Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding

The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C¹ submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the Cr (r?1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C¹ but not neatly embedded, the other in which S is not of class C¹.     

Structural stability of vector fields with shadowing

Let X be a C¹ vector field without singularities. In this paper, we show that X is in the C¹ interior of the set of vector fields with the shadowing property if and only if X satisfies both Axiom A and the strong transversality condition; that is, X is structurally stable.     

Embedding solenoids in foliations

In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H¹(L;R) not equal to 0 and if the foliation is a product foliation in some saturated open neighborhood U of L, then there exists a foliation F^′ on M which is C¹-close to F, and F^′ has an uncountable set of solenoidal minimal sets contained in U that are pairwise non-homeomorphic. If H¹(L;R) is 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighborhood. Thus, our result can be thought of as an instability result complementing the stability results of Reeb, Thurston and Langevin and Rosenberg.     

On critical orbits and sectional hyperbolicity of the nonwandering set for flows

In this note, we introduce the notion of nonuniformly sectional hyperbolic set and use it to prove that any C¹-open set which contains a residual subset of vector fields with nonuniformly sectional hyperbolic critical set also contains a residual subset of vector fields with sectional hyperbolic nonwandering set. This not only extends Theorem A of Castro [11], but using suspensions we recover it.     

Structurally stable perturbations of polynomials in the Riemann sphere

The perturbations of complex polynomials of one variable are considered in a wider class than the holomorphic one. It is proved that under certain conditions on a polynomial p   of the plane, the Cr$C^r$ conjugacy class of a map f   in a C¹$C^1$ neighborhood of p depends only on the geometric structure of the critical set of f. This provides the first class of examples of structurally stable maps with critical points and nontrivial nonwandering set in dimension greater than one.     

An example on movable approximations of a minimal set in a continuous flow

In the present paper the study of flows on n-manifolds in particular in dimension three, e.g., R³, is motivated by the following question. Let A be a compact invariant set in a flow on X. Does every neighbourhood of A contain a movable invariant set M containing A? It is known that a stable solenoid in a flow on a 3-manifold has approximating periodic orbits in each of its neighbourhoods. The solenoid with the approximating orbits form a movable set, although the solenoid is not movable. Not many such examples are known. The main part of the paper consists of constructing an example of a set in R³ that is not stable, is not a solenoid, and is approximated by Denjoy-like invariant sets instead of periodic orbits. As in the case of a solenoid, the constructed set is an inverse limit of its approximating sets. This gives a partial answer to the above question.     

A λ-lemma for foliations

We show that a C⁰ codimension one foliation with C¹ leaves F of a closed manifold is minimal if there are a foliation G transverse to F, and a diffeomorphism f preserving both foliations, such that every leaf of F intersects every leaf of G and f expands G. We use this result to study of Anosov actions on closed manifolds.     

Characterizing omega-limit sets which are closed orbits

Let X be a vector field in a compact n-manifold M, n?2. Given Σ⊂M we say that q∈M satisfies (P)_Σ if the closure of the positive orbit of X through q does not intersect Σ, but, however, there is an open interval I with q as a boundary point such that every positive orbit through I intersects Σ. Among those q having saddle-type hyperbolic omega-limit set ω(q) the ones with ω(q) being a closed orbit satisfy (P)_Σ for some closed subset Σ. The converse is true for n=2 but not for n?4. Here we prove the converse for n=3. Moreover, we prove for n=3 that if ω(q) is a singular-hyperbolic set [C. Morales, M. Pacifico, E. Pujals, On C¹ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris Sér. I 26 (1998) 81-86], [C. Morales, M. Pacifico, E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2) 160 (2) (2004) 375-432], then ω(q) is a closed orbit if and only if q satisfies (P)_Σ for some Σ closed. This result improves [S. Bautista, Sobre conjuntos hiperbólicos-singulares (On singular-hyperbolic sets), thesis Uiversidade Federal do Rio de Janeiro, 2005 (in Portuguese)] and [C. Morales, M. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001) 359-378].     

Periodic orbits in complex Abel equations

This paper is devoted to prove two unexpected properties of the Abel equation dz/dt=z³+B(t)z²+C(t)z, where B and C are smooth, 2π-periodic complex valuated functions, t∈R and z∈C. The first one is that there is no upper bound for its number of isolated 2π-periodic solutions. In contrast, recall that if the functions B and C are real valuated then the number of complex 2π-periodic solutions is at most three. The second property is that there are examples of the above equation with B and C being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of B and C. This result is also in contrast with the characterization of the center variety for the examples of Abel equations dz/dt=A(t)z³+B(t)z² studied in the literature, where the center variety is located in a finite number of connected components.     

Chaos and the shadowing property

We present, as a simpler alternative for the results of [P. Ko?cielniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl. 310 (2005) 188-196; P. Ko?cielniak, M. Mazur, On C⁰ genericity of various shadowing properties, Discrete Contin. Dynam. Syst. 12 (2005) 523-530], an elementary proof of C⁰ genericity of the periodic shadowing property. We also characterize chaotic behavior (in the sense of being semiconjugated to a shift map) of shadowing systems.     

Orthonormal expansions of invariant densities for expanding maps

We give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding C^ω Markov maps. This is a classical problem, with one of the standard approaches being Ulam's method [Problems in Modern Mathematics, Interscience, New York, 1960] of phase space discretisation.Our method hinges instead on the expansion of the density function with respect to an L² orthonormal basis, and the computation of the expansion coefficients in terms of the periodic orbits of the expanding map. The efficiency of the method, and its extension to C^k expanding maps, are also discussed.     

Invariant fibrations of geodesic flows

Let (Σ,g) be a compact C² finslerian 3-manifold. If the geodesic flow of g is completely integrable, and the singular set is a tamely-embedded polyhedron, then π₁(Σ) is almost polycyclic. On the other hand, if Σ is a compact, irreducible 3-manifold and π₁(Σ) is infinite polycyclic while π₂(Σ) is trivial, then Σ admits an analytic riemannian metric whose geodesic flow is completely integrable and singular set is a real-analytic variety. Additional results in higher dimensions are proven.     

The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C^∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C^∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.     

Embedding smooth and formal diffeomorphisms through the Jordan-Chevalley decomposition

In [Xiang Zhang, The embedding flows of C^∞ hyperbolic diffeomorphisms, J. Differential Equations 250 (5) (2011) 2283-2298] Zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field. We present a new and more conceptual proof of such result using the Jordan-Chevalley decomposition in algebraic groups and the properties of the exponential operator.We characterize the hyperbolic smooth (resp. formal) diffeomorphisms that are embedded in a smooth (resp. formal) flow. We introduce a criterion showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable. This solves a conjecture of Zhang. The criterion is optimal, we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions.     

Local structure of ideal shapes of knots

Relatively extremal knots are the relative minima of the ropelength functional in the C¹ topology. They are the relative maxima of the thickness (normal injectivity radius) functional on the set of curves of fixed length, and they include the ideal knots. We prove that a C1,1 relatively extremal knot in Rn either has constant maximal (generalized) curvature, or its thickness is equal to half of the double critical self distance. This local result also applies to the links. Our main approach is to show that the shortest curves with bounded curvature and C¹ boundary conditions in Rn contain CLC (circle-line-circle) curves, if they do not have constant maximal curvature.     

Partial hyperbolicity far from homoclinic bifurcations

We prove that any diffeomorphism of a compact manifold can be C¹-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in Crovisier (in press) [10].     

Nonexistence of Bonatti-Langevin examples of Anosov flows on closed four manifolds

Bonatti and Langevin constructed an Anosov flow on a closed 3-manifold with a transverse torus intersecting all orbits except one [C. Bonatti, R. Langevin, Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension, Ergodic Theory Dynam. Systems 14 (4) (1994), 633-643]. We shall prove that these flows cannot be constructed on closed 4-manifolds. More precisely, there are no Anosov flows on closed 4-manifolds with a closed, incompressible, transverse submanifold intersecting all orbits except finitely many closed ones. The proof relies on the analysis of the trace of the weak invariant foliations of the flow on the transverse submanifold.