Related papers: Robust criterion for the existence of nonhyperboli…

We study expansive measures for continuous flows without fixed points on compact metric spaces. We provide a new characterization of expansive measures through dynamical balls that, in contrast to the dynamical balls considered in [\emph{J.…

We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f-invariant set, except in some singular set of neutral points. We prove that if there exists some…

We study the entropy and Lyapunov exponents of invariant measures $\mu$ for smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result is an inequality relating the discontinuities of these functions. One consequence is…

In this paper, we provide an example of a partially hyperbolic diffeomorphism with any finite number of physical measures when some Lyapunov exponent is 0 on the center.

We consider linear cocycles acting on Banach spaces which satisfy the assumptions of the multiplicative ergodic theorem. A cocycle is nonuniformly hyperbolic if all Lyapunov exponents are non-zero, which is equivalent to the existence of a…

We prove a generalization of a so called "invariance principle" for partially hyperbolic diffeomorphisms: if an invariant probability measure has all its center Lyapunov exponents equal to zero then the measure admits a center…

Let $f$ be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the stable and unstable bundles of $f$ are jointly integrable if and only if every periodic…

Let $f$ be a $C^2$ diffeomorphism on compact Riemannian manifold $M$ with partially hyperbolic splitting $$ TM=E^u\oplus E_1^c\oplus\cdots\oplus E_k^c \oplus E^s, $$ where $E^u$ is uniformly expanding, $E^s$ is uniformly contracting, and…

For a class of partially hyperbolic $C^k$, $k>1$ diffeomorphisms with circle center leaves we prove existence and finiteness of physical (or Sinai-Ruelle-Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient…

We prove for $C^\infty$ non-singular flows on three-dimensional compact manifolds with positive entropy, there are at most finitely many ergodic measures of maximal entropy. This result extends the notable work of Buzzi-Crovisier-Sarig…

We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal…

We show that a stably ergodic diffeomorphism can be $C^1$ approximated by a diffeomorphism having stably non-zero Lyapunov exponents.

We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns--Wilkinson and Avila--Santamaria--Viana. Combining this new technique with other constructions, we prove…

We show that for every compact 3-manifold $M$ there exists an open subset of $\diff ^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures which are hyperbolic while their supports are disjoint and…

For an expansive homeomorphism, we investigate the relationship among dimension, entropy, and Lyapunov exponents. Motivated by Young's formula for surface diffeomorphisms, which links dimension and measure-theoretic entropy with hyperbolic…

Bochi-Katok-Rodriguez Hertz proposed in [BKH21] a program on the flexibility of Lyapunov exponents for conservative Anosov diffeomorphisms, and obtained partial results in this direction. For conservative Anosov diffeomorphisms with strong…

We present several results suggesting that the concept of $C^1$-inverse limit stability is free of singularity theory. We describe an example of a $C^1$-inverse stable endomorphism which is robustly transitive with persistent critical set.…

We prove that any $C^{1+\alpha}$ transitive conservative partially hyperbolic diffeomorphism of a closed 3-manifold with virtually solvable fundamental group is ergodic. Consequently, in light of \cite{FP-classify}, this establishes the…

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…

We obtain some results of existence and continuity of physical measures through equilibrium states and apply these to non-uniformly expanding transformations on compact manifolds with non-flat critical sets, obtaining sufficient conditions…