Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA numerical study of Gibbs $u$-measures for partially hyperbolic diffeomorphisms on $mathbb T^3$
70
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider a hyperbolic automorphism $Acolonmathbb T^3tomathbb T^3$ of the 3-torus whose 2-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold $A$ into two one-parameter families of Anosov diffeomorphisms --- a conservative family and a dissipative one. For diffeomorphisms in these families we numerically calculate the strong unstable manifold of the fixed point. Our calculations strongly suggest that the strong unstable manifold is dense in $mathbb T^3$. Further, we calculate push-forwards of the Lebesgue measure on a local strong unstable manifold. These numeric data indicate that the sequence of push-forwards converges to the SRB measure.
rate research
Read More
Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f, mathcal{W}^{c})$ and the growth rate of periodic center leaves $p^{c}(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $Lambda$ is center topologically mixing then $f|_{Lambda}$ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that $ h(f)leq h(f,mathcal{W}^{c})+p^{c}(f)$. Moreover, if the center foliation $mathcal{W}^{c}$ is of dimension one, we obtain an equality $h(f)= p^{c}(f)$.
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 growth in the center direction and uniform exponential growth along the unstable foliation. Our result applies to partially hyperbolic diffeomorphisms which are Lyapunov stable in the center direction. It applies to another important class of systems which do have subexponential growth in the center direction, for which we develop a technique to use exponential mixing property of the systems to get uniform distribution of unstable manifolds. A primary example is the translations on homogenous spaces which may have center of arbitrary dimension and of polynomial orbit growth.
Let $f$ be a partially hyperbolic diffeomorphism. $f$ is called has the quasi-shadowing property if for any pseudo orbit ${x_k}_{kin mathbb{Z}}$, there is a sequence ${y_k}_{kin mathbb{Z}}$ tracing it in which $y_{k+1}$ lies in the local center leaf of $f(y_k)$ for any $kin mathbb{Z}$. $f$ is called topologically quasi-stable if for any homeomorphism $g$ $C^0$-close to $f$, there exist a continuous map $pi$ and a motion $tau$ along the center foliation such that $picirc g=taucirc fcircpi$. In this paper we prove that if $f$ is dynamically coherent then it has quasi-shadowing and topological quasi-stability properties.
In these notes we prove that the $s$ or $u$-states of cocycles over partially hyperbolic maps are closed in the space of invariant measures.
In this paper, we study the existence of SRB measures for infinite dimensional dynamical systems in a Banach space. We show that if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has an SRB measure.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
