Compact families and typical entropy invariants of measure-preserving actions
Valery V. Ryzhikov
备注：
in Russian. The article is intended for Volume 82:1 of Proceedings of the Moscow Mathematical Society, dedicated to the 80th anniversaries of V.I. Oseledets and A.M. Stepin and the memory of A.M. Stepin, who left us in November 2020
For a compact set of actions, an invariant of Kushnirenko's entropy type is chosen in such a way that on this set it is equal to zero, but will be infinity for typical actions. As a consequence, we show that typical measure-preserving transformations are not isomorphic to geometric shape exchange transformations. This problem arose in connection with the result of Chaika and Davis about the atypical nature of IETs.
We study dynamics of area-preserving maps in a neighbourhood of an elliptic fixed point. We describe simplified normal forms for a fixed point of co-dimension 3. We also construct normal forms for a generic three-parameter family which contains such degeneracy and use the normal form theory to describe generic bifurcations of periodic orbits in these families.
We derive simplified normal forms for an area-preserving map in a neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points appear in generic two-parameter families of area-preserving maps. We also derive a simplified normal form for a generic two-parametric unfolding. The normal forms are used to analyse bifurcations of $n$-periodic orbits. In particular, for $n\ge6$ we find regions of parameters where the normal form has "meandering'' invariant curves.
We study parametrised families of piecewise expanding interval mappings $T_a \colon [0,1] \to [0,1]$ with absolutely continuous invariant measures $μ_a$ and give sufficient conditions for a point $X(a)$ to be typical with respect to $(T_a, μ_a)$ for almost all parameters $a$. This is similar to a result by D. Schnellmann, but with different assumptions.
Using the novel notion of parablender, P. Berger proved that the existence of finitely many attractors is not Kolmogorov typical in parametric families of diffeomorphisms. Here, motivated by the concept of Newhouse domains we define Berger domains for families of diffeomorphisms. As an application, we show that the coexistence of infinitely many attracting invariant smooth circles is Kolmogorov typical in certain non-sectionally dissipative Berger domains of parametric families in dimension three or greater.
We will introduce the notion of Emergence for a dynamical system, and we will conjecture the local typicality of super-polynomial ones. Then, as part of this program, we will provide sufficient conditions for an open set of Cd-families of Cr-dynamics to contain a Baire generic set formed by families displaying infinitely many sinks at every parameter, for all 1 \le d \le r with d finite (and r possibly infinite) for two different topologies on families. In particular the case d=r=1 is new.
Let us denote $λ$ the Lebesgue measure on $[0,1]$, put$$ C(λ)=\{f\in C([0,1]);\ \forall~A\subset [0,1], A~\text{Borel}:\ λ(A)=λ(f^{-1}(A))\}.$$ We endow the set $C(λ)$ by the uniform metric $ρ$ and investigate dynamical properties of typical maps in the complete metric space $(C(λ),ρ)$.
We consider the rotation number $ρ(t)$ of a diffeomorphism $f_t=R_t\circ f$, where $R_t$ is the rotation by $t$ and $f$ is an orientation preserving $C^\infty$ diffeomorphism of the circle $S^1$. We shall show that if $ρ(t)$ is irrational $$\limsup_{t'\to t}(ρ(t')-ρ(t))/(t'-t)\geq 1.$$
Let $I\subset\mathbb{R}$ be an interval and $T_a:[0,1]\to[0,1]$, $a\in I$, a one-parameter family of piecewise expanding maps such that for each $a\in I$ the map $T_a$ admits a unique absolutely continuous invariant probability measure $μ_a$. We establish sufficient conditions on such a one-parameter family such that a given point $x\in[0,1]$ is typical for $μ_a$ for a full Lebesgue measure set of parameters $a$, i.e. $$ \frac{1}{n}\sum_{i=0}^{n-1}δ_{T_a^i(x)} \overset{\text{weak-}*}{\longrightarrow}μ_a,\qquad\text{as} n\to\infty, $$ for Lebesgue almost every $a\in I$. In particular, we consid…
This paper gives a complete classification of the possible ergodic decompositions for certain open families of volume-preserving partially hyperbolic diffeomorphisms. These families include systems with compact center leaves and perturbations of Anosov flows under conditions on the dimensions of the invariant subbundles. The paper further shows that the non-open accessibility classes form a $C^1$ lamination and gives results about the accessibility classes of non-volume-preserving systems. Note: this document has been modified slightly from earlier preprints. The numbering of sections was chan…
We obtain estimates relating the phase space and the parameter space of analytic families of unimodal maps. Using those estimates, we show that typical analytic unimodal maps admit a quasiquadratic renormalization. This reduces the study of the statistical properties of typical unimodal maps to the quasiquadratic case which had been studied in \cite {AM2}. The estimates proved here correspond exactly to the Phase-Parameter relation proved in \cite {AM} in the quadratic case, and allows one to obtain sharp estimates on the dynamics of typical unimodal maps which were available only in the quadr…
For a nonlinear Anosov diffeomorphism of the 2-torus, we present examples of measures so that the group of $μ$-preserving diffeomorphisms is, up to zero-entropy transformations, cyclic. For families of equilibrium states $μ$, we strengthen this to show that the group of $μ$-preserving diffeomorphism is virtually cyclic.
This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three-parameter families of a class of non$-$smooth vector fields are studied and the tridimensional bifurcation diagrams are exhibited. Our main results describe the unfolding of the so called fold-cusp singularity by means of the variation of 3 parameters.
We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of any genus are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations.
Topological structure of minimal sets is studied for a dynamical system $(E,F)$ given by a fibre-preserving, in general non-invertible, continuous selfmap $F$ of a graph bundle $E$. These systems include, as a very particular case, quasiperiodically forced circle homeomorphisms. Let $M$ be a minimal set of $F$ with full projection onto the base space $B$ of the bundle. We show that $M$ is nowhere dense or has nonempty interior depending on whether the set of so called endpoints of $M$ is dense in $M$ or is empty. If $M$ is nowhere dense, we prove that either a typical fibre of $M$ is a Cantor …
We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there exists an open subset of the set of smooth area-preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points.
We show that for a large space of volume preserving partially hyperbolic diffeomorphisms of the 3-torus with non-compact central leaves the central foliation generically is non-absolutely continuous.
We study the dynamics of strongly dissipative Hénon-like maps, around the first bifurcation parameter $a^*$ at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove that $a^*$ is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under positive iteration. A key ingredient is that $a^*$ corresponds to "non-recurrence of every critical point", reminiscent of Misiurewicz parameters in one-dimensional dynamics. Adapting on the one hand Benedicks & Carleson's parameter exclusi…
We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in $[0,1]$), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map. We also obtain several applications. Near the time on…
For every countable abelian group $G$ we find the set of all its subgroups $H$ ($H\leq G$) such that a typical measure-preserving $H$-action on a standard atomless probability space $(X,\mathcal{F}, μ)$ can be extended to a free measure-preserving $G$-action on $(X,\mathcal{F}, μ)$. The description of all such pairs $H\leq G$ was made in purely group terms, in the language of the dual $\hat{G}$, and $G$-actions with discrete spectrum. As an application, we answer a question when a typical $H$-action can be extended to a $G$-action with some dynamic property, or to a $G$-action at all. In parti…
