Surface Diffeomorphisms Research Articles

Approximation property on entropies for surface diffeomorphisms

In this paper, we prove that for any C 1 C^1 surface diffeomorphism f f with positive topological entropy, there exists a diffeomorphism g g arbitrarily close (in the C 1 C^1 topology) to f f exhibiting a horseshoe Λ \Lambda , such that the topological entropy of g g restricted on Λ \Lambda can arbitrarily approximate the topological entropy of f f . This extends a classical result of Katok for C 1 + α ( α > 0 ) C^{1+\alpha }(\alpha >0) surface diffeomorphisms.

On periodic mapping data of a two-dimensional torus with one saddle orbit

Periodic data of diffeomorphisms with regular dynamics on surfaces were studied using zeta functions in a series of already classical works by such authors as P. Blanchard, J. Franks, S. Narasimhan, S. Batterson and others. The description of periodic data for gradient-like diffeomorphisms of surfaces were given in the work of A. Bezdenezhnykh and V. Grines by means of the classification of periodic surface transformations obtained by J. Nielsen. V. Grines, O. Pochinka, S. Van Strien showed that the topological classification of arbitrary Morse-Smale diffeomorphisms on surfaces is based on the problem of calculating periodic data of diffeomorphisms with a single saddle periodic orbit. Namely, the construction of filtering for Morse-Smale diffeomorphisms makes it possible to reduce the problem of studying periodic surface diffeomorphism data to the problem of calculating periodic diffeomorphism data with a single saddle periodic orbit. T. Medvedev, E. Nozdrinova, O. Pochinka solved this problem in a general formulation, that is, the periods of source orbits are calculated from a known period of the sink and saddle orbits. However, these formulas do not allow to determine the feasibility of the obtained periodic data on the surface of this kind. In an exhaustive way, the realizability problem is solved only on a sphere. In this paper we establish a complete list of periodic data of diffeomorphisms of a two-dimensional torus with one saddle orbit, provided that at least one nodal point of the map is fixed.

Classification of one-dimensional attractors of diffeomorphisms of surfaces by means of pseudo-anosov homeomorphisms

In the present paper axiom diffeomorphisms of closed 2-manifolds of genus whose nonwandering set contains perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of the paper is the following. Two diffeomorphisms from the given class are topologically conjugate on attractors if and only if corresponding pseudo-Anosov homeomorphisms are topologically conjugate by means of homeomorphism that maps a certain subset of one pseudo-Anosov map onto the certain subset of the other pseudo-Anosov map.

STABLE SETS OF CERTAIN NON-UNIFORMLY HYPERBOLIC HORSESHOES HAVE THE EXPECTED DIMENSION

AbstractWe 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.

Classification of One-Dimensional Attractors of Diffeomorphisms of Surfaces by Means of Pseudo-Anosov Homeomorphisms

Classification of One-Dimensional Attractors of Diffeomorphisms of Surfaces by Means of Pseudo-Anosov Homeomorphisms

On the growth rate of periodic orbits for vector fields

We establish the relationship between the growth rate of periodic orbits and the topological entropy for C1 generic vector fields: this extends a classical result of Katok for C1+α(α>0) surface diffeomorphisms to C1 generic vector fields of any dimension. The main difficulty comes from the existence of singularities and the shear of the flow.

Torsion and linking number for a surface diffeomorphism

For a $$\mathcal{{C}}^1$$ diffeomorphism $$f:\mathbb {R}^2\rightarrow \mathbb {R}^2$$ isotopic to the identity, we prove that for any value $$l\in \mathbb {R}$$ of the linking number at finite time of the orbits of two points there exists at least a point whose torsion at the same finite time equals $$l\in \mathbb {R}$$ . As an outcome, we give a much simplier proof of a theorem by Matsumoto and Nakayama concerning torsion of measures on $$\mathbb {T}^2$$ . In addition, in the framework of twist maps, we generalize a known result concerning the linking number of periodic points: indeed, we estimate such value for any couple of points for which the limit of the linking number exists.

Invariant graphs for chaotically driven maps

This paper investigates the geometrical structures of invariant graphs of skew product systems of the form driven by a hyperbolic base map (e.g. a baker map or an Anosov surface diffeomorphism) and with monotone increasing fibre maps having negative Schwarzian derivatives. We recall a classification, with respect to the number and to the Lyapunov exponents of invariant graphs, for this class of systems. Our major goal here is to describe the structure of invariant graphs and study the properties of the pinching set, the set of points where the values of all of the invariant graphs coincide. In Keller and Otani (2018 Stoch. Dyn. 18 1850009), the authors studied skew product systems driven by a generalized baker map with the restrictive assumption that depend on only through the stable coordinate x of θ. Our aim is to relax this assumption and construct a fibre-wise conjugation between the original system and a new system for which the fibre maps depend only on the stable coordinate of the drive. As an application of this construction we prove that, when S is an Anosov diffeomorphism, the pinching set is a union of global unstable fibres for S.

A calculation of periodic data of surface diffeomorphisms with one saddle orbit.

In the paper it is proved that any orientable surface admits an orientation-preserving diffeomorphism with one saddle orbit. It distinguishes in principle the considered class of systems from source-sink diffeomorphisms existing only on the sphere. It is shown that diffeomorphisms with one saddle orbit of a positive type on any surface have exactly three node orbits. In addition, all possible types of periodic data for such diffeomorphisms are established. Namely, formulas are found expressing the periods of the sources through the periods of the sink and the saddle.

Exceptional Sets for Nonuniformly Hyperbolic Diffeomorphisms

For a surface diffeomorphism, a compact invariant locally maximal set W and some subset $$A\subset W$$ we study the A-exceptional set, that is, the set of points whose orbits do not accumulate at A. We show that if the Hausdorff dimension of A is smaller than the Hausdorff dimension d of some ergodic hyperbolic measure, then the topological entropy of the exceptional set is at least the entropy of this measure and its Hausdorff dimension is at least d. Particular consequences occur when there is some a priori defined hyperbolic structure on W and, for example, if there exists an SRB measure.

Strongly dissipative surface diffeomorphisms

We introduce a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. For that type of systems one can associate to the dynamics a reduced one-dimensional model and it is proved a type of C^\infty -closing lemma on the support of every ergodic measure. We also show that this class contains Hénon maps with Jacobian in (–1/4, 1/4).

Homological stability and stable moduli of flat manifold bundles

We prove that the group homology of the diffeomorphism group of #gSn×Sn\\int(D2n) as a discrete group is independent of g in a range, provided that n>2. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford–Morita–Miller classes lift to a secondary R/Z-invariants for flat (#gSn×Sn)-bundles provided g≫0.

Stable homology of surface diffeomorphism groups made discrete

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that $C^{\infty}$-diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology of $C^{\infty}$-diffeomorphims of surfaces as discrete groups is the same as homology of certain infinite loop space related to Haefliger's classifying space of foliations of codimension 2. We use this infinite loop space to obtain new results about (non)triviality of characteristic classes of flat surface bundles.

The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors

Extending work of Hochman, we study the almost-Borel structure, i.e., the nonatomic invariant probability measures, of symbolic systems and surface diffeomorphisms. We first classify Markov shifts and characterize them as strictly universal with respect to a natural family of classes of Borel systems. We then study their continuous factors showing that a low entropy part is almost-Borel isomorphic to a Markov shift but that the remaining part is much more diverse, even for finite-to-one factors. However, we exhibit a new condition which we call 'Bowen type‘ which gives complete control of those factors. This last result applies to and was motivated by the symbolic covers of Sarig. We find complete numeric invariants for Borel isomorphism of C^{1+} surface diffeomorphisms modulo zero entropy measures; for those admitting a totally ergodic measure of positive (not necessarily maximal) entropy, we get a classification up to almost-Borel isomorphism.

Stable ergodicity and accessibility for certain partially hyperbolic diffeomorphisms with bidimensional center leaves

We consider classes of partially hyperbolic diffeomorphism f:M\to M with splitting TM=E^s\oplus E^c\oplus E^u and \dim E^c=2 . These classes include for instance (perturbations of) the product of Anosov and conservative surface diffeomorphisms, skew products of surface diffeomorphisms over Anosov, partially hyperbolic symplectomorphisms on manifolds of dimension four with bidimensional center foliation whose center leaves are all compact. We prove that accessibility holds in these classes for C^1 open and C^r dense subsets and moreover they are stably ergodic.

The existence connected characteristic space at the gradient-like diffeomorphisms of surfaces

The existence connected characteristic space at the gradient-like diffeomorphisms of surfaces

Surface diffeomorphisms with connected but not path-connected minimal sets containing arcs

In 1955, Gottschalk and Hedlund introduced in their book that Jones constructed a minimal homeomorphism whose minimal set is connectd but not path-connected and contains infinitely many arcs. However the homeomorphism is defined only on this set. In 1991, Walker first constructed a homeomorphism of $S^1\times \mathbf{R}$ with such a minimal set. In this paper, we will show that Walker's homeomorphism cannot be a diffeomorphism (Theorem 2). Furthermore, we will construct a $C^\infty$ diffeomorphism of $S^1\times \mathbf{R}$ with a compact connected but not path-connected minimal set containing arcs (Theorem 1) by using the approximation by conjugation method.

An effective version of Katok's horseshoe theorem for conservative <i>C</i><sup>2</sup> surface diffeomorphisms

For area preserving C2 surface diffeomorphisms, we give an explicit finite information condition on the exponential growth of the number of Bowen's (n, δ)-balls needed to cover a positive proportion of the space, that is sufficient to guarantee positive topological entropy. This can be seen as an effective version of Katok's horseshoe theorem in the conservative setting. We also show that the analogous result is false in dimension larger than 3.

On Möbius orthogonality for subshifts of finite type with positive topological entropy

We prove that Mobius orthogonality does not hold for subshifts of finite type with positive topological entropy. This, in particular, shows that all C1+alpha surface diffeomorphisms with positive e ...

Sinks with relatively large immediate basins and a refinement of Mañé’s C1 generic dichotomy

A C1 perturbation theorem creating sinks with relatively large immediate basins of attraction under the existence of a non-atomic ergodic measure admitting at most small positive Lyapunov exponents is provided. As an application, we prove a refinement of Mañé’s C1 generic dichotomy for surface diffeomorphisms, that is, C1 generically they have either (I) hyperbolicity; or (II) (by taking the inverse if necessary) infinitely many attracting periodic orbits each of which has either (a) relatively large immediate basins of attraction or (b) a pathological feature (i.e. the maximal and minimum norms of the derivatives along the periodic orbits increase and decrease exponentially, respectively, at the periods by a uniform rate). Moreover, the property (a) of (II) is considered from numerical viewpoints in the context of ‘observability’.