Related papers: Dynamics forced by surface trellises
Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of…
A blender is a hyperbolic basic set such that the projection of its stable/unstable set onto some center subspace has a higher topological dimension than the set itself. We prove that, for any $C^s$ symplectic diffeomorphism (where…
We construct an explicit example of family of non-uniformly hyperbolic diffeomorphisms, at the boundary of the set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this…
Topological dynamics constitutes the study of asymptotic properties of orbits under flows or maps on the Hausdorff phase space. Hyperbolic dynamics is the study of differentiable flows or maps that are usually characterized by the presence…
A diffeomorphism $f$ has a $C^1$-robust homoclinic tangency if there is a $C^1$-neighbourhood $\cU$ of $f$ such that every diffeomorphism in $g\in \cU$ has a hyperbolic set $\La_g$, depending continuously on $g$, such that the stable and…
We consider partially hyperbolic diffeomorphisms $f$ with a one-dimensional central direction such that the unstable entropy exceeds the stable entropy. Our main result proves that such maps have a finite number of ergodic measures of…
Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate…
A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and…
We consider some smooth maps on a bouquet of circles. For these maps we can compute the number of fixed points, the existence of periodic points and an exact formula for topological entropy. We use Lefschetz fixed point theory and actions…
We show that heterodimensional cycles can be born at the bifurcations of a pair of homoclinic loops to a saddle-focus equilibrium for flows in dimension 4 and higher. In addition to the classical heterodimensional connection between two…
We prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms, the ones exhibiting a homoclinic tangency are C^1 dense. This represents a step towards the global understanding of dynamics of surface…
A number of techniques have been developed to perturb the dynamics of $C^1$-diffeomorphisms and to modify the properties of their periodic orbits. For instance, one can locally linearize the dynamics, change the tangent dynamics, or create…
A convenient measure of a map or flow's chaotic action is the topological entropy. In many cases, the entropy has a homological origin: it is forced by the topology of the space. For example, in simple toral maps, the topological entropy is…
Counting periodic orbits of endomorphisms on the 2-torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The…
A topological classification of many classes of dynamical systems with regular dynamics in low dimensions is often reduced to combinatorial invariants. In dimension 3 combinatorial invariants are proved to be insufficient even for simplest…
We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there…
Les travaux pr\'esent\'es dans ce m\'emoire portent sur la dynamique de diff\'eomorphismes de vari\'et\'es compactes. Pour l'\'etude des propri\'et\'es g\'en\'eriques ou pour la construction d'exemples, il est souvent utile de savoir…
It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom $A$ and the topology of the ambient manifold. In the given article, this statement is considered for the class $\mathbb G(M^2)$…
We study $C^r$ ($5 \le r \le \infty$) diffeomorphisms on closed manifolds of dimension at least three with a heteroclinic cycle between two hyperbolic periodic points. At each point, the unstable direction is one dimensional, and the stable…
Particles whose shapes couple translation to rotation display a rich array of behaviors as they sediment at low Reynolds number. We introduce a unifying perspective in which the possible dynamical regimes and bifurcations between them can…