We study the time of $n$th return of orbits to some given (union of) rectangle(s) of a Markov partition of an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure (associated to a Holderian potential). As a by-product, we derive precise large deviation estimates and a central limit theorem for Birkhoff averages of Poincare cycles. We emphasize that we look at the limiting behavior in term of number of visits (the size of the visited set is kept fixed). Our approach relies on the spectral properties of a one-parameter family of induced transfer operators on unstable leaves crossing the visited set.
We introduce Smale-Vietoris diffeomorphisms that include the classical DE-mappings with Smale solenoids. We describe the correspondence between basic sets of axiom A Smale-Vietoris diffeomorphisms and basic sets of nonsingular axiom A endomorphisms. For Smale-Vietoris diffeomorphisms of 3-manifolds, we prove the uniqueness of nontrivial solenoidal basic set. We construct a bifurcation between different types of solenoidal basic sets which can be considered as a destruction (or birth) of Smale solenoid.
In this paper, we study the structural stability of three-dimensional diffeomorphisms with source-sink dynamics. Here the role of source and sink is played by one-dimensional hyperbolic repeller and attractor. It is well known that in the case when the repeller and the attractor are solenoids (not embedded in the surface), the diffeomorphism is not structurally stable. The author proves that in the case when the attractor and the repeller are canonically embedded in a surface, the diffeomorphism is also not structurally stable.
This article is devoted to the study of the historic set, which was introduced by Ruelle, of Birkhoff averges in some nonuniformly hyperbolic systems via Pesin theory. Particularly, we give a conditional variational principle for historic sets. Our results can be applied (i) to the diffeomorphisms on surfaces, (ii) to the nonuniformly hyperbolic diffeomorphisms described by Katok and several other classes of diffeomorphisms derived from Anosov systems.
Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $ϕ\colon X \to \mathbb{R}$. We provide an abstract criterion, called \emph{control at any scale with a long sparse tail} for a point $x\in X$ and the map $ϕ$, that guarantees that any weak$\ast$ limit measure $μ$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}δ(f^i(x))$ is such that $μ$-almost every point $y$ has a dense orbit in $X$ and the Birkhoff average of $ϕ$ along the orbit of $y$ is zero. As an illustration of the strength of this criterion, we prove that the diffeomorphisms with nonh…
In this paper we mainly study the dynamical complexity of Birkhoff ergodic average under the simultaneous observation of any number of continuous functions. These results can be as generalizations of [6,35] etc. to study Birkhorff ergodic average from one (or finite) observable function to any number of observable functions from the dimensional perspective. For any topological dynamical system with $g-$almost product property and uniform separation property, we show that any {\it jointly-irregular set}(i.e., the intersection of a series of $ϕ-$irregular sets over several continuous functions) …
We show generic $C^\infty$ hyperbolic flows (Axiom A and no cycles, but not transitive Anosov) commute with no $C^\infty$-diffeomorphism other than a time-t map of the flow itself. Kinematic expansivity, a substantial weakening of expansivity, implies that $C^0$ flows have quasi-discrete $C^0$-centralizer, and additional conditions broader than transitivity then give discrete $C^0$-centralizer. We also prove centralizer-rigidity: a diffeomorphism commuting with a generic hyperbolic flow is determined by its values on any open set.
We prove that for a polynomial diffeomorphism of C^2, uniform hyperbolicity on the set of saddle periodic points implies that saddle points are dense in the Julia set. In particular f satisfies Smale's Axiom A on C^2 .
