Related papers: Formula of Entropy along Unstable Foliations for $…

For any C1 diffeomorphism with dominated splitting we consider a nonempty set of invariant measures which describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the…

Let $f$ be a $C^2$ diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure $\mu$. We relate those entropies to covering numbers in order to give a new upper bound on…

The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper…

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 study the unstable entropy of $C^1$ diffeomorphisms with dominated splittings. Our main result shows that when the zero Lyapunov exponent has multiplicity one, the center direction contributes no entropy, and the unstable entropy…

A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the…

In this paper, we consider certain partially hyperbolic diffeomorphisms with center of arbitrary dimension and obtain continuity properties of the topological entropy under $C^1$ perturbations. The systems considered have subexponential…

For $C^1$ diffeomorphisms, we prove that the Pesin's entropy formula holds for some invariant measure supported on any topological attractor that admits a dominated splitting without mixed behavior. We also prove Shub's entropy conjecture…

We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states…

Let $M$ be a compact manifold and $f:\,M\to M$ be a $C^1$ diffeomorphism on $M$. If $\mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $\mu$ $a.\,\,e.\,\,x\in M,$ there is a…

In this paper, we introduce the unstable topological pressure for C^1-smooth partially hyperbolic diffeomorphisms with sub-additive potentials. Moreover, without any additional assumption, we have established the expected variational…

For a $C^{r}$ $(r>1)$ diffeomorphism on a compact manifold that admits a dominated splitting, this paper establishes the upper semi-continuity of the entropy map. More precisely, this paper establishes the upper semi-continuity of the…

For $C^1$ diffeomorphisms with continuous invariant splitting without domination, we prove the existence of (un)stable manifold under the hyperbolicity of invariant measures.

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional,…

Let $\mathcal{F}$ be a $C^2$ random partially hyperbolic dynamical system. For the unstable foliation, the corresponding unstable metric entropy, unstable topological entropy and unstable pressure via the dynamics of $\mathcal{F}$ on the…

We develop the nonuniformly hyperbolic theory for $C^1$ diffeomorphisms admitting continuous invariant splitting without domination. This framework includes stable manifold theorems, shadowing and closing lemmas, the existence of horseshoes…

Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\rm top}(f)\geq\frac{\lambda^{+}(f)}{r}$ where $\lambda^{+}(f):=\lim_{n\to+\infty}\frac{1}{n}\max_{x\in M}\log \left\|Df^{n}_{x}\right\|$. We establish an…

In this paper we consider $C^{1}$ diffeomorphisms on compact Riemannian manifolds of any dimension that admit a dominated splitting $E^{cs} \oplus E^{cu}.$ We prove that if the Lyapunov exponents along $E^{cu}$ are positive for Lebesgue…

In the study of non-equilibrium statistical mechanics, Ruelle derived explicit formulae for entropy production of smooth dynamical systems. The vanishing or strict positivity of entropy production is determined by the {\it entropy formula…

Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting on a compact Riemanian manifold $M$ without boundary. We state and prove several sufficient conditions for the topological entropy of $f$ to be positive. The…