Invariant Borel Probability Measure Research Articles

On some generalizations of skew-shifts on

In this paper we investigate maps of the two-torus $\mathbb{T}^{2}$ of the form $T(x,y)=(x+\unicode[STIX]{x1D714},g(x)+f(y))$ for Diophantine $\unicode[STIX]{x1D714}\in \mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\rightarrow \mathbb{T}$, where each $g$ is strictly monotone and of degree 2 and each $f$ is an orientation-preserving circle homeomorphism. For our class of $f$ and $g$, we show that $T$ is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two $T$-invariant graphs. One of the graphs is a strange non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in $\mathbb{T}^{2}$. Only a low-regularity assumption (Lipschitz) is needed on the maps $f$ and $g$, and the results are robust with respect to Lipschitz-small perturbations of $f$ and $g$.

Invariant random subgroups of linear groups

Let Γ < GL n (F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty topology. An invariant random subgroup (IRS) of Γ is a conjugation invariant Borel probability measure on Sub(Γ). An IRS is called non-trivial if it does not have an atom in the trivial group, i.e. if it is non-trivial almost surely. We denote by IRS0(Γ) the collection of all non-trivial IRS on Γ. Theorem 0.1: With the above notation, there exists a free subgroup F < Γ and a non-discrete group topology on Γ such that for every μ ∈ IRS0(Γ) the following properties hold: Φ: (Sub(Γ), μ) → (Sub(F),Φ*μ) Δ → Δ ∩ F is an F-invariant isomorphism of probability spaces. A more technical version of this theorem is valid for general countable linear groups. We say that an action of Γ on a probability space, by measure preserving transformations, is almost surely non-free (ASNF) if almost all point stabilizers are non-trivial. Corollary 0.2: Let Γ be as in the Theorem above. Then the product of finitely many ASNF Γ-spaces, with the diagonal Γ action, is ASNF. Corollary 0.3: Let Γ then Δ = , μ almost surely.

Measurable sensitivity via Furstenberg families

Let $(X, T)$ be a topological dynamical system, and $\mu$ be a $T$-invariant Borel probability measure on $X$. Let $\mathcal{F}$ be a family of subsets of $\mathbb{Z}_+$. We introduce notions of $\mathcal{F}$-sensitivity for $\mu$ and block $\mathcal{F}$-sensitivity for $\mu$. Let $\mathcal{F}_t$ (resp. $\mathcal{F}_{ip}$) be the families consisting of thick sets (resp. IP-sets). The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either $\mathcal{F}_{t}$-sensitive for $\mu$ or an almost one-to-one extension of its maximal equicontinous factor. (2) a minimal system is either block $\mathcal{F}_{t}$-sensitive for $\mu$ or a proximal extension of its maximal equicontinous factor. (3) a minimal system is either block $\mathcal{F}_{ip}$-sensitive for $\mu$ or an almost one-to-one extension of its $\infty$-step nilfactor. We also introduce the notion of topological $l$-sensitivity, and construct a minimal system which is $l$-sensitive but not $(l+1)$-sensitive for $l\in\mathbb{N}$.

Dimension, recurrence via entropy and Lyapunov exponents for map with singularities

Let $f:M\rightarrow M$ be a $C^{1}$ self-map of a smooth Riemannian manifold $M$ and $\unicode[STIX]{x1D707}$ be an $f$-invariant ergodic Borel probability measure with a compact support $\unicode[STIX]{x1D6EC}$. We prove that if $f$ is Hölder mild on the intersection of the singularity set and $\unicode[STIX]{x1D6EC}$, then the pointwise dimension of $\unicode[STIX]{x1D707}$ can be controlled by the Lyapunov exponents of $\unicode[STIX]{x1D707}$ with respect to $f$ and the entropy of $f$. Moreover, we establish the distinction of the Hausdorff dimension of the critical points sets of maps between the $C^{1,\unicode[STIX]{x1D6FC}}$ continuity and Hölder mildness conditions. Consequently, this shows that the Hölder mildness condition is much weaker than the $C^{1,\unicode[STIX]{x1D6FC}}$ continuity condition. As applications of our result, if we study the recurrence rate of $f$ instead of the pointwise dimension of $\unicode[STIX]{x1D707}$, then we deduce that the analogous relation exists between recurrence rate, entropy and Lyapunov exponents.

Invariant measures for piecewise continuous maps

We say that f:[0,1]→[0,1] is a piecewise continuous interval map if there exists a partition 0=x0<x1<⋯<xd<xd+1=1 of [0,1] such that f|(xi−1,xi) is continuous and the lateral limits w0+=limx→0+⁡f(x), wd+1−=limx→1−⁡f(x), wi−=limx→xi−⁡f(x) and wi+=limx→xi+⁡f(x) exist for each i. We prove that every piecewise continuous interval map without connections admits an invariant Borel probability measure. We also prove that every injective piecewise continuous interval map with no connections and no periodic orbits is topologically semiconjugate to an interval exchange transformation.

Incomparable actions of free groups

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\unicode[STIX]{x1D707}$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $\unicode[STIX]{x1D6E4}$, there is a Borel sequence $(\cdot _{r})_{r\in \mathbb{R}}$ of free actions of $\unicode[STIX]{x1D6E4}$ on $X$, generating subequivalence relations $E_{r}$ of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic, with the further property that $(E_{r})_{r\in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $\unicode[STIX]{x1D707}$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic.

Classifying crossed product C*-algebras

I combine recent results in the structure theory of nuclear ${\rm C}^*$-algebras and in topological dynamics to classify certain types of crossed products in terms of their Elliott invariants. In particular, transformation group ${\rm C}^*$-algebras associated to free minimal ${\Bbb Z}^{d}$-actions on the Cantor set with compact space of ergodic measures are classified by their ordered ${\rm K}$-theory. In fact, the respective statement holds for finite dimensional compact metrizable spaces, provided that projections of the crossed products separate tracial states. Moreover, ${\rm C}^*$-algebras associated to certain minimal homeomorphisms of spheres $S^{2n+1}$ are only determined by their spaces of invariant Borel probability measures (without a condition on the space of ergodic measures). Finally, I show that for a large collection of classifiable ${\rm C}^*$-algebras, crossed products by ${\Bbb Z}^{d}$-actions are generically again classifiable.

Equilibrium Measures at Temperature Zero for Hénon-Like Maps at the First Bifurcation

We develop a thermodynamic formalism for a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any $t\in\mathbb R$ we prove the existence of an invariant Borel probability measure which minimizes the free energy associated with a noncontinuous geometric potential $-t\log J^u$, where $J^u$ denotes the Jacobian in the unstable direction. We characterize accumulation points of these measures as $t\to\pm\infty$ in terms of the unstable Lyapunov exponent.

Real bounds and Lyapunov exponents

We prove that a $C^3$ critical circle map without periodic points has zero Lyapunov exponentwith respect to its unique invariant Borel probability measure. Moreover, no critical point of such a mapsatisfies the Collet-Eckmann condition. This result is proved directly from the well-known real a-priori bounds,without using Pesin's theory. We also show how our methods yield an analogous result for infinitely renormalizableunimodal maps of any combinatorial type. Finally we discuss an application of these facts to the study of neutralmeasures of certain rational maps of the Riemann sphere.

On the distribution of distances in homogeneous compact metric spaces

We provide a simple proof that in any homogeneous, compact metric space of diameter D, if one finds the average distance A achieved in X with respect to some isometry invariant Borel probability measure, thenD2≤A≤D. This result applies equally to vertex-transitive graphs and to compact, connected, homogeneous Riemannian manifolds.

Equilibrium states and invariant measures for random dynamical systems

Random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniformly continuous and contractive are considered. A non-degeneracy and a consistency conditions for such systems, which admit some proper Markov partitions of connected spaces, are introduced, and further sufficient conditions for them are provided. It is shown that every uniformly continuous Markov system associated with a continuous random dynamical system is consistent if it has a dominating Markov chain. A necessary and sufficient condition for the existence of an invariant Borel probability measure for such a non-degenerate system with a dominating Markov chain and a finite (16) is given. The condition is also sufficient if the non-degeneracy is weakened with the consistency condition. A further sufficient condition for the existence of an invariant measure for such a consistent system which involves only the properties of the dominating Markov chain is provided. In particular, it implies that every such a consistent system with a finite Markov partition and a finite (16) has an invariant Borel probability measure. A bijective map between these measures and equilibrium states associated with such a system is established in the non-degenerate case. Some properties of the map and the measures are given.

Topological dynamics of unordered Ramsey structures

We investigate the connections between Ramsey properties of Fraisse classes K and the universal minimal flow M(GK) of the automorphism group GK of their Fraisse limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class K has finite Ramsey degree for embeddings, then this degree equals the size of M(GK). We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if K is a relational Ramsey class and GK is amenable, then M(GK) admits a unique invariant Borel probability measure that is concentrated on a unique generic orbit.

Strong ergodicity of systems with the average shadowing property

Let X be a compact metric space and f : X → X be a continuous map. It is proved that if f has the average shadowing property and an invariant Borel probability measure with full support, then for all n ≥ 1, the map f × f ×⋅ ⋅ ⋅× f (n times) and hence f n are totally strongly ergodic.

The asymptotic average shadowing property and strong ergodicity

Let X be a compact metric space and f:X→X be a continuous map. In this paper, we prove that if f has the asymptotic average shadowing property (Abbrev. AASP) and an invariant Borel probability measure with full support or the positive upper Banach density recurrent points of f are dense in X, then for all n⩾1, f×f×⋯×f(n times) and fn are totally strongly ergodic. Moreover, we also give some sufficient conditions for an interval map having the AASP to be Li-Yorke chaotic.

Variational principles for relative local pressure with subadditive potentials

We prove two relative local variational principles of topological pressure functions \documentclass[12pt]{minimal}\begin{document}$P(T,\mathcal {F},\mathcal {U},y)$\end{document}P(T,F,U,y) and \documentclass[12pt]{minimal}\begin{document}$P(T,\mathcal {F},\mathcal {U}|Y)$\end{document}P(T,F,U|Y) for a given factor map π: (X, T) → (Y, S) between two topological dynamical systems, an open cover \documentclass[12pt]{minimal}\begin{document}$\mathcal {U}$\end{document}U of X and a subadditive potential \documentclass[12pt]{minimal}\begin{document}$\mathcal {F}$\end{document}F in \documentclass[12pt]{minimal}\begin{document}$C(X,\mathbb {R})$\end{document}C(X,R). By proving the upper semi-continuity and affinity of the entropy maps \documentclass[12pt]{minimal}\begin{document}$\mu \rightarrow h_{\mu }(T,\mathcal {U}\mid Y)$\end{document}μ→hμ(T,U∣Y) and \documentclass[12pt]{minimal}\begin{document}$\mu \rightarrow h^+_{\mu }(T,\mathcal {U}\mid Y)$\end{document}μ→hμ+(T,U∣Y) on the space of all invariant Borel probability measures, we show that the relative local topological pressure with subadditive potentials determines the local measure-theoretic conditional entropies.

Absolutely continuous random power series in reciprocals of Pisot numbers

We construct a continuum of shift invariant Borel probability measures on {−1,1}N such that corresponding random power series in reciprocals of Pisot numbers are absolutely continuous with respect to the Lebesgue measure.

The speed with which an orbit approaches a limit point

If is a continuous map of a compact metric space , and if is a sequence of positive reals converging to 0, we investigate the properties of the set . We show that is a dense subset of for every when x is a recurrent point, even though can be disjoint with the orbit of x for some . Under the assumption that f has an invariant non-atomic Borel probability measure , we prove results to the effect that (i) there is a uniform upper limit to the speed with which the orbit of each x can approach y for -almost every , (ii) if is ergodic with full support and if is the set of points having dense orbits, then for -almost every and for every there is a uniform upper limit to the speed with which the orbit of x can approach y. Next, using as a useful tool in proofs, we establish the following. If f is totally transitive and X is infinite, then there is a dense subset which is a countable union of Cantor sets such that and for any two distinct and any two distinct . If f is a transitive map enjoying a certain type of continuity in the backward direction, then f has a residual set of points with dense backward orbits.

On distortion in groups of homeomorphisms

Let $X$ be a path-connected topological space admitting a universal cover.Let Homeo$(X, a)$ denote the group of homeomorphisms of $X$ preservinga degree one cohomology class $ a$. We investigate the distortion in Homeo$(X, a)$. Let $g\in$ Homeo$(X,a)$. We define a Nielsen-type equivalence relation on the space of$g$-invariant Borel probability measures on $X$ and prove that if ahomeomorphism $g$ admits two nonequivalent invariant measures then itis undistorted. We also define a local rotation number of a homeomorphismgeneralizing the notion of the rotation of a homeomorphism of the circle.Then we prove that a homeomorphism is undistorted if its rotation number isnonconstant.

Uniform ergodic theorems for discontinuous skew-product flows and applications to Schrödinger equations

Motivated by linear Schrödinger equations with almost periodic potentials and phase transitions over almost periodic lattices, we introduce the so-called skew-product quasi-flows (SPQFs), which may admit both temporal and spatial discontinuity. In this paper we establish two basic theorems for SPQFs. One is an extension of the Bogoliubov–Krylov theorem for the existence of invariant Borel probability measures and the other is the uniform ergodic theorems. As applications, it will be shown that such a Schrödinger equation admits a well-defined rotation number.

About boundary values in $A(\Omega)$

Assume that Ω C ℂ n is a balanced bounded domain with a holomorphic support function (e.g. strictly pseudoconvex domain with C 2 boundary). We denote ∥f∥ 2 z := ∫ 1 0 |f(e 2πit z)| 2 dt. Let e > 0 and σ be a circular invariant Borel probability measure on ∂Ω. If g ∈ A(Ω) and h is a continuous function on ∂Ω with |g| < h on ∂Ω, then we construct nonconstant functions f 1 , f 2 ∈ A(Ω) with ∥g + f 1 ∥ z ≤ ∥h∥ z , |(g + f 2 )(z)| ≤ max |λ|=1 h(λz) for z ∈ ∂Ω and Additionally if Ω is a circular, bounded, strictly convex domain with C 2 boundary, then we give the construction of f 3 ∈ O(Ω), the holomorphic function with: ∥h ― |g + f * 3 |∥ z = 0 for all z ∈ ∂Ω, where f * denotes the radial limit of f. We also construct f 4 ∈ A(Ω) with ∥g + f 4 ∥ z = ∥h∥ z for z ∈ ∂Ω. In all cases we can make f i arbitrarily small on a given compact subset F C Ω and make it vanish to a given order at the point 0.