Title: Surface dynamics and 3-dimensional topology and geometry
Abstract.
A basic construction in topology is the “mapping torus”: given a homeomorphism f of a space S, a new space M(f,S) is constructed by taking the product of S with the unit interval and gluing the two ends Sx{0} and Sx{1} by the map f. When f is a homeomorphism of a surface S, M(f,S) is a 3-manifold and Thurston proved theorems which state that if for any simple closed curve c, its f-iterates are never isotopic to c, then M(f,S) admits a hyperbolic metric (i.e., a complete metric with constant sectional curvature equal to -1) of finite volume. When f is a homeomorphism of the disk D with many periodic orbits — such as the Smale horseshoe — we can consider the mapping tori M(f,D-P), where P is an f-periodic orbit. Topologically, M(f,D-P) is simply a solid torus minus the knot obtained following the orbit P under the suspension flow. We will discuss a family of hyperbolic manifolds M(f,D-P), where f is the horseshoe and P are horseshoe periodic orbits, and show how understanding the dynamics of the horseshoe helps clarify a mysterious mismatch between convergence of disk homeomorphisms in the uniform topology and convergence of geometric 3-manifolds in the Gromov-Hausdorff topology.
Title: Weakly contracting on average RDSs
Abstract.
In this talk, I will present some fundamental properties of random dynamical systems (RDSs) that are weakly contracting on average. This condition will be shown to imply several probabilistic consequences, including the uniqueness of the stationary measure, decay of correlations for Lipschitz observables, concentration inequalities, and an almost sure central limit theorem. I will also illustrate examples to which these results apply, including RDSs on the circle, systems induced by the projective action of linear cocycles, and classically contractive systems of Hutchinson type.
Title: Polya urns with dynamics
Abstract.
I'll describe some simple probabilistic Polya urns models which are governed, via the ``stochastic approximation method'', by deterministic dynamical systems with need not be simple, but which may nevertheless be tractable to some degree. This leads to interesting consequences for the underlying urn model. (Work in progress with Yuri Lima.)
Title: Proximal actions on the circle
Abstract.
In this talk, we discuss the dynamical behavior of random actions determined by proximal semigroups of homeomorphisms of the circle. Our main goal is to examine the similarities between this class and the class generated by transformations induced by linear actions on the circle, presenting a version of Oseledets theorem, which determines the asymptotic behavior of the orbits of the proximal system. Under additional regularity assumption on the semigroup of transformations, we provide a characterization of the extremal Lyapunov exponents - contraction/expansion rate of the orbits of the action - in terms of the stationary measures of the system. This is a joint work with G. Salcedo.
Title: SPR property for a class of skew product maps
Abstract.
For skew products with 2-dimensional fibers, we establish a relation between the continuity of the fiber entropy and the continuity of the fiber Lyapunov exponents. This result extends the theorem for surfaces proved by Buzzi–Crovisier–Sarig. As a consequence, we are able to obtain classes of skew products that satisfy the strong positive recurrence property (SPR); in particular, these systems have a finite number of measures of maximal entropy, all exponentially mixing with good statistical properties. This is a joint work with Mauricio Poletti (UFC) and Filiphe Veiga (UFMG).
Title: Full flexibility of entropies among ergodic measures
Abstract.
For partially hyperbolic diffeomorphisms with one-dimensional center direction, we prove the following flexibility-type result: for each center Lyapunov exponent in the interior of the spectrum (including value 0), every possible entropy value can be achieved by some ergodic measure. Our hypotheses involve minimal foliations and blender-horseshoes. The list of examples our results apply includes fibered-by-circles, flow-type, some Derived from Anosov diffeomorphisms, and some anomalous (non-dynamically coherent) diffeomorphisms.
This is joint work with L.J. Díaz, M Rams, and J Zhang.
Title: Robust heterodimensional cycles of co-index two via split blender machines
Abstract.
For simplicity, we restrict our attention to diffeomorphisms in manifolds of dimension 4. We consider heterodimensional cycles of co-index two, that is, cycles associated to saddles whose indices (i.e, dimension of the stable bundle) have dimensions 1 and 3. We introduce a class of such cycles that we call non-escaping. We show that such cycles simultaneously generate robust heterodimensional cycles of co-indices one (simultaneously associated to saddles of indices 1 and 2 and of indices 2 and 3) and 3. We present a simple setting in which these cycles arise, discuss further scenarios (including cocycles of three dimensional matrices), and outline some of the key ingredients involved (different types of blender-like sets).
This is joint work with Barrientos, Ki, Lizana, and Pérez.
Title: An ergodic spectral decomposition theorem for Singular Star Flows
Abstract.
For Axiom A diffeomorphisms and flows, the celebrated Spectral Decomposition Theorem of Smale states that the non-wandering set decomposes into a finite disjoint union of isolated compact invariant sets, each of which is the homoclinic class of a periodic orbit. For singular star flows which can be seen as “Axiom A flows with singularities”, this result remains open and is known as the Spectral Decomposition Conjecture. In this talk, we will provide a positive answer to an ergodic version of this conjecture: C^1 open and densely, singular star flows with positive topological entropy can only have finitely many ergodic measures of maximal entropy. More generally, we obtain the finiteness of equilibrium states for any Holder continuous potential functions satisfying a mild, yet optimal, condition. We also show that C^1 open and densely, star flows are almost expansive, and the topological pressure of continuous functions varies continuously with respect to the vector field in C^1 topology. This corresponds to a joint work with Fan Yang and Jiagang Yang.
Title: Computer-assisted proof of robust transitivity.
Abstract.
A smooth dynamical system is transitive if it has a dense orbit, loosely meaning that it has some chaos in a topological sense. If this property holds for all diffeomorphisms in a C¹-neighborhood, we say that systems in this neighborhood are robustly transitive. By Bonatti, Diaz and Pujals (2003), robustly transitive diffeomorphisms are volume hyperbolic, and thus they have positive topological entropy, being chaotic in a strict sense and in a robust way. Robust properties are key in classifying smooth dynamical systems, and they are also desirable to model applications.
We develop a computer-assisted strategy to prove robust transitivity in dimension 3 that includes the proof of the existence of a blender. We present a family of D.A. (derived-from-Anosov) systems on the 3-torus for which this strategy applies. This work in preparation is a collaboration with Andy Hammerlindl and Warwick Tucker.
Title: Zip Shift as Extended Symbolic Dynamics: Some of its Applications and Future Perspectives
Abstract.
We will introduce zip shift maps as local homeomorphisms that extend the dynamics of the two-sided shift. Using examples of finite-to-one horseshoe maps, we'll show how these systems can be fully codified, up to topological conjugacy, using symbolic dynamics. We will also explore some applications in ergodic theory, topological dynamics, and cellular automata, and discuss potential future directions for their development and use in these fields.