth Classification

In this note, we study smooth classification of all germs of 0-resonant diffeomorphisms on ℝ3 having generic nonlinear parts. We prove that, for the Poincaré type diffeomorphisms, except for one germ, any two such germs are at least C3 conjugated if and only if their linear parts are similar; and for the non-Poincaré ones, any such two germs can be C∞ conjugated provided that they have similar linear parts.

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