scholarly journals On Anosov diffeomorphisms with asymptotically conformal periodic data

2009 ◽  
Vol 29 (1) ◽  
pp. 117-136 ◽  
Author(s):  
BORIS KALININ ◽  
VICTORIA SADOVSKAYA
Keyword(s):  
Periodic Points ◽  
Finite Cover ◽  
Rigidity Result ◽  
Anosov Diffeomorphisms ◽  
Local Rigidity ◽  
Return Maps ◽  
Periodic Data ◽  
Conformal Structures ◽  
Derivatives Of ◽  
Volume Preserving

AbstractWe consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We prove various properties of such systems, including strong pinching, C1+β smoothness of the Anosov splitting, and C1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume-preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of 𝕋4; as a corollary, we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.

Local rigidity of Anosov higher-rank lattice actions

1998 ◽  
Vol 18 (3) ◽  
pp. 687-702 ◽  
Author(s):  
NANTIAN QIAN ◽  
CHENGBO YUE
Keyword(s):  
Abelian Group ◽  
Lyapunov Functional ◽  
Rigidity Result ◽  
Absolute Value ◽  
Local Rigidity ◽  
Higher Rank ◽  
Standard Action ◽  
Lattice Actions

Let $\rho_0$ be the standard action of a higher-rank lattice $\Gamma$ on a torus by automorphisms induced by a homomorphism $\pi_0:\Gamma\to SL(n,{\Bbb Z})$. Assume that there exists an abelian group ${\cal A}\subset \Gamma$ such that $\pi_0({\cal A})$ satisfies the following conditions: (1) ${\cal A}$ is ${\Bbb R}$-diagonalizable; (2) there exists an element $a\in {\cal A}$, such that none of its eigenvalues $\lambda_1,\dots,\lambda_n$ has unit absolute value, and for all $i,j,k=1,\dots,n$, $|\lambda_i\lambda_j|\neq|\lambda_k|$; (3) for each Lyapunov functional $\chi_i$, there exist finitely many elements $a_j\in {\cal A}$ such that $E_{\chi_i}=\cap_{j} E^u(a_j)$ (see \S1 for definitions). Then $\rho_0$ is locally rigid. This local rigidity result differs from earlier ones in that it does not require a certain one-dimensionality condition.

scholarly journals Classification of rough transformations of a circle from a modern point of view

2018 ◽  
Vol 20 (4) ◽  
pp. 408-418
Author(s):  
A. E. Kolobyanina ◽  
E. V. Nozdrinova ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Point Of View ◽  
Topological Classification ◽  
Topological Invariants ◽  
Modern Theory ◽  
Ambient Manifold ◽  
Periodic Data

In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.

scholarly journals Revisiting Leighton’s theorem with the Haar measure

2020 ◽  
pp. 1-9
Author(s):  
DANIEL J. WOODHOUSE
Keyword(s):  
Haar Measure ◽  
Free Groups ◽  
Finite Cover ◽  
Covering Theorem ◽  
Rigidity Result ◽  
Graph Covering ◽  
Finite Graphs ◽  
Universal Covers ◽  
Line Patterns

Abstract Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

scholarly journals Hyperbolicity in the volume-preserving scenario

2012 ◽  
Vol 33 (6) ◽  
pp. 1644-1666 ◽  
Author(s):  
ALEXANDER ARBIETO ◽  
THIAGO CATALAN
Keyword(s):  
Analogous Result ◽  
Periodic Points ◽  
Axiom A ◽  
Volume Preserving

AbstractExtending a result of Mañé, Hayashi proved that every diffeomorphism$f$which has a$C^1$-neighborhood$\mathcal {U}$where all periodic points of any$g\in \mathcal {U}$are hyperbolic is an Axiom A diffeomorphism. Here we prove the analogous result in the volume-preserving scenario.

scholarly journals Center foliation: absolute continuity, disintegration and rigidity

2014 ◽  
Vol 36 (1) ◽  
pp. 256-275 ◽  
Author(s):  
RÉGIS VARÃO
Keyword(s):  
Absolute Continuity ◽  
Geodesic Flows ◽  
Absolutely Continuous ◽  
Rigidity Result ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Smooth Conjugacy ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism ◽  
Volume Preserving

In this paper we address the issues of absolute continuity for the center foliation, as well as the disintegration on the non-absolute continuous case and rigidity of volume-preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov automorphism on $\mathbb{T}^{3}$. It is shown that the disintegration of volume on center leaves for these diffeomorphisms may be neither atomic nor Lebesgue, in contrast to the dichotomy (Lebesgue or atomic) obtained by Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. Preprint, 2012, arXiv:1110.2365v2] for perturbations of time-one of geodesic flow. In the case of atomic disintegration of volume on the center leaves of an Anosov diffeomorphism on $\mathbb{T}^{3}$, we show that it has to be one atom per leaf. Moreover, we show that not even a $C^{1}$ center foliation implies a rigidity result. However, for a volume-preserving partially hyperbolic diffeomorphism isotopic to a linear Anosov automorphism, assuming the center foliation is $C^{1}$ and transversely absolutely continuous with bounded Jacobians, we obtain smooth conjugacy to its linearization.

scholarly journals Renormalization for Lorenz maps of monotone combinatorial types

2017 ◽  
Vol 39 (1) ◽  
pp. 132-158
Author(s):  
DENIS GAIDASHEV
Keyword(s):  
Critical Point ◽  
A Priori ◽  
Periodic Points ◽  
Unit Interval ◽  
Return Maps ◽  
Lorenz Maps

Lorenz maps are maps of the unit interval with one critical point of order $\unicode[STIX]{x1D70C}>1$ and a discontinuity at that point. They appear as return maps of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics and a sufficiently flat critical point, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.

scholarly journals A lower bound for topological entropy of generic non-Anosov symplectic diffeomorphisms

2013 ◽  
Vol 34 (5) ◽  
pp. 1503-1524 ◽  
Author(s):  
THIAGO CATALAN ◽  
ALI TAHZIBI
Keyword(s):  
Lyapunov Exponent ◽  
Lower Bound ◽  
Topological Entropy ◽  
Upper Semicontinuity ◽  
Periodic Points ◽  
Surface Diffeomorphisms ◽  
Symplectic Diffeomorphisms ◽  
Symbolic Extension ◽  
Symplectic Diffeomorphism ◽  
Volume Preserving

AbstractWe prove that a${C}^{1} $generic symplectic diffeomorphism is either Anosov or its topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of its periodic points. We also prove that${C}^{1} $generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and, finally, we give examples of volume preserving surface diffeomorphisms which are not points of upper semicontinuity of the entropy function in the${C}^{1} $topology.

scholarly journals Transitivity and the centre for maps of the circle

1986 ◽  
Vol 6 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Ethan M. Coven ◽  
Irene Mulvey
Keyword(s):  
Periodic Points ◽  
Periodic Data ◽  
Continuous Maps ◽  
Single Power

AbstractWe study the dynamics of continuous maps of the circle with periodic points. We show that the centre is the closure of the periodic points and that the depth of the centre is at most two. We also characterize the property that every power is transitive in terms of transitivity of a single power and some periodic data.

Periodic points and finite group actions on shifts of finite type

1993 ◽  
Vol 13 (3) ◽  
pp. 485-514 ◽  
Author(s):  
Ulf-Rainer Fiebig
Keyword(s):  
Finite Group ◽  
Zeta Function ◽  
Finite Type ◽  
Ordered Set ◽  
Zeta Functions ◽  
Orbit Space ◽  
Periodic Points ◽  
Shift Of Finite Type ◽  
Finite Group Actions ◽  
Periodic Data

AbstractLet G be an abstract finite group. For an action α of G on a shift of finite type (SFT) S we introduce the periodic data of α, a computable finite-ordered set of complex polynomials. We show that two actions of G on possibly different SFTs are conjugate on periodic points iff their periodic data coincide. For each subgroup H of G the points fixed by α|H (the restriction of α to H) form a subsystem of S, which is of finite type. Our result shows that the zeta functions of these subsystems determine the conjugacy class (on periodic points) of α up to finitely many possibilities.The orbit space of a finite skew action on an SFT S, endowed with the homeomorphism induced by S, is shown to have a zeta function equal to the zeta function of an SFT which is a left-closing quotient of S. We show that this zeta function equals the zeta function of S iff the skew action is inert with respect to a certain power of S.Finally we consider functions of the periodic data as for example gyration numbers.

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