GEOMETRICAL AND MEASURE-THEORETIC STRUCTURES OF MAPS WITH A MOSTLY EXPANDING CENTER

In this article we study physical measures for $\operatorname {C}^{1+\alpha }$ partially hyperbolic diffeomorphisms with a mostly expanding center. We show that every diffeomorphism with a mostly expanding center direction exhibits a geometrical-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how physical measures bifurcate as the diffeomorphism changes under $C^1$ topology. Moreover, for each diffeomorphism with a mostly expanding center, there exists a $C^1$ neighbourhood, such that diffeomorphism among a $C^1$ residual subset of this neighbourhood admits finitely many physical measures, whose basins have full volume. We also show that the physical measures for diffeomorphisms with a mostly expanding center satisfy exponential decay of correlation for any Hölder observes. In particular, we prove that every $C^2$ , partially hyperbolic, accessible diffeomorphism with 1-dimensional center and nonvanishing center exponent has exponential decay of correlations for Hölder functions.

On super-exponential divergence of periodic points for partially hyperbolic systems

<p style='text-indent:20px;'>We say that a diffeomorphism <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> is super-exponentially divergent if for every <inline-formula><tex-math id="M2">\begin{document}$ b&gt;1 $\end{document}</tex-math></inline-formula> the lower limit of <inline-formula><tex-math id="M3">\begin{document}$ \#\mbox{Per}_n(f)/b^n $\end{document}</tex-math></inline-formula> diverges to infinity, where <inline-formula><tex-math id="M4">\begin{document}$ \mbox{Per}_n(f) $\end{document}</tex-math></inline-formula> is the set of all periodic points of <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> with period <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>. This property is stronger than the usual super-exponential growth of the number of periodic points. We show that for any <inline-formula><tex-math id="M7">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional smooth closed manifold <inline-formula><tex-math id="M8">\begin{document}$ M $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M9">\begin{document}$ n\ge 3 $\end{document}</tex-math></inline-formula>, there exists a non-empty open subset <inline-formula><tex-math id="M10">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M11">\begin{document}$ \mbox{Diff}^1(M) $\end{document}</tex-math></inline-formula> such that diffeomorphisms with super-exponentially divergent property form a dense subset of <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{O} $\end{document}</tex-math></inline-formula> in the <inline-formula><tex-math id="M13">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-topology. A relevant result about the growth rate of the lower limit of the number of periodic points for diffeomorphisms in a <inline-formula><tex-math id="M14">\begin{document}$ C^r $\end{document}</tex-math></inline-formula>-residual subset of <inline-formula><tex-math id="M15">\begin{document}$ \mbox{Diff}^r(M)\ (1\le r\le \infty) $\end{document}</tex-math></inline-formula> is also shown.</p>

On the rotation sets of generic homeomorphisms on the torus

We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$ , $d\ge 2$ . In the conservative setting, we prove that there exists a Baire residual subset of the set $\text {Homeo}_{0, \lambda }(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$ , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$ -generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

Equi-Cliquishness and The Hahn Property

We study the joint continuity of mappings of two variables. In particular, we show that for a Baire space X, a second countable space Y and a metric space Z, a map f : X × Y → Z has the Hahn property (i.e., there is a residual subset A of X such that A × Y ⊆ C(f)) if and only if f is locally equi-cliquish with respect to y and {x ∈ X: fx is continuous} is a residual subset of X.

A note on a Residual Subset of Lipschitz Functions on Metric Spaces

Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of real-valued Lipschitz functions with non-zero pointwise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous to a result proved for real-valued Lipschitz maps defined on ℝ2 by Alberti et al.

On an extension of Mycielski’s theorem and invariant scrambled sets

Let $(X,f)$ be a dynamical system, where $X$ is a perfect Polish space and $f:X\rightarrow X$ is a continuous map. In this paper we study the invariant dependent sets of a given relation string ${\it\alpha}=\{R_{1},R_{2},\ldots \}$ on $X$. To do so, we need the relation string ${\it\alpha}$ to satisfy some dynamical properties, and we say that ${\it\alpha}$ is $f$-invariant (see Definition 3.1). We show that if ${\it\alpha}=\{R_{1},R_{2},\ldots \}$ is an $f$-invariant relation string and $R_{n}\subset X^{n}$ is a residual subset for each $n\geq 1$, then there exists a dense Mycielski subset $B\subset X$ such that the invariant subset $\bigcup _{i=0}^{\infty }f^{i}B$ is a dependent set of $R_{n}$ for each $n\geq 1$ (see Theorems 5.4 and 5.5). This result extends Mycielski’s theorem (see Theorem A) when $X$ is a perfect Polish space (see Corollary 5.6). Furthermore, in two applications of the main results, we simplify the proofs of known results on chaotic sets in an elegant way.

Separate and Joint Properties of some Analogues of Pointwise Discontinuity

We study separate and joint properties of pointwise discontinuity, simple continuity and mild continuity of functions of two variables. In particular, it is shown that for a Baire space X, aBaire space Y which has a countable pseudobase and for a metric space Z, a function ƒ : X×Y → Z is pointwise discontinuous if and only if f satisfies (α, β)-condition and condition (C), and M = {x ∊ X : C(ƒx) = Y } is a residual subset of X. In addition, a characterization of simple continuity for mappings of one and two variables is given

Contributions to the geometric and ergodic theory of conservative flows

We prove the following dichotomy for vector fields in a $C^1$-residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and $C^1$-stably ergodic flow can be $C^1$-approximated by another volume-preserving flow which is non-uniformly hyperbolic.

Convexities and Existence of the Farthest Point

Five counterexamples are given, which show relations among the new convexities and some important convexities in Banach space. Under the assumption that Banach space is nearly very convex, we give a sufficient condition that bounded, weakly closed subset of has the farthest points. We also give a sufficient condition that the farthest point map is single valued in a residual subset of when is very convex.

Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

We prove that there is a residual subset 𝒮 in Diff1(M) such that, for everyf∈𝒮, any homoclinic class offcontaining saddles of different indices (dimension of the unstable bundle) contains also an uncountable support of an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure off.