C1 Diffeomorphisms Research Articles

On normal numbers and self-similar measures

Let {fi(x)=si⋅x+ti} be a self-similar IFS on R and let β>1 be a Pisot number. We prove that if log⁡|si|log⁡β∉Q for some i then for every C1 diffeomorphism g and every non-atomic self-similar measure μ, the measure gμ is supported on numbers that are normal in base β.

(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.

Local stable and unstable sets for positive entropy C1 dynamical systems

For any C1 diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent. The main line of our approach to this result is under the setting of topological dynamical systems, which is also applicable to infinite-dimensional C1 dynamical systems.

The approximation of uniform hyperbolicity for C1 diffeomorphisms with hyperbolic measures

Let f be a C1-diffeomorphism which preserves an ergodic hyperbolic measure μ with positive measure-theoretic entropy. Assume that the Oseledec splittingTxM=E1(x)⊕⋯⊕Es(x)⊕Es+1(x)⊕⋯⊕El(x) is dominated on the Oseledec basin Γ. We give extensions of Katok's Horseshoes construction. Moreover there is a dominated splitting corresponding to Oseledec subspace on horseshoes. Based on this, we also establish the continuity of the sub-additive topological pressure of the singular value potential.

On Some Sufficient Hyperbolicity Conditions

Abstract—We consider an arbitrary C1 diffeomorphism f that acts from an open subset U of a Riemannian manifold M of dimension m, m ≥ 2, to f(U) ⊂ M. Let A be a compact f-invariant (i.e., f(A) = A) subset in U. We propose various sufficient conditions under which A is a hyperbolic set of f.

Entropy Spectrum of Lyapunov Exponents for Nonhyperbolic Step Skew-Products and Elliptic Cocycles

We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of N, $${N\ge2}$$ , symbols and with C1 diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. Examples of such systems arise from the projective action of $${2\times 2}$$ matrix cocycles and our results apply to an open and dense subset of elliptic $${\mathrm{SL}(2,\mathbb{R})}$$ cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov exponent. The results are formulated in terms of Legendre–Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov exponent. The level set of the zero exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skew-products arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov exponent.

Local perturbations of conservative C1 diffeomorphisms

A number of techniques have been developed to perturb the dynamics of C1-diffeomorphisms and to modify the properties of their periodic orbits. For instance, one can locally linearize the dynamics, change the tangent dynamics, or create local homoclinic orbits. These techniques have been crucial for the understanding of C1 dynamics, but their most precise forms have mostly been shown in the dissipative setting. This work extends these results to volume-preserving and especially symplectic systems. These tools underlie our study of the entropy of C1-diffeomorphisms in Buzzi et al (2016 (arXiv:1606.01765)). We also give an application to the approximation of transitive invariant sets without genericity assumptions.

The entropy conjecture for dominated splitting with multi 1D centers via upper semi-continuity of the metric entropy

Let f be a C1 diffeomorphism on a compact manifold M, be a compact invariant subset with a dominated splitting such that dim and for any invariant probability measure ν, the Lyapunov exponents of ν are non-negative along Ecu and non-positive along Ecs. Then the entropy conjecture is true in this setting as a consequence of the upper semi-continuity of the metric entropy.

On Pesin's entropy formula for dominated splittings without mixed behavior

For C1 diffeomorphisms, we prove that the Pesin's entropy formula holds for some invariant measure supported on any topological attractor that admits a dominated splitting without mixed behavior. We also prove Shub's entropy conjecture for diffeomorphisms having such kind of splittings.

Denjoy C1 diffeomorphisms of the circle and McDuff’s question

In this expository article, after the basic theory of orientation preserving C1 diffeomorphisms of the circle, we present D. McDuff’s theorem on the lengths of the complementary intervals of the unique Cantor minimal set of a Denjoy C1 diffeomorphism of the circle. This leads to a question posed by Dusa McDuff which is related to the solvability of a cohomological equation on the Cantor set.

Formula omitted] variational problems for maps and the Aronsson PDE system

By employing Aronssonʼs absolute minimizers of L∞ functionals, we prove that absolutely minimizing maps u:Rn→RN solve a “tangential” Aronsson PDE system. By following Sheffield and Smart (2012) [24], we derive Δ∞ with respect to the dual operator norm and show that such maps miss information along a hyperplane when compared to tight maps. We recover the lost term which causes non-uniqueness and derive the complete Aronsson system which has discontinuous coefficients. In particular, the Euclidean ∞-Laplacian is Δ∞u=Du⊗Du:D2u+|Du|2[Du]⊥Δu where [Du]⊥ is the projection on the null space of Du⊤. We demonstrate C∞ solutions having interfaces along which the rank of their gradient is discontinuous and propose a modification with C0 coefficients which admits varifold solutions. Away from the interfaces, Aronsson maps satisfy a structural property of local splitting to 2 phases, a horizontal and a vertical; horizontally they possess gradient flows similar to the scalar case and vertically solve a linear system coupled by a scalar Hamilton Jacobi PDE. We also construct singular ∞-harmonic local C1 diffeomorphisms and singular Aronsson maps.

There are no C1-stable intersections of regular Cantor sets

We prove that there are no stable intersections of regular Cantor sets in the C 1 topology: given any pair (K, K′) of regular Cantor sets, we can find, arbitrarily close to it in the C 1 topology, pairs $ \left( {\tilde{K},\tilde{K}'} \right) $ of regular Cantor sets with $ \tilde{K} \cap \tilde{K}' = \emptyset $ . Moreover, for generic pairs (K, K′) of C 1-regular Cantor sets, the arithmetic difference $ K - K' = \left\{ {x - y:x \in K,y \in K'} \right\} = \left\{ {t \in \mathbb{R}:K \cap \left( {K' + t} \right) \ne \emptyset } \right\} $ has empty interior (and so is a Cantor set). This is very different from the situation in the C 2 topology: according to a theorem by Moreira and Yoccoz, typical pairs (K, K′) of C 2-regular Cantor sets whose sum of Hausdorff dimensions is larger than 1 are such that their arithmetic difference K, K′ is the closure of its interior. We also show that there is a generic set $ \mathcal{R} $ of C 1 diffeomorphisms of M such that, for every $ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\psi }} \in \mathcal{R} $ , there are no tangencies between leaves of the stable and unstable foliations of Λ1, Λ2, for any horseshoes Λ1, Λ2 of $ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\psi }} $ .

Non-uniformly hyperbolic periodic points and uniform hyperbolicity for C1 diffeomorphisms

We prove that for C1 generic diffeomorphisms, every isolated compact invariant set Λ which satisfies a mild condition on the hyperbolicity of periodic points in Λ (called the L-NUH condition, see definition 1.1) is hyperbolic. In parallel, we prove that for C1 diffeomorphisms, every compact invariant set which satisfies Katok's periodic closing property and the L-NUH condition on periodic points is hyperbolic, which is a generalized result of Castro et al (2007 Nonlinearity 20 75-85) for C2 case with a periodic closing property (called periodic shadowing property in Castro et al (2007 Nonlinearity 20 75-85)).

Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains

In the first part of this article we give intrinsic characterizations of the classes of Lipschitz and C1 domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversal vector field. We also show that if the geometric measure theoretic unit normal of the domain is continuous, then the domain in question is of class C1. In the second part of the article, we study the invariance of various classes of domains of locally finite perimeter under bi-Lipschitz and C1 diffeomorphisms of the Euclidean space. In particular, we prove that the class of bounded regular SKT domains (previously called chord-arc domains with vanishing constant, in the literature) is stable under C1 diffeomorphisms. A number of other applications are also presented.

New examples of Cantor sets in S1 that are not C1-minimal

Although every Cantor subset of the circle (S1) is the minimal set of some homeomorphism of S1, not every such set is minimal for a C1 diffeomorphism of S1. In this work, we construct new examples of Cantor sets in S1 that are not minimal for any C1-diffeomorphim of S1.

Fixed points of abelian actions on S2

AbstractWe prove that if ${\mathcal F}$ is a finitely generated abelian group of orientation preserving C1 diffeomorphisms of $\mathbb {R}^2$ which leaves invariant a compact set then there is a common fixed point for all elements of ${\mathcal F}$. We also show that if ${\mathcal F}$ is any abelian subgroup of orientation preserving C1 diffeomorphisms of S2 then there is a common fixed point for all elements of a subgroup of ${\mathcal F}$ with index at most two.

Hyperbolicity, heterodimensional cycles and Lyapunov exponents for partially hyperbolic dynamics

We prove a dichotomy of C2 partially hyperbolic sets with one-dimensional center direction admitting no zero Lyapunov exponents, either hyperbolicity over the supports of ergodic measures or approximation by a heterodimensional cycle. This provides a partial result to the C1 Palis Conjecture that claims a dichotomy, hyperbolicity or homoclinic bifurcations in a dense subset of the space of C1 diffeomorphisms. Moreover, a theorem of Mane applied in the proof is modified to have an additional property concerning the Hausdorff distance between a periodic orbit and the support of a hyperbolic ergodic measure.

Horseshoe and entropy for C1 surface diffeomorphisms

Let M be a two-dimensional closed Riemannian manifoldand denote by Diff1(M) the set of C1 diffeomorphisms on M. Then, the C1 interior of {f∊Diff1(M):h(f) = 0} is equal tothe C1 interior of the closure of the Morse-Smale systemsand equal to the C1 interior of the set of diffeomorphisms having no horseshoe.

Uniform (projective) hyperbolicity or no hyperbolicity: A dichotomy for generic conservative maps

We show that the Lyapunov exponents of volume preserving C1 diffeomorphisms of a compact manifold are continuous at a given diffeomorphism if and only if the Oseledets splitting is either dominated or trivial. It follows that for a C1-residual subset of volume preserving diffeomorphisms the Oseledets splitting is either dominated or trivial.We obtain analogous results in the setting of symplectic diffeomorphisms, where the conclusion is actually stronger: dominated splitting is replaced by partial hyperbolicity. We also obtain versions of these results for continuous cocycles with values in some matrix groups.In the text we give the precise statements of these results and the ideas of the proofs. The complete proofs will appear in [4].

A SUSPENSION OF AN ORIENTATION PRESERVING DIFFEOMORPHISM OF D2 WITH A HYPERBOLIC FIXED POINT AND UNIVERSAL TEMPLATE

For an orientation preserving homeomorphism φ of the disk into itself, a suspension of a finite union of periodic orbits P of φ represents a link type in the 3-sphere S3. Let φ be a C1 diffeomorphism, and p a hyperbolic fixed point of φ with a homoclinic point. If all the homoclinic points for p are transeverse, then for infinitely many n>0, φn induces all link types, that is, for each link type L in S3, there exists a finite union of periodic orbits [Formula: see text] of φn such that a suspension of [Formula: see text] of φn represents L.