Search

A class of planar diffeomorphims is formulated, with infinitely many coexisting Smale horseshoes, where the Lebesgue measure of the parameters with such strange dynamics is infinite. On each horseshoe, there exists a uniformly hyperbolic invariant set, on which the map is topologically conjugate to the two-sided full-shift on two symbols. Moreover, the topological entropy is infinite in certain parameter regions. [ABSTRACT FROM AUTHOR]

We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer's metric. Namely: (1) we resolve the Kapovich-Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also find, as a corollary, that the group of area-preserving diffeomorphisms of the open disc, equipped with an area form of finite area, is not perfect. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology. [ABSTRACT FROM AUTHOR]

Let S 1 , S 2 , S 3 {S_{1},S_{2},S_{3}} be oriented non-flat minimal surfaces immersed in ℝ n + 1 {{\mathbb{R}}^{n+1}} ( n ≥ 2 ) {(n\geq 2)} with the Gauss maps G 1 , G 2 , G 3 {G_{1},G_{2},G_{3}} into ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} , respectively. Assume that there are conformal diffeomorphisms Φ 2 , Φ 3 {\Phi_{2},\Phi_{3}} of S 1 {S_{1}} onto S 2 , S 3 {S_{2},S_{3}} respectively and let f 1 = G 1 {f^{1}=G_{1}} , f 2 = G 2 ∘ Φ 2 {f^{2}=G_{2}\circ\Phi_{2}} , f 3 = G 3 ∘ Φ 3 {f^{3}=G_{3}\circ\Phi_{3}} . In this paper, we will show that f 1 , f 2 , f 3 {f^{1},f^{2},f^{3}} are algebraic dependence, i.e., f 1 ∧ f 2 ∧ f 3 ≡ 0 {f^{1}\wedge f^{2}\wedge f^{3}\equiv 0} , if they have the same inverse images for a few hyperplanes of ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} in general position with some certain conditions. [ABSTRACT FROM AUTHOR]

Motivated by the classical concept of Newhouse domains (an open set of diffeomorphisms having a dense set of systems with homoclinic tangencies), we introduce formally the novel notion of Berger domains. Namely, a Berger domain is an open set of parametric families of diffeomorphisms having a dense set of families with persistent homoclinic tangencies. The original construction of Berger provides examples of such domains such that the tangencies are of codimension one and associated with sectional dissipative periodic points with real multipliers. We show that Berger domains can be constructed associated with homoclinic tangencies of arbitrarily large codimension and any type of hyperbolic periodic points. As an application, we find new Kolmogorov locally typical phenomena in dimension three or greater. Namely, we prove that generic families in a certain class of Berger domains have the phenomenon of coexistence of infinitely many attracting invariant circles for any parameter. [ABSTRACT FROM AUTHOR]

We propose a two‐scale finite element method designed for heterogeneous microstructures. Our approach exploits domain diffeomorphisms between the microscopic structures to gain computational efficiency. By using a conveniently constructed pullback operator, we are able to model the different microscopic domains as macroscopically dependent deformations of a reference domain. This allows for a relatively simple finite element framework to approximate the underlying system of partial differential equations with a parallel computational structure. We apply this technique to a model problem where we focus on transport in plant tissues. We illustrate the accuracy of the implementation with convergence benchmarks and show satisfactory parallelization speed‐ups. We further highlight the effect of the heterogeneous microscopic structure on the output of the two‐scale systems. Our implementation (publicly available on GitHub) builds on the deal.II FEM library. Application of this technique allows for an increased capacity of microscopic detail in multiscale modeling, while keeping running costs manageable. [ABSTRACT FROM AUTHOR]

Let M=(M,\omega) be either S^2 \times S^2 or \mathbb {C}P^2\# \overline {\mathbb {C}P^2} endowed with any symplectic form \omega. Suppose a finite cyclic group \mathbb {Z}_n is acting effectively on (M,\omega) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism \mathbb {Z}_n\hookrightarrow Ham(M,\omega). In this paper, we investigate the homotopy type of the group Symp^{\mathbb {Z}_n}(M,\omega) of equivariant symplectomorphisms. We prove that for some infinite families of \mathbb {Z}_n actions satisfying certain inequalities involving the order n and the symplectic cohomology class [\omega ], the actions extend to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on 4-manifolds, and on the Chen-Wilczyński classification of smooth \mathbb {Z}_n-actions on Hirzebruch surfaces. [ABSTRACT FROM AUTHOR]

We show that a cocycle over a hyperbolic system with constant periodic data has a dominated splitting whenever the periodic data indicates it should. This implies global periodic data rigidity of generic Anosov automorphisms of . Further, our approach also works when the periodic data is narrow, that is, sufficiently close to constant. We can show global periodic data rigidity for certain non-linear Anosov diffeomorphisms in a neighborhood of an irreducible Anosov automorphism with simple spectrum. [ABSTRACT FROM AUTHOR]

A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold endowed with a volume density. We study the geometry of Fréchet manifolds of weighted nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians. When the ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic nonlinear flags. [ABSTRACT FROM AUTHOR]

We prove that a transitive uniformly quasiconformal Anosov diffeomorphism with a two-dimensional stable or unstable distribution has a global product structure. As an application, we remove a topological assumption in a global rigidity theorem for uniformly quasiconformal Anosov diffeomorphisms of Kalinin-Sadovskaya [13]. [ABSTRACT FROM AUTHOR]

Gompf showed that for |$K$| in a certain family of double-twist knots, the swallow-follow operation makes |$1/n$| -surgery on |$K \# -K$| into a cork boundary. We derive a general Floer-theoretic condition on |$K$| under which this is the case. Our formalism allows us to produce many further examples of corks, partially answering a question of Gompf. Unlike Gompf's method, our proof does not rely on any closed 4-manifold invariants or effective embeddings, and also generalizes to other diffeomorphisms. [ABSTRACT FROM AUTHOR]

In this paper we consider C 1 surface diffeomorphisms and study the existence of phase transitions, here expressed by the non-analiticity of the pressure function associated to smooth and geometric-type potentials. We prove that the space of C 1 -surface diffeomorphisms admitting phase transitions is a C 1 -Baire generic subset of the space of non-Anosov diffeomorphisms. In particular, if S is a compact surface which is not homeomorphic to the 2-torus then a C 1 -generic diffeomorphism on S has phase transitions. We obtain similar statements in the context of C 1 -volume preserving diffeomorphisms. Finally, we prove that a C 2 -surface diffeomorphism exhibiting a dominated splitting admits phase transitions if and only if has some non-hyperbolic periodic point. [ABSTRACT FROM AUTHOR]

Let \operatorname {Diff}_{\partial }(D^{n}) be the topological group of diffeomorphisms of D^{n} which agree with the identity near the boundary. In this short note, we compute the fundamental group \pi _1 \operatorname {Diff}_{\partial }(D^{4k}) for k\geq 3. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for an expansive partially hyperbolic diffeomorphism with one-dimensional center to be (topologically) Anosov. [ABSTRACT FROM AUTHOR]

Due to severe deformation, noise, and occlusion, the registration problem of non-rigid point sets in rolling formed metal workpieces poses challenges, and the demand for real-time data storage and registration during the rolling forming process makes this problem even more prominent. This paper proposes an enhanced nonrigid point set registration algorithm based on the Coherent Point Drift (CPD) framework, introducing novel methods to improve accuracy and efficiency. A refined local distance calculation method combining spatial distance has been proposed to improve matching accuracy. In contrast, an optimized shape context method introduces a new driving force criterion to expedite initial registration and reduce subsequent errors. Leveraging the Expectation-Maximization (EM) algorithm, the approach iteratively solves point correspondences, demonstrating robustness in handling complex scenarios like non-rigid deformation and noise. Experimental validation using real production datasets shows superior accuracy and efficiency over classical algorithms, showcasing a practical solution for non-rigid point set registration challenges in roll forming applications. [ABSTRACT FROM AUTHOR]

We prove that the singular support of an element in the derived category of sheaves is γ-coisotropic, a notion defined in [Vit22]. We prove that this implies that it is involutive in the sense of Kashiwara-Schapira, but being γ-coisotropic has the advantage to be invariant by symplectic homeomorphisms (while involutivity is only invariant by C1 diffeomorphisms) and we give an example of an involutive set that is not γ-coisotropic. Along the way we prove a number of results relating the singular support and the spectral norm γ and raise a number of new questions. [ABSTRACT FROM AUTHOR]

For diffeomorphisms with hyperbolic sets, the Anosov Closing Lemma ensures the existence of periodic orbits in the neighbourhood of orbits that return close enough to themselves. Moreover, it defines how the distance between the corresponding points of an initial orbit and the constructed periodic orbits is controlled. In the essential, this article presents proof of the estimate of this distance. The Anosov Closing Lemma is crucial in the statement of Livschitz Theorem that, based only on the periodic data, provides a necessary and sufficient condition so that cohomological equations have sufficiently regular solutions, H¨older solutions. It is one of the main tools to obtain global data of a cohomological nature based only on periodic data. As suggested by Katok and Hasselblat in [2], it is demonstrated, in detail and the cohomology context, the Livschitz Theorem for hyperbolic diffeomorphisms, where the mentioned distance control inequality is essential. [ABSTRACT FROM AUTHOR]

We prove that for any two Riemannian metrics |$\sigma _{1}, \sigma _{2}$| on the unit disk, a homeomorphism |$\partial \mathbb{D}\to \partial \mathbb{D}$| extends to at most one quasiconformal minimal diffeomorphism |$(\mathbb{D},\sigma _{1})\to (\mathbb{D},\sigma _{2})$| with |$L^{1}$| Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the |$L^{1}$| assumption in variable curvature. The key input for our proof is the uniqueness of solutions for a certain Plateau problem in a product of trees. [ABSTRACT FROM AUTHOR]

We construct a braided monoidal functor |$J_4$| from Bobtcheva and Piergallini's category |$4\textrm {HB}$| of connected 4-dimensional 2-handlebodies (up to 2-deformations) to an arbitrary unimodular ribbon category |$\mathscr {C}$|⁠ , which is not required to be semisimple. The main example of target category is provided by |${H}\textrm{-mod}$|⁠ , the category of left modules over a unimodular ribbon Hopf algebra |$H$|⁠. The source category |$4\textrm {HB}$| is freely generated, as a braided monoidal category, by a Bobtcheva--Piergallini Hopf (BPH) algebra object, and this is sent by the Kerler–Lyubashenko functor |$J_4$| to the end |$\int _{X \in \mathscr {C}} X \otimes X^*$| in |$\mathscr {C}$|⁠ , which is given by the adjoint representation in the case of |${H}\textrm{-mod}$|⁠. When |$\mathscr {C}$| is factorizable, we show that the construction only depends on the boundary and signature of handlebodies and thus projects to a functor |$J_3^{\sigma }$| defined on Kerler's category |$3\textrm {Cob}^{\sigma }$| of connected framed 3-dimensional cobordisms. When |$H^*$| is not semisimple and |$H$| is not factorizable, our functor |$J_4$| has the potential of detecting diffeomorphisms that are not 2-deformations. [ABSTRACT FROM AUTHOR]

We study \mathbb {R}^k \times \mathbb {Z}^\ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program. [ABSTRACT FROM AUTHOR]

To a graph, Hausel and Proudfoot associate two complex manifolds, B and D, which behave, respectively, like moduli of local systems on a Riemann surface and moduli of Higgs bundles. For instance, B is a moduli space of microlocal sheaves, which generalize local systems, and D carries the structure of a complex integrable system. We show the Euler characteristics of these varieties count spanning subtrees of the graph, and the point-count over a finite field for B is a generating polynomial for spanning subgraphs. This polynomial satisfies a deletion-contraction relation, which we lift to a deletion-contraction exact triangle for the cohomology of B. There is a corresponding triangle for D. Finally, we prove that B and D are diffeomorphic, the diffeomorphism carries the weight filtration on the cohomology of B to the perverse Leray filtration on the cohomology of D, and all these structures are compatible with the deletion-contraction triangles. [ABSTRACT FROM AUTHOR]

We study the local structure of vector fields on that preserve the Martinet -form . We classify their singularities up to diffeomorphisms that preserve the form , as well as their transverse unfoldings. We are thus able to provide a fairly complete list of the bifurcations such vector fields undergo. [ABSTRACT FROM AUTHOR]

In [1] the Theorems 1, 2 and 3, as well as Proposition 1, are incorrect as they are stated. To make them correct it suffices to add the extra condition D (x , 0 , λ) = 0 to the expressions (1.1) and (1.4). The same holds for Definition 2. [ABSTRACT FROM AUTHOR]

We prove several rigidity results about the centralizer of a smooth diffeomorphism, concentrating on two families of examples: diffeomorphisms with transitive centralizer, and perturbations of isometric extensions of Anosov diffeomorphisms of nilmanifolds. We classify all smooth diffeomorphisms with transitive centralizer: they are exactly the maps that preserve a principal fiber bundle structure, acting minimally on the fibers and trivially on the base. We also show that for any smooth, accessible isometric extension |$f_{0}\colon M\to M$| of an Anosov diffeomorphism of a nilmanifold, subject to a spectral bunching condition, any |$f\in \textrm{Diff}^{\infty }(M)$| sufficiently |$C^{1}$| -close to |$f_{0}$| has centralizer a Lie group. If the dimension of this Lie group equals the dimension of the fiber, then |$f$| is a principal fiber bundle morphism covering an Anosov diffeomorphism. Using the results of this paper, we classify the centralizer of any partially hyperbolic diffeomorphism of a |$3$| -dimensional, nontoral nilmanifold: either the centralizer is virtually trivial, or the diffeomorphism is an isometric extension of an Anosov diffeomorphism, and the centralizer is virtually |${{\mathbb{Z}}}\times{{\mathbb{T}}}$|⁠. [ABSTRACT FROM AUTHOR]

We consider a generic symplectic partially hyperbolic diffeomorphism close to direct/skew products of symplectic Anosov diffeomorphisms with area-preserving diffeomorphisms and prove that every hyperbolic periodic point has transverse homoclinic points. [ABSTRACT FROM AUTHOR]

We prove that a partially hyperbolic attractor for a C 1 vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is C 2 , and the center bundle admits the sectionally expanding condition w.r.t. Gibbs u-states, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms. [ABSTRACT FROM AUTHOR]

Let F be a Morse–Bott foliation on the solid torus T = S 1 × D 2 into 2-tori parallel to the boundary and one singular central circle. Gluing two copies of T by some diffeomorphism between their boundaries, one gets a lens space L p , q with a Morse–Bott foliation F p , q obtained from F on each copy of T and thus consisting of two singular circles and parallel 2-tori. In the previous paper Khokliuk and Maksymenko (J Homotopy Relat Struct 18:313–356. https://doi.org/10.1007/s40062-023-00328-z, 2024) there were computed weak homotopy types of the groups D lp (F p , q) of leaf preserving (i.e. leaving invariant each leaf) diffeomorphisms of such foliations. In the present paper it is shown that the inclusion of these groups into the corresponding group D + fol (F p , q) of foliated (i.e. sending leaves to leaves) diffeomorphisms which do not interchange singular circles are homotopy equivalences. [ABSTRACT FROM AUTHOR]

Asaoka & Irie [3] recently proved a $ C^{\infty} $ closing lemma of Hamiltonian diffeomorphisms of closed surfaces. We reformulated their techniques into a more general perturbation lemma for area-preserving diffeomorphism and proved a $ C^{\infty} $ closing lemma for area-preserving diffeomorphisms on a torus $ \mathbb{T}^2 $ that is isotopic to identity. i.e., we show that the set of periodic orbits is dense for a generic diffeomorphism isotopic to identity area-preserving diffeomorphism on $ \mathbb{T}^2 $. The main tool is the flux vector of area-preserving diffeomorphisms, which may not be zero as in the Hamiltonian cases. [ABSTRACT FROM AUTHOR]

We study periodic points for area-preserving maps on surfaces, particularly some global properties related to the action functional. We generalize recent works of Hutchings [14] and Weiler [17] , proving the existence of periodic orbits with certain action values. [ABSTRACT FROM AUTHOR]

We prove a Hölder-type inequality for Hamiltonian diffeomorphisms relating the |$C^0$| norm, the |$C^0$| norm of the derivative, and the Hofer/spectral norm. We obtain as a consequence that sufficiently fast convergence in Hofer/spectral metric forces |$C^0$| convergence. The second theme of our paper is the study of pseudo-rotations that arise from the Anosov–Katok method. As an application of our Hölder-type inequality, we prove a |$C^0$| rigidity result for such pseudo-rotations. [ABSTRACT FROM AUTHOR]

The present paper is devoted to a lower bound for the number of critical points of the Lyapunov function for Morse–Smale 3-diffeomorphisms with fixed points with pairwise distinct indices. It is known that, in the presence of a single noncompact heteroclinic curve, the supporting manifold of the diffeomorphisms under consideration is a 3-sphere, and the class of topological conjugacy of such a diffeomorphism is completely determined by the equivalence class (there exist infinitely many of them) of the Hopf knot , which is a knot in the generating class of the fundamental group of the manifold . Moreover, any Hopf knot is realized by some diffeomorphism of the class under consideration. It is known that the diffeomorphisms defined by the standard Hopf knot have an energy function, which is a Lyapunov function whose set of critical points coincides with the chain recurrent set. However, the set of critical points of any Lyapunov function of a diffeomorphism with a nonstandard Hopf knot is strictly greater than the chain recurrent set of the diffeomorphism. In the present paper, for the diffeomorphisms defined by generalized Mazur knots, a quasi-energy function has been constructed, which is a Lyapunov function with a minimum number of critical points. [ABSTRACT FROM AUTHOR]

We show that the cobordism maps on Khovanov homology can distinguish smooth surfaces in the 4-ball that are exotically knotted (i.e., isotopic through ambient homeomorphisms but not ambient diffeomorphisms). We develop new techniques for distinguishing cobordism maps on Khovanov homology, drawing on knot symmetries and braid factorizations. We also show that Plamenevskaya's transverse invariant in Khovanov homology is preserved by maps induced by positive ascending cobordisms. [ABSTRACT FROM AUTHOR]

We show that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism then it also admits an Anosov flow. Moreover, we give a complete classification of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds as well as partially hyperbolic diffeomorphisms in Seifert manifolds inducing pseudo-Anosov dynamics in the base. This classification is given in terms of the structure of their center (branching) foliations and the notion of collapsed Anosov flows. [ABSTRACT FROM AUTHOR]

We study minimisers of the p-conformal energy functionals, E p (f) : = ∫ D K p (z , f) d z , f | S = f 0 | S , defined for self mappings f : D → D with finite distortion and prescribed boundary values f 0 . Here K (z , f) = ‖ D f (z) ‖ 2 J (z , f) = 1 + | μ f (z) | 2 1 - | μ f (z) | 2 is the pointwise distortion functional and μ f (z) is the Beltrami coefficient of f. We show that for quasisymmetric boundary data the limiting regimes p → ∞ recover the classical Teichmüller theory of extremal quasiconformal mappings (in part a result of Ahlfors), and for p → 1 recovers the harmonic mapping theory. Critical points of E p always satisfy the inner-variational distributional equation 2 p ∫ D K p μ f ¯ 1 + | μ f | 2 φ z ¯ d z = ∫ D K p φ z d z , ∀ φ ∈ C 0 ∞ (D). We establish the existence of minimisers in the a priori regularity class W 1 , 2 p p + 1 (D) and show these minimisers have a pseudo-inverse - a continuous W 1 , 2 (D) surjection of D with (h ∘ f) (z) = z almost everywhere. We then give a sufficient condition to ensure C ∞ (D) smoothness of solutions to the distributional equation. For instance K (z , f) ∈ L loc p + 1 (D) is enough to imply the solutions to the distributional equation are local diffeomorphisms. Further K (w , h) ∈ L 1 (D) will imply h is a homeomorphism, and together these results yield a diffeomorphic minimiser. We show such higher regularity assumptions to be necessary for critical points of the inner variational equation. [ABSTRACT FROM AUTHOR]

We prove that if two germs of plane curves (C, 0) and (C ′ , 0) with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then C is complex isomorphic to C ′ or to C ′ ¯ . A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component that is non-factorable. Moreover, we provide a general overview of the different classifications of plane curve singularities. [ABSTRACT FROM AUTHOR]

We prove three formulas for computing the topological pressure of $ C^1 $ C 1 -generic conservative diffeomorphism with no dominated splitting and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there are no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there are no equilibrium states. For $ C^1 $ C 1 generic conservative diffeomorphism on compact surfaces with no dominated splitting and $ \phi _m(x):=-\frac {1}{m}\log \Vert D_x f^m\Vert, m \in \mathbb {N} $ ϕ m (x) := − 1 m log ⁡ ‖ D x f m ‖ , m ∈ N , we show that there exist equilibrium states with zero entropy and there exists a transition point $ t_0 $ t 0 for the one parameter family $ \lbrace t \phi _m\rbrace _{t\geq ~0} $ { t ϕ m } t ≥ 0 , such that there is no equilibrium states for $ t \in [0, t_0) $ t ∈ [ 0 , t 0) and there is an equilibrium state for $ t \in [t_0,+\infty) $ t ∈ [ t 0 , + ∞). [ABSTRACT FROM AUTHOR]

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds Ug,1n:=#g(Sn×Sn+1)∖int(D2n+1)$U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}(D^{2n+1})$, for large g$g$ and n$n$, up to degree n−3$n-3$. The answer is that it is a free graded commutative algebra on an appropriate set of Miller–Morita–Mumford classes. Our proof goes through the classical three‐step procedure: (a) compute the cohomology of the homotopy automorphisms, (b) use surgery to compare this to block diffeomorphisms, and (c) use pseudoisotopy theory and algebraic K$K$‐theory to get at actual diffeomorphism groups. [ABSTRACT FROM AUTHOR]

Let f and g be two Anosov diffeomorphisms on \mathbb {T}^3 with three-subbundles partially hyperbolic splittings where the weak stable subbundles are considered as center subbundles. Assume that f is conjugate to g and the conjugacy preserves the strong stable foliation, then their center Lyapunov exponents of corresponding periodic points coincide. This is the converse of the main result of Gogolev and Guysinsky [Discrete Contin. Dyn. Syst. 22 (2008), pp. 183–200]. Moreover, we get the same result for partially hyperbolic diffeomorphisms derived from Anosov on \mathbb {T}^3. [ABSTRACT FROM AUTHOR]

A C ∞ smooth surface diffeomorphism admits an SRB measure if and only if the set { x , lim sup n 1 n log ∥ d x f n ∥ > 0 } has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost everywhere. We also obtain similar results for C r surface diffeomorphisms with + ∞ > r > 1 . [ABSTRACT FROM AUTHOR]

Let f$f$ be a C1+ε$C^{1+\varepsilon }$ diffeomorphism of the closed annulus A$A$ that preserves orientation and the boundary components, and f∼$\widetilde{f}$ be a lift of f$f$ to its universal covering space. Assume that A$A$ is a Birkhoff region of instability for f$f$, and the rotation set of f∼$\widetilde{f}$ is a nondegenerate interval. Then there exists an open f$f$‐invariant essential annulus A∗$A^*$ whose frontier intersects both boundary components of A$A$, and points z+$z^+$ and z−$z^-$ in A∗$A^*$, such that the positive (resp., negative) orbit of z+$z^+$ converges to a set contained in the upper (resp., lower) boundary component of A∗$A^*$ and the positive (resp., negative) orbit of z−$z^-$ converges to a set contained in the lower (resp., upper) boundary component of A∗$A^*$. This extends a celebrated result originally proved by Mather in the context of area‐preserving twist diffeomorphisms. [ABSTRACT FROM AUTHOR]

This work concerns inverse boundary value problems for the time-harmonic Maxwell's equations on differential 1-forms. We formulate the boundary value problem on a 3-dimensional compact and simply connected Riemannian manifold M with boundary ∂ M endowed with a Riemannian metric g. Assuming that the electric permittivity ε and magnetic permeability μ are real-valued anisotropic (i.e (1, 1)-tensors), we aim to determine certain metrics induced by these parameters, denoted by ε ^ and μ ^ at ∂ M . We show that the knowledge of the impedance and admittance maps determines the tangential entries of ε ^ and μ ^ at ∂ M in their boundary normal coordinates, although the background volume form cannot be determined in such coordinates due to a non-uniqueness occuring from diffeomorphisms that fix the boundary. Then, we prove that in some cases, we can also recover the normal components of μ ^ up to a conformal multiple at ∂ M in boundary normal coordinates for ε ^ . Last, we build an inductive proof to show that if ε ^ and μ ^ are determined at ∂ M in boundary normal coordinates for ε ^ , then the same follows for their normal derivatives of all orders at ∂ M . [ABSTRACT FROM AUTHOR]

We investigate the geometry of the central extension |$\widehat{\mathcal D}_{\mu}(S^{2})$| of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with an |$L^{2}$| -metric, for which geodesics correspond to solutions of the incompressible Euler equation with Coriolis force. In particular, we calculate the Misiołek curvature of this group. This value is related to the existence of a conjugate point and its positivity directly implies the positivity of the sectional curvature. [ABSTRACT FROM AUTHOR]

We prove a finite smooth version of the entropic continuity of Lyapunov exponents proved recently by Buzzi, Crovisier, and Sarig for C ∞ surface diffeomorphisms (Buzzi et al., Invent Math 230(2):767–849, 2022). As a consequence, we show that any C r , r > 1 , smooth surface diffeomorphism f with h top (f) > 1 r lim sup n 1 n log + ‖ d f n ‖ ∞ admits a measure of maximal entropy. We also prove the C r continuity of the topological entropy at f. [ABSTRACT FROM AUTHOR]

Einstein presented the Hole Argument against General Covariance, understood as invariance with respect to a change in coordinates, as a consequence of his initial failure to obtain covariant equations that, in the weak static limit, contain Newton's law. Fortunately, about two years later, Einstein returned to General Covariance, and found these famous equations of gravity. However, the rejection of his Hole Argument carries a totally different vision of space-time. Its substantivalism notion, which is an essential ingredient in Newtonian theory and also in his special theory of relativity, has to be replaced, following Descartes and Leibniz's relationalism, by a set of "point-coincidences". [ABSTRACT FROM AUTHOR]

We introduce a notion of residual diffeomorphism covariance in quantum Kantowski–Sachs (KS) describing the interior of a Schwarzschild black hole. We solve for the family of Hamiltonian constraint operators satisfying the associated covariance condition, as well as parity invariance, preservation of the Bohr Hilbert space of the Loop Quantum KS and a correct (naïve) classical limit. We further explore the imposition of minimality for the number of terms and compare the solution with those of other Hamiltonian constraints proposed for the Loop Quantum KS in the literature. In particular, we discuss a lapse that was recently commonly chosen due to the resulting decoupling of the evolution of the two degrees of freedom and the exact solubility of the model. We show that such a choice of lapse can indeed be quantized as an operator that is densely defined on the Bohr Hilbert space and that any such operator must include an infinite number of shift operators. [ABSTRACT FROM AUTHOR]

In this article, a family of diffeomorphisms with growing horseshoes contained in global attracting regions is presented, where the dimension of the unstable direction can be any fixed integer and a growing horseshoe means that the number of the folds of the horseshoe is increasing as a parameter is varied. Moreover, it is demonstrated that the horseshoe-like attractors are observable for certain parameters. [ABSTRACT FROM AUTHOR]

With the rapid development of microtechnology, the low-dimensional materials are fabricated with nontrivial topological structures, and then the action of geometric properties on the effective dynamics receives increasing attention. It is an effective theory that the quantum mechanics of low-dimensional curved systems can be given by using the thin-layer quantization approach, in which the geometric potential and the geometric momentum have been demonstrated experimentally. In the present paper, the thin-layer quantization formalism is first recalled, its fundamental calculation framework is first clarified, and the geometric quantum effects result from the diffeomorphism transformation and the rotation transformation connecting the local frames of different points. The results are helpful to understand the gravitational gauge and emerging gauge, and to image the geometries implied in the artificial gauge. In the particular quantum systems, the novel physical phenomena are briefly recalled that are induced by geometry, such as resonation tunneling, quantum block, quantum Hall effect, quantum Hall viscosity, quantum spin Hall effect, effective monopole magnetic field and so on. The results will shed light from a different angle on the relationships between the geometry and the novel physical phenomenon. [ABSTRACT FROM AUTHOR]

Mechanical systems subject to constraints play a essential role in the field of control engineering, profoundly influencing the design and performance of control strategies. Consequently, there is a compelling need to explore diverse control methods to effectively tackle the complex task of stabilizing nonlinear systems while ensuring the constraints are not violated. In this context, this paper proposes a design procedure for position-constrained controllers in robot manipulators. The solution relies on the construction of a diffeomorphism (a differentiable coordinate transformation) that maps all the trajectories of the constrained dynamics into an unconstrained one. The controller design is carried out in the unconstrained dynamics without dealing directly with the constraints. The proposed family of controllers employ an explicit control law which circumvents the need for additional time-consuming computation for feasibility and/or optimization. Moreover, the proposed controller is parametrized by a class of diffeomorphisms which can be selected by the designer. Exponential stability in constrained and unconstrained position states is achieved, in the certain case. For the uncertain case, the controller is augmented through sliding modes guaranteeing finite-time convergence towards the manifold and keeping the exponential convergence within the manifold dynamics. The approach is validated through experiments in an actual 2 DOF lightweight robot manipulator. • A diffeomorphism-based control law for MIMO constrained nonlinear systems. • The explicit controller ensures (local) exponential convergence of the all states. • A robustification of the controller to deal with uncertainties via sliding modes. [ABSTRACT FROM AUTHOR]

We present the topological transitivity of a class of diffeomorphisms on the thickened torus, including the partially hyperbolic example introduced by Ittai Kan in 1994, which is well known for the first systems with the intermingled basins phenomenon. [ABSTRACT FROM AUTHOR]

We introduce continuum-wise hyperbolicity , a generalization of hyperbolicity with respect to the continuum theory. We discuss similarities and differences between topological hyperbolicity and continuum-wise hyperbolicity. A shadowing lemma for cw-hyperbolic homeomorphisms is proved in the form of the L-shadowing property and a Spectral Decomposition is obtained in this scenario. In the proof we generalize the construction of Fathi [16] of a hyperbolic metric using only cw-expansivity, obtaining a hyperbolic cw-metric. We also introduce cwN-hyperbolicity, exhibit examples of these systems for arbitrarily large N ∈ N and obtain further dynamical properties of these systems such as finiteness of periodic points with the same period. We prove that homeomorphisms of S 2 that are induced by topologically hyperbolic homeomorphisms of T 2 are continuum-wise-hyperbolic and topologically conjugate to linear cw-Anosov diffeomorphisms of S 2 , being in particular cw2-hyperbolic. [ABSTRACT FROM AUTHOR]