Probability Invariant Measure Research Articles

Asymptotic distribution of hitting times for critical maps of the circle

It is well known that the renormalization group transformation $\mathcal{R}$ has a unique fixed point $f_{cr}$ in the space of critical $C^{3}$-circle homeomorphisms with one cubic critical point $x_{cr}$ and the golden mean rotation number $\overline{\rho}:=\frac{\sqrt{5}-1}{2}.$ Denote by $Cr(\overline{\rho})$ the set of all critical circle maps $C^{1}$-conjugated to $f_{cr}.$ Let $f\in Cr(\overline{\rho})$ and let $\mu:=\mu_{f}$ be the unique probability invariant measure of $f.$ Fix $\theta \in(0,1).$ For each $n\geq1$ define $c_{n}:=c_{n}(\theta)$ such that $\mu([x_{cr},c_{n}])=\theta\cdot\mu([x_{cr},f^{q_{n}}(x_{cr})]),$ where $q_{n}$ is the first return time of the linear rotation $f_{\overline{\rho}}.$ We study convergence in law of rescaled point process of time hitting. We show that the limit distribution is singular w.r.t. the Lebesgue measure.

Random non-hyperbolic exponential maps

We consider random iteration of exponential entire functions, i.e., of the form ℂ ∍ z ↦ fλ(z) ≔ λez ∈ ℂ, λ ∈ ℂ {0}. Assuming that λ is in a bounded closed interval [A, B] ⊂ ℝ with A > 1/e, we deal with random iteration of the maps fλ governed by an invertible measurable map θ: Ω → Ω preserving a probability ergodic measure m on Ω, where Ω is a measurable space. The link from Ω to exponential maps is then given by an arbitrary measurable function η: Ω ↦ [A, B]. We in fact work on the cylinder space Q ≔ ℂ/ ∼, where ∼ is the natural equivalence relation: z ∼ w if and only if w−z is an integral multiple of 2πi. We prove that then for every t > 1 there exists a unique random conformal measure v(t) for the random conformal dynamical system on Q. We further prove that this measure is supported on the, appropriately defined, radial Julia set. Next, we show that there exists a unique random probability invariant measure μ(t) absolutely continuous with respect to v(t). In fact μ(t) is equivalent with v(t). Then we turn to geometry. We define an expected topological pressure \({\cal E}{\rm{P}}(t) \in \mathbb{R}\) and show that its only zero h coincides with the Hausdorff dimension of m-almost every fiber radial Julia set Jr (ω) ⊂ Q, ω ∈ Ω. We show that h ∈ (1, 2) and that the omega-limit set of Lebesgue almost every point in Q is contained in the real line ℝ. Finally, we entirely transfer our results to the original random dynamical system on ℂ. As our preliminary result, we show that all fiber Julia sets coincide with the entire complex plane ℂ.

The thermodynamic formalism and exponents of singularity of invariant measure of circle maps with a single break

Let $T \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, be a circle homeomorphism with one break point $x_{b}$, at which $ T'(x) $ has a discontinuity of the first kind and both one-sided derivatives at the point $x_{b} $ are strictly positive. Assume that the rotation number $\rho_{T}$ is irrational and its decomposition into a continued fraction beginning from a certain place coincides with the golden mean, i.e., $\rho_{T}=[m_{1}, m_{2}, \ldots, m_{l}, \, m_{l + 1}, \ldots] $, $ m_{s} = 1$, $s> l> 0$. Since the rotation number is irrational, the map $ T $ is strictly ergodic, that is, possesses a unique probability invariant measure $\mu_{T}$. A.A. Dzhalilov and K.M. Khanin proved that the probability invariant measure $ \mu_{G} $ of any circle homeomorphism $ G \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0$, with one break point $ x_{b} $ and the irrational rotation number $ \rho_{G} $ is singular with respect to the Lebesgue measure $ \lambda $ on the circle, i.e., there is a measurable subset of $ A \subset S^{1} $ such that $ \mu_ {G} (A) = 1 $ and $ \lambda (A) = 0$. We will construct a thermodynamic formalism for homeomorphisms $ T_{b} \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, with one break at the point $ x_{b} $ and rotation number equal to the golden mean, i.e., $ \rho_{T}:= \frac {\sqrt{5} -1}{2} $. Using the constructed thermodynamic formalism, we study the exponents of singularity of the invariant measure $ \mu_{T} $ of homeomorphism $ T $.

Long Time Behavior and Global Dynamics of Simplified Von Karman Plate Without Rotational Inertia Driven by White Noise

Without the assumption that the coefficient of weak damping is large enough, the existence of the global random attractors for simplified Von Karman plate without rotational inertia driven by either additive white noise or multiplicative white noise are proved. Instead of the classical splitting method, the techniques to verify the asymptotic compactness rely on stabilization estimation of the system. Furthermore, a clear relationship between in-plane components of the external force that act on the edge of the plate and the expectation of radius of the global random attractors can be obtained from the theoretical results. Based on the relationship between global random attractor and random probability invariant measure, the global dynamics of the plates are analyzed numerically. With increasing the in-plane components of the external force that act on the edge of the plate, global D-bifurcation, secondary global D-bifurcation and complex local dynamical behavior occur in motion of the system. Moreover, increasing the intensity of white noise leads to the dynamical behavior becoming simple. The results on global dynamics reveal that random snap-through which seems to be a complex dynamics intuitively is essentially a simple dynamical behavior.

ℬ-free sets and dynamics

Given B ⊂ N \mathscr {B}\subset \mathbb {N} , let η = η B ∈ { 0 , 1 } Z \eta =\eta _{\mathscr {B}}\in \{0,1\}^{\mathbb {Z}} be the characteristic function of the set F B := Z ∖ ⋃ b ∈ B b Z \mathcal {F}_\mathscr {B}:=\mathbb {Z}\setminus \bigcup _{b\in \mathscr {B}}b\mathbb {Z} of B \mathscr {B} -free numbers. The B \mathscr {B} -free shift ( X η , S ) (X_\eta ,S) , its hereditary closure ( X ~ η , S ) (\widetilde {X}_\eta ,S) , and (still larger) the B \mathscr {B} -admissible shift ( X B , S ) (X_{\mathscr {B}},S) are examined. Originated by Sarnak in 2010 for B \mathscr {B} being the set of square-free numbers, the dynamics of B \mathscr {B} -free shifts was discussed by several authors for B \mathscr {B} being Erdös; i.e., when B \mathscr {B} is infinite, its elements are pairwise coprime, and ∑ b ∈ B 1 / b > ∞ \sum _{b\in \mathscr {B}}1/b>\infty : in the Erdös case, we have X η = X ~ η = X B X_\eta =\widetilde {X}_\eta =X_{\mathscr {B}} . It is proved that X η X_\eta has a unique minimal subset, which turns out to be a Toeplitz dynamical system. Furthermore, a B \mathscr {B} -free shift is proximal if and only if B \mathscr {B} contains an infinite coprime subset. It is also shown that for B \mathscr {B} with light tails, i.e., d ¯ ( ∑ b > K b Z ) → 0 \overline {d}(\sum _{b>K}b\mathbb {Z})\to 0 as K → ∞ K\to \infty , proximality is the same as heredity. For each B \mathscr {B} , it is shown that η \eta is a quasi-generic point for some natural S S -invariant measure ν η \nu _\eta on X η X_\eta . A special role is played by subshifts given by B \mathscr {B} which are taut, i.e., when δ ( F B ) > δ ( F B ∖ { b } ) \boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}})>\boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}\setminus \{b\}}) for each b ∈ B b\in \mathscr {B} ( δ \boldsymbol {\delta } stands for the logarithmic density). The taut class contains the light tail case; hence all Erdös sets and a characterization of taut sets B \mathscr {B} in terms of the support of ν η \nu _\eta are given. Moreover, for any B \mathscr {B} there exists a taut B ′ \mathscr {B}’ with ν η B = ν η B ′ \nu _{\eta _{\mathscr {B}}}=\nu _{\eta _{\mathscr {B}’}} . For taut sets B , B ′ \mathscr {B},\mathscr {B}’ , it holds that X B = X B ′ X_\mathscr {B}=X_{\mathscr {B}’} if and only if B = B ′ \mathscr {B}=\mathscr {B}’ . For each B \mathscr {B} , it is proved that there exists a taut B ′ \mathscr {B}’ such that ( X ~ η B ′ , S ) (\widetilde {X}_{\eta _{\mathscr {B}’}},S) is a subsystem of ( X ~ η B , S ) (\widetilde {X}_{\eta _{\mathscr {B}}},S) and X ~ η B ′ \widetilde {X}_{\eta _{\mathscr {B}’}} is a quasi-attractor. In particular, all invariant measures for ( X ~ η B , S ) (\widetilde {X}_{\eta _{\mathscr {B}}},S) are supported by X ~ η B ′ \widetilde {X}_{\eta _{\mathscr {B}’}} . Moreover, the system ( X ~ η , S ) (\widetilde {X}_\eta ,S) is shown to be intrinsically ergodic for an arbitrary B \mathscr {B} . A description of all probability invariant measures for ( X ~ η , S ) (\widetilde {X}_\eta ,S) is given. The topological entropies of ( X ~ η , S ) (\widetilde {X}_\eta ,S) and ( X B , S ) (X_\mathscr {B},S) are shown to be the same and equal to d ¯ ( F B ) \overline {d}(\mathcal {F}_\mathscr {B}) ( d ¯ \overline {d} stands for the upper density). Finally, some applications in number theory on gaps between consecutive B \mathscr {B} -free numbers are given, and some of these results are applied to the set of abundant numbers.

Multifractal analysis of non-uniformly contracting iterated function systems

Let X = [0,1]. Given a non-uniformly contracting conformal iterated function system (IFS) and a family of positive Dini continuous potential functions , the triple system , under some conditions, determines uniquely a probability invariant measure, denoted by μ. In this paper, we study the pressure function of the system and multifractal structure of μ. We prove that the pressure function is Gateaux differentiable and the multifractal formalism holds, if the IFS has non-overlapping.

Simultaneous uniformization for uniformly quasisymmetric circle dynamical systems

In the 1960s Bers showed how to uniformize simultaneously two Riemann surfaces of the same finite analytic type by using a single quasi-Fuchsian group of the first kind. In this paper, we show how to uniformize simultaneously two uniformly quasisymmetric circle endomorphisms of the same degree by a unique normalized branched covering of the Riemann sphere of the same degree such that this branched covering has a unique normalized quasicircle as an invariant limit set. We use this simultaneous uniformization to define a transformation between their spaces of probability invariant measures and formulate several equivalent conjectures to the uniqueness conjecture for symmetric invariant probability measures. In a subsequent paper, we study these conjectures.

Real analyticity of Jacobian of invariant measures for hyperbolic meromorphic functions

We prove that for a hyperbolic meromorphic function f having a rapid derivative growth, if HD (J(f))>1, then the Jacobian Dμϕ of a probability invariant measure μϕ on J(f), equivalent to a conformal measure mϕ, has a real analytic extension on a neighbourhood of J(f)∖ f−1(∞) in ℂ. If, in addition, f satisfies a balanced derivative growth condition with constant exponents, then this extension is bounded in a neighbourhood of every pole of f.

Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order

AbstractWorking with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamic formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynamic formalism, a version of Bowen’s formula expressing the Hausdorff dimension of the radial Julia set as the zero of the pressure function and, moreover, the real analyticity of this dimension, is proved.

On the Uniqueness Property for Invariant Measures

It is shown that the uniqueness property of probability invariant measures is preserved under the operation of Cartesian products. A similar question is investigated for inductive limits of nonzero -finite invariant measures.

Rigidity of Conformal Iterated Function Systems

The paper extends the rigidity of the mixing expanding repellers theorem of D. Sullivan announced at the 1986 IMC. We show that, for a regular conformal, satisfying the ‘Open Set Condition’, iterated function system of countably many holomorphic contractions of an open connected subset of a complex plane, the Radon–Nikodym derivative dμ/dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and μ is the unique probability invariant measure equivalent with m. Next, we introduce the concept of nonlinearity for iterated function systems of countably many holomorphic contractions. Several necessary and sufficient conditions for nonlinearity are established. We prove the following rigidity result: If h, the topological conjugacy between two nonlinear systems F and G, transports the conformal measure mF to the equivalence class of the conformal measure mG, then h has a conformal extension on an open neighbourhood of the limit set of the system F. Finally, we prove that the hyperbolic system associated to a given parabolic system of countably many holomorphic contractions is nonlinear, which allows us to extend our rigidity result to the case of parabolic systems.

A Chaotic Continuous Map Generates All Probability Distributions

Let ƒ be a continuous map of the compact unit interval I = [0, 1], such that ƒ2, the second iterate of ƒ, is topologically transitive in I. If for some x and y in I and any t in I there exists lim(1/n) # {i ≤ n; |ƒi(x) − ƒi(y)| < t} for n → ∞, denote it by φxy(t). In the paper we consider the class F(ƒ) if all φxy. The main results are that F(ƒ) is convex and pointwise closed. Using this we show that F(ƒ) is always bigger than the class D(ƒ) of probability distributions generated analogously by single trajectories (and corresponding to the class of probability invariant measures of ƒ), and prove that there are universal generators of probability distributions, i.e., maps ƒ such that F(ƒ) is the class M of all non-decreasing functions I ⇒ I (contrary to this, D(ƒ) ⊃ M for no ƒ). These results can be extended to more general continuous maps. One of the possible applications is to use the size of F(ƒ) as a measure of the degree of chaos of ƒ.

Lyapunov characteristic exponents are nonnegative

We prove that, for an arbitrary rational map f f on the Riemann sphere and an arbitrary probability invariant measure on the Julia set, Lyapunov characteristic exponents are nonnegative a.e. In particular log ⁡ | f ′ | \log |f’| is integrable. An analogous theorem is proved for smooth maps of an interval with all critical points being nonflat. This allows us to fill a gap in the proof of Denker and Urbański’s theorem that there exists a probability conformal measure on the Julia set with exponent equal to the supremum of the Hausdorff dimensions of probability invariant measures with positive entropy.

SKEW PRODUCT ACTIONS OF SEMI-DIRECT PRODUCT GROUPS

Introduction Let f be a homomorphism from a second countable locally compact group G into the continuous automorphisms of a second countable compact group K. Let g ∗ k = f(g)(k) and suppose further the mapping given by (g, k) → g ∗ k is continuous from G × K into K, g ∈ G, k ∈ K. Then the cartesian product K × G becomes a group with multiplication given by (k1, g1)(k2, g2) = (k1(g1 ∗ k2), g1g2), where (ki, gi) ∈ K × G, i = 1, 2. We denote this semi-direct product group by K ×s G. Let (S, μ), a standard Borel space, be an ergodic Borel K ×s G-space with a probability invariant measure μ. In this paper we state necessary and sufficient conditions so the quotient Borel K ×s G-mapping from S into the space of K-orbits in S has relative discrete spectrum. These conditions are in terms of the stabilizers and the natural action of G on the dual K of K. (We remind the reader K is the set of unitary equivalence classes of irreducible unitary representations of K). Choose a Borel subset S of S that meets each K-orbit in S exactly once. Let p : (S, μ) → (S, μ) be a mapping given by p(s) = s where s is the unique point in S that meets the K-orbit s · K of s and μ = μ ◦ p−1.