Examples Of Diffeomorphisms Research Articles

Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity * *Y Cao is partially supported by National Key R&D Program of China(2022YFA1005802) and NSFC 12371194. Z Mi is partially supported by National Key R&D Program of China(2022YFA1007800) and NSFC 12271260. J Yang was partially supported by CNPq, FAPERJ, PRONEX and NNSFC Grants 12271538, 12071202 and 12161141002.

We study the partially hyperbolic diffeomorphisms whose centre direction admits the u-definite property in the sense that all the central Lyapunov exponents of each ergodic Gibbs u-state are either all positive or all negative. We prove that for this kind of partially hyperbolic diffeomorphisms, there are finitely many physical measures, whose basins cover a full Lebesgue measure subset of the ambient space. We also provide examples of diffeomorphisms whose centres are u-definite, where the supports of different ergodic Gibbs u-states have nontrivial intersections.

Homoclinic tangencies leading to robust heterodimensional cycles

We consider \(C^r\) (\(r\geqslant 1\)) diffeomorphisms f defined on manifolds of dimension \(\geqslant 3\) with homoclinic tangencies associated to saddles. Under generic properties, we show that if the saddle is homoclinically related to a blender then the diffeomorphism f can be \(C^r\) approximated by diffeomorphisms with \(C^1\) robust heterodimensional cycles. As an application, we show that the classic Simon–Asaoka’s examples of diffeomorphisms with \(C^1\) robust homoclinic tangencies also display \(C^1\) robust heterodimensional cycles. In a second application, we consider homoclinic tangencies associated to hyperbolic sets. When the entropy of these sets is large enough we obtain \(C^1\) robust cycles after \(C^1\) perturbations.

(Un)distorted diffeomorphisms in different regularities

We build the first examples of diffeomorphisms that are distorted in a group of C1 diffeomorphisms yet undistorted in the corresponding group of C2 diffeomorphisms. This explicit construction is performed for the closed interval.

Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams

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.

Stable Minimality of Expanding Foliations

We prove that generically in \({\text {Diff}}^{1}_{m}(M)\), if an expanding f-invariant foliation W of dimension u is minimal and there is a periodic point of unstable index u, the foliation is stably minimal. By this we mean there is a \(C^{1}\)-neighborhood \({\mathcal {U}}\) of f such that for all \(C^{2}\)-diffeomorphisms \(g\in {\mathcal {U}}\), the g-invariant continuation of W is minimal. In particular, all such g are topologically mixing. Moreover, all such g have a hyperbolic ergodic component of the volume measure m which is essentially dense. This component is, in fact, Bernoulli. We provide new examples of diffeomorphisms with stably minimal expanding foliations which are not partially hyperbolic.

Parallel connections and bundles of arrangements

Let A be a complex hyperplane arrangement, and let X be a modular element of arbitrary rank in the intersection lattice of A . Projection along X restricts to a fiber bundle projection of the complement of A to the complement of the localization A X of A at X. We identify the fiber as the decone of a realization of the complete principal truncation of the underlying matroid of A along the flat corresponding to X. We then generalize to this setting several properties of strictly linear fibrations, the case in which X has corank one, including the triviality of the monodromy action on the cohomology of the fiber. This gives a topological realization of results of Stanley, Brylawsky, and Terao on modular factorization. We also show that (generalized) parallel connection of matroids corresponds to pullback of fiber bundles, clarifying the notion that all examples of diffeomorphisms of complements of inequivalent arrangements result from the triviality of the restriction of the Hopf bundle to the complement of a hyperplane. The modular fibration theorem also yields a new method for identifying K( π,1) arrangements of rank greater than three. We exhibit new families of K( π,1) arrangements, providing more evidence for the conjecture that factored arrangements of arbitrary rank are K( π,1).

Cyclic reversing k-symmetry groups

We consider discrete invertible dynamical systems L with the property that the kth iterate Lk possesses (reversing) symmetries mat are not possessed by L. A map U is called a (reversing) k-symmetry of L if k is the smallest positive integer for which U is a (reversing) symmetry of Lk. In this paper we discuss the particular case that L possesses a cyclic reversing k-symmetry group. We derive a decomposition property of maps that possess a cyclic reversing k-symmetry group and we classify the occurrence of such groups in invertible dynamical systems. We discuss the occurrence of nonsimultaneously linearizable nonisomorphic reversing k-symmetry groups in maps possessing cyclic reversing k-symmetry groups, illustrated by an example of a diffeomorphism on the plane R2. We also construct examples of diffeomorphisms with cyclic reversing k-symmetry groups on the circle S1, on the two-torus T2, and on the cylinder S1*R.

HYPERBOLIC ATTRACTORS OF DIFFEOMORPHISMS OF ORIENTABLE SURFACES

In this second part of the article an enumeration algorithm is given for the codes of one-dimensional hyperbolic attractors of diffeomorphisms of orientable surfaces introduced in the first part. Using this algorithm new examples of diffeomorphisms with attractors and pseudo-Anosov diffeomorphisms are constructed. Bibliography: 11 titles.

Wild unstable manifolds

LET M be a compact smooth manifold, f: M + M a diffeomorphism. An energy function for f is a Morse function M + R which decreases along non-periodic orbits and has critical points only at the periodic points off. An analogous concept for vector fields was introduced by Meyer [8], who proved that Morse-Smale vector fields on compact manifolds always have energy functions. Shub[lZ] and Takens [15] have since asserted, without proof, that the theorem is also true for Morse-Smale diffeomorphisms. In this paper we show, using an example on the three-sphere, that this conjecture is false. The example is given in 02, after a section consisting of definitions and a construction of energy functions for Morse-Smale diffeomorphisms of surfaces. The distinguishing characteristic of the counterexample on S’ is an unstable manifold whose closure is a wildly embedded arc: under certain hypotheses it is shown that such phenomena preclude the existence of energy functions. In investigating further examples of diffeomorphisms with such wild phase portraits we see that no possible extension of the notion of labeled diagram will serve to classify conjugacy classes of Morse-Smale diffeomorphisms in dimension 3. These examples are discussed in 63. All our examples are gradient-like: there is no complicated behavior in the sense of [lo]. Note. The major results of Shub and Takens in the articles referenced above are still true. An alternative proof of Shub’s theorem has appeared in [13] and elsewhere, and Takens’ arguments work just as well using non-degenerate Liapunov functions, as defined in 91.