Low emergence in one-dimensional dynamics
We show that every $C^2$ immersion of the circle $\mathbb{S}^1$ into itself has low emergence.
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arXiv subjects
Bruno Santiago
Publications and source records attributed to Bruno Santiago.
U-Gibbs measure rigidity for partially hyperbolic endomorphisms on surfaces
We prove that, for a $C^2$ partially hyperbolic endomorphism of the 2-torus which is strongly transitive, given an ergodic $u$-Gibbs measure that has positive center Lyapunov exponent and has full support, then either the map is special (has only one unstable direction per point), or the measure is the unique absolutely continuous invariant measure. We can apply this result in many settings, in particular obtaining uniqueness of $u$-Gibbs measures for every non-special perturbation of irreducible linear expanding maps of the torus with simple spectrum. This gives new open sets of partially hyperbolic systems displaying a unique $u$-Gibbs measure.
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Entropy rigidity of $u$-Gibbs measures
We obtain new entropy rigidity results for $u$-Gibbs measures by showing that whenever a $u$-Gibbs measure of a partially hyperbolic diffeomorphism admits an unstable Margulis family, the unstable Jacobian data of the system must to be constant. We apply our result to center isometries and flow type diffeomorphisms showing that if a measure of maximal entropy is also $u$-Gibbs then Jacobian periodic data along the unstable bundle are constant. In the case of smooth jointly integrable partially hyperbolic diffeomorphisms of $\mathbb{T}^3$, assuming that there exists some $u$-Gibbs measure which is also a measure of maximal unstable entropy, we obtain smooth conjugacy along the center-unstable foliation and uniqueness of $u$-Gibbs measures in this case.
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Rigidity of $\mathbf{\textit{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on $\mathbb{T}^3$
We show that within a $C^1$-neighbourhood $\mathcal{U}$ of the set of volume preserving Anosov diffeomorphisms on the three-torus $\mathbb{T}^3$ which are strongly partially hyperbolic with expanding center, any $f\in\mathcal{U}\cap\operatorname{Diff}^2(\mathbb{T}^3)$ satisfies the dichotomy: either the strong stable and unstable bundles $E^s$ and $E^u$ of $f$ are jointly integrable, or any fully supported $u$-Gibbs measure of $f$ is SRB.
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Dirac physical measures on saddle-type fixed points
In this article we study some statistical aspects of surface diffeomorphisms. We first show that for a $C^1$ generic diffeomorphism, a Dirac invariant measure whose \emph{statistical basin of attraction} is dense in some open set and has positive Lebesgue measure, must be supported in the orbit of a sink. We then construct an example of a $C^1$-diffeomorphism having a Dirac invariant measure, supported on a hyperbolic fixed point of saddle type, whose statistical basin of attraction is a nowhere dense set with positive Lebesgue measure. Our technique can be applied also to construct a $C^1$ diffeomorphism whose set of points with historic behaviour has positive measure and is nowhere dense.
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The disintegration of measures with periodic repetitive pattern
In the context of locally constant skew-products over the shift with circle fiber maps we introduce the notion of measures with periodic repetitive pattern, inspired by \cite{GorIlyKleNal:05} and which includes the non-hyperbolic measures they construct. We prove that these measures have atomic disintegration along the central fibers.
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Existence of common zeros for commuting vector fields on $3$-manifolds II. Solving global difficulties
We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such that $X$ does not vanish on the boundary of $U$ and has a non vanishing Poincaré-Hopf index in $U$, then $X$ and $Y$ have a common zero inside $U$. We prove this conjecture when $X$ and $Y$ are of class $C^3$ and every periodic orbit of $Y$ along which $X$ and $Y$ are collinear is partially hyperbolic. We also prove the conjecture, still in the $C^3$ setting, assuming that the flow $Y$ leaves invariant a transverse plane field. These results shed new light on the $C^3$ case of the conjecture.
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On the centralizer of vector fields: criteria of triviality and genericity results
In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold $M^d$ has a `small' centralizer. In the $C^1$ case, we give two criteria, one of which is $C^1$-generic, which guarantees that the centralizer of a $C^1$-generic vector field is indeed small, namely \textit{collinear}. The other criterion states that a $C^1$ \textit{separating} flow has a collinear $C^1$-centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as \textit{quasi-triviality}. In particular, the $C^1$-centralizer of a $C^1$-generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a $C^1$-generic vector field, which includes $C^1$-generic Axiom A (or sectional Axiom A) vector fields and $C^1$-generic vector fields with countably many chain recurrent classes. For sufficiently regular vector fields, we also obtain various criteria which ensure that the centralizer is \textit{trivial} (as small as it can be), and we show that in higher regularity, collinearity and triviality of the $C^d$-centralizer are equivalent properties for a generic vector field in the $C^d$ topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity $C^2$, the $C^1$-centralizer is trivial.
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Weak* and entropy approximation of nonhyperbolic measures: a geometrical approach
We study $C^1$-robustly transitive and nonhyperbolic diffeomorphisms having a partially hyperbolic splitting with one-dimensional central bundle whose strong un-/stable foliations are both minimal. {In dimension $3$, an important class of examples of such systems is given by those with a simple closed periodic curve tangent to the central bundle.} We prove that there is a $C^1$-open and dense subset of such diffeomorphisms such that every nonhyperbolic ergodic measure (i.e. with zero central exponent) can be approximated in the weak$\ast$ topology and in entropy by measures supported in basic sets with positive (negative) central Lyapunov exponent. Our method also allows to show how entropy changes across measures with central Lyapunov exponent close to zero. We also prove that any nonhyperbolic ergodic measure is in the intersection of the convex hulls of the measures with positive central exponent and with negative central exponent.
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Entropy, Lyapunov exponents, and rigidity of group actions
This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teresópolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems 16, 1996] and recent the work of the main author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [arXiv:1608.04995; arXiv:1610.09997]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures.
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Existence of common zeros for commuting vector fields on $3$-manifolds
In $64$ E. Lima proved that commuting vector fields on surfaces with non-zero Euler characteristic have common zeros. Such statement is empty in dimension $3$, since all the Euler characteristics vanish. Nevertheless, \cite{Bonatti_analiticos} proposed a local version, replacing the Euler characteristic by the Poincaré-Hopf index of a vector field $X$ in a region $U$, denoted by $\operatorname{Ind}(X,U)$; he asked: \emph{Given commuting vector fields $X,Y$ and a region $U$ where $\operatorname{Ind}(X,U)\neq 0$, does $U$ contain a common zero of $X$ and $Y$?} \cite{Bonatti_analiticos} gave a positive answer in the case where $X$ and $Y$ are real analytic. In this paper, we prove the existence of common zeros for commuting $C^1$ vector fields $X$, $Y$ on a $3$-manifold, in any region $U$ such that $\operatorname{Ind}(X,U)\neq 0$, assuming that the set of collinearity of $X$ and $Y$ is contained in a smooth surface. This is a strong indication that the results in \cite{Bonatti_analiticos} should hold for $C^1$-vector fields.
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On weakly hyperbolic iterated functions systems
We study weakly hyperbolic iterated function systems on compact spaces, as defined by Edalat, but in the more general setting of a compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint, and prove that the measure theoretical attractor is ergodic. We also define weakly hyperbolic iterated functions systems for complete spaces and compact parameter space, and prove that this definition extends the one given by Edalat. Furthermore, we study the question of existence of the attractors in this setting. Finally, we prove a version of the results of Barnsley and Vince about drawing the attractor (also called the chaos game), for the case of compact parameter space.
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Dirac physical measures for generic diffeomorphisms
We prove that, for a $C^1$ generic diffeomorphism, the only Dirac physical measures with dense statistical basin are those supported on sinks.
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Mixing-like properties for some generic and robust dynamics
We show that the set of Bernoulli measures of an isolated topologically mixing homoclinic class of a generic diffeomorphism is a dense subset of the set of invariant measures supported on the class. For this, we introduce the large periods property and show that this is a robust property for these classes. We also show that the whole manifold is a homoclinic class for an open and dense subset of the set of robustly transitive diffeomorphisms far away from homoclinic tangencies. In particular, using results from Abdenur and Crovisier, we obtain that every diffeomorphism in this subset is robustly topologically mixing.
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On Araujo's Theorem for flows
Araujo proved in his thesis \cite{A} that a $C^1$ generic surface diffeomorphism has either infinitely many sinks (i.e. attracting periodic orbits) or finitely many hyperbolic attractors with full Lebesgue measure basin. The goal of this paper is to extend this result to $C^1$ vector fields on compact connected boundaryless manifolds $M$ of dimension 3 (three-dimensional flows for short). More precisely, we shall prove that a $C^1$ generic three-dimensional flow without singularities has either infinitely many sinks or finitely many hyperbolic attractors with full Lebesgue measure basin.
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