F-invariant Measure Research Articles

Symbolic dynamics for pointwise hyperbolic systems on open regions

Abstract Under certain conditions, we construct a countable Markov partition for pointwise hyperbolic diffeomorphism $f:M\rightarrow M$ on an open invariant subset $O\subset M$ , which allows the Lyapunov exponents to be zero. From this partition, we define a symbolic extension that is finite-to-one and onto a subset of O that carries the same finite f-invariant measures as O. Our method relies upon shadowing theory of a recurrent-pointwise-pseudo-orbit that we introduce. As a canonical application, we estimate the number of closed orbits for f.

Dimension estimates and approximation in non-uniformly hyperbolic systems

Abstract Let $f: M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and $\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of $\mu $ , which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If $\mu $ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets $\{\Lambda _n\}_{n\geq 1}$ . The limit behaviour of the Carathéodory singular dimension of $\Lambda _n$ on the unstable manifold with respect to the super-additive singular valued potential is also studied.

Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces* *This work is partially supported by NSFC (12271386, 11871361), and the second author is partially supported by Qinglan project from Jiangsu Province.

Let f : X → X be an invertible Lipschitz transformation on a compact metric space X. Given a Hölder continuous invertible operator cocycles on a Banach space and an f-invariant ergodic measure, this paper establishes the Hölder continuity of Oseledets subspaces over a compact set of arbitrarily large measure. This extends a result in [V Araujo, A I Bufetov and S Filip, On Hölder-continuity of Oseledets subspaces, J. Lond. Math. Soc. 2016, 93 : 194–218]. for invertible operator cocycles on a Banach space. This paper also proves the Hölder continuity in the non-invertible case. Finally, some applications are given in the end of this paper.

On entropy and pressure via preimage structure for random dynamical systems

In this paper, entropy, pressure and equilibrium states are investigated for noninvertible random dynamical systems via the preimage structure. For a continuous random dynamical system F on a compact metric space and an F-invariant measure μ, the folding entropy Fμ(F) and the preimage entropy hm,μ(F) are introduced, and it is shown that these two measure-theoretic versions of entropies coincide with each other if F has uniform separation of preimages. Some fundamental properties for hm,μ(F) such as Shannon-McMillan-Breiman Theorem, Kolmogorov-Sinai Theorem and upper semi-continuity are established. From the topological viewpoint, the preimage pressures Pm(F,⋅), Pp(F,⋅) and Pp˜(F,⋅) (resp. topological preimage entropy hm(F), hp(F) and hp˜(F)) are introduced via the preimage structure of single points. Some basic properties are discussed, and particularly, variational principles are obtained. For a C1 nondegenerate random partially hyperbolic dynamical system F on a closed Riemannian manifold, we propose the notions of metric stable entropy hm,μs(F) and stable pressures Pms(F,⋅) and Pps(F,⋅) (resp. topological stable entropies hms(F) and hps(F)) via the preimage structure of local stable manifolds. Several basic properties are obtained and variational principles are established particularly. At the end, as an application of variational principles, p-equilibrium states (resp. s-equilibrium states) and p-tangent functionals (resp. s-tangent functionals) are studied.

Non-statistical rational maps

We show that in the family of degree $$d\geqslant 2$$ rational maps of the Riemann sphere, the closure of strictly postcritically finite maps contains a (relatively) Baire generic subset of maps displaying maximal non-statistical behavior: for a map f in this generic subset, the set of accumulation points of the sequence of empirical measures of almost every point in the phase space, is equal to the largest possible one, that is the set of all f-invariant measures. We also introduce an abstract setting to study non-statistical dynamics and sufficient conditions for their existence in a general family of maps. These sufficient conditions are related to the notions of “statistical instability” and “statistical bifurcation” for which we give a general formalization in the first section. The proof of our result in the family of rational maps is based on a transversality argument which allows us to control the behavior of the orbits of critical points for maps close to strictly postcritically finite rational maps. Using this property and doing careful perturbations, we show that the closure of strictly postcritically finite maps satisfies the sufficient conditions introduced in the abstract setting.

Entropy via preimage structure

To measure the complexity of a “non-invertible” continuous map f on a compact metric space via the preimage structure, Hurley introduced the notion of pointwise topological preimage entropies hm(f) and hp(f) in 1995. A natural question was: Can one introduce the counterpart of hm(f) or hp(f) from the measure-theoretic point of view for an f-invariant measure μ and obtain properties analogous to that for the classical entropies? Cheng and Newhouse made the first step on this topic and introduced the notions of preimage entropies hpre(f) and hpre,μ(f) and investigated their properties in 2005. Recently, Wu and Zhu proposed a notion of pointwise metric preimage entropy hm,μ(f) and obtained some properties for f with uniform separation of preimages.In this paper, we give a complete answer to the above question for general continuous maps. The Shannon-McMillan-Breiman theorem for hm,μ(f) and the variational principle relating hm,μ(f) and hm(f) are obtained. It is shown that hm(f)=hpre(f) and the three types of measure-theoretic entropy-like invariants hpre,μ(f), hm,μ(f) and the folding entropy Fμ(f) (introduced by Ruelle) via the preimage structure coincide with each other. We also show that the preimage entropy variational principle can be realized as a variant of the conditional variational principle. These results answer several open questions asked by Cheng-Newhouse and Downarowicz. The upper semi-continuity of hm,μ(f) with respect to invariant measures with the same separation rate is obtained. Moreover, the preimage pressure and preimage equilibrium states are investigated.

Pressure and exponential rate over periodic orbits

For a measure μ preserved by a C1+α0 (α0>0) diffeomorphism f and a continuous function ϕ on the manifold, we study the relationship between the exponential growth rate of ∑eSnϕ(x) over the orbits of some periodic points and the free energy hμ(f)+∫ϕdμ (in certain cases, it is equal to the measure theoretic pressure) or the topological pressure. When μ is an ergodic hyperbolic measure, we prove that the exponential growth rate coincides with the free energy (measure theoretic pressure). And we also verify the equality of the exponential growth rate and the topological pressure when the manifold is 2-dimensional. However, for the higher-dimensional manifold, we show an inequality between the exponential growth rate and the topological pressure. For an ergodic hyperbolic measure ω, we also prove that there is a ω-full measured set Λ˜ such that for every f-invariant measure supported on Λ˜, the exponential growth rate equals to the free energy. And moreover, we prove that there is another ω-full measured set Δ˜ such that for every f-invariant measure supported on Δ˜, the exponential growth rate equals to the topological pressure.

Shadowing Property and Invariant Measures Having Full Supports

We consider the shadowing property from the measure theoretical view point. Let f be a homeomorphism of a compact metric space. Then f has the shadowing property on the chain recurrent set if and only if there exists a shadowable f-invariant measure having full support. Suppose that f does not have the shadowing property. Then, if there exists a f-invariant measure having full support, then, generically, every f-invariant measure is not shadowable.

The pressure function for infinite equilibrium measures

Assume that (X, f) is a dynamical system and φ: X → [−∞, ∞) is a potential such that the f-invariant measure μφ equivalent to the φ-conformal measure is infinite, but that there is an inducing scheme F = fτ with a finite measure $$\mu_\phi^-$$ and polynomial tails $$\mu_\phi^-$$ (τ ≥ n) = O(n−β), β ∈ (0, 1). We give conditions under which the pressure of f for a perturbed potential φ + sψ relates to the pressure of the induced system as $$P(\phi + s\psi ) = (CP{(\overline {\phi + s\psi )} )^{1/\beta }}(1 + o(1)),$$ together with estimates for the o(1)-error term. This extends results from Sarig [S06] to the setting of infinite equilibrium states. We give several examples of such systems, thus improving on the results of Lopes [L93] for the Pomeau-Manneville map with potential φt = −tlogf′, as well as on the results by Bruin and Todd [BTo09, BTo12] on countably piecewise linear unimodal Fibonacci maps. In addition, limit properties of the family of measures μφ+sψ as s → 0 are studied and statistical properties (correlation coefficients and arcsine laws) under the limit measure are derived.

Geometry and entropy of generalized rotation sets

For a continuous map f on a compact metric space we study the geometry and entropy of the generalized rotation set Rot(Φ). Here Φ = (ϕ1, ..., ϕ m ) is a m-dimensional continuous potential and Rot(Φ) is the set of all µ-integrals of Φ and µ runs over all f-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of ℝm. We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set K in ℝm a potential Φ = Φ(K) with Rot(Φ) = K. Next, we study the relation between Rot(Φ) and the set of all statistical limits Rot Pt (Φ). We show that in general these sets differ but also provide criteria that guarantee Rot(Φ) = Rot Pt (Φ). Finally, we study the entropy function w ↦ H(w),w ∈ Rot(Φ). We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems H(w) is determined by the growth rate of those hyperbolic periodic orbits whose Φ-integrals are close to w. We also show that for systems with strong thermodynamic properties (sub-shifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function w ↦ H(w) is real-analytic in the interior of the rotation set.

On Invariant Measure of the Circle Maps

Let f be piecewise smooth circle homeomorphisms with break points and the rotation number ρ is irrational. We provide a necessary condition for the absolute continuity of f-invariant measure with respect to Lebesgue measure.

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \({\mathbb{C}\mathbb{P}^k}\) and ν be an f-invariant measure with positive Lyapunov exponents (λ1, . . . , λk). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by \({{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}\), which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of fn in \({\mathbb{C}\mathbb{P}^k}\) . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \({\mathbb{C}\mathbb{P}^k}\) and a normalization theorem for the ν-generic inverse branches of fn.

Flows on graphs and dynamical systems

Let f be a homeomorphism of a compact manifold M. The goal of the paper is the construction of f-invariant measures. To obtain an approximation of such measures we use notion of symbolic image with respect to a partition of M. Symbolic image G is a directed graph which is a finite approximation of the mapping f, the diameter of the partition is an evaluation of the approximation. A flow on the symbolic image is a probability distribution on the edges satisfying Kirchhoff low at each vertex: the incoming flow equals outgoing one. The flow is an approximation to some invariant measure. By considering a sequence of subdivisions one gets a sequence of symbolic images Gk and flows that tend to an invariant measure as partition diameter goes to zero.

Basins of measures on inverse limit spaces for the induced homeomorphism

AbstractLet f:X→X be continuous and onto, where X is a compact metric space. Let $Y:=\invlim {X,f}$ be the inverse limit and F:Y →Y the induced homeomorphism. Suppose that μ is an f-invariant measure, and let m be the measure induced on Y by (μ,μ,…). We show that B is a basin of μ if and only if π−11(B) is a basin of m. From this it follows that if μ is an SRB measure for f on X, then the induced measure m on Y is an inverse-limit SRB measure for F. Conversely, if m is an inverse-limit SRB measure for F on Y, then the induced measure μ on X is an SRB measure for f.

Density of periodic points, invariant measures and almost equicontinuous points of cellular automata

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of such automata converges in Cesàro mean to an invariant measure μ c . Moreover the dynamical system (cellular automaton F, invariant measure μ c ) has still the μ c -almost equicontinuity property and the set of periodic points is dense in the topological support of the measure μ c . We also show that the density of periodic points in the topological support of a measure μ occurs for each μ-almost equicontinuous cellular automaton when μ is an invariant and shift ergodic measure. Finally using most of these results we give a non-trivial example of a couple ( μ-equicontinuous cellular automaton F, shift and F-invariant measure μ) such that the restriction of F to the topological support of μ has no equicontinuous points.

Module shifts and measure rigidity in linear cellular automata

Suppose R is a finite commutative ring of prime characteristic, A is a finite R-module, M:=Z^D x N^E, and F is an R-linear cellular automaton on A^M. If mu is an F-invariant measure which is multiply shift-mixing in a certain way, then we show that mu must be the Haar measure on a coset of some submodule shift of A^M. Under certain conditions, this means mu must be the uniform Bernoulli measure on A^M.

Maximizing measures for endomorphisms of the circle

We study a given fixed continuous function and an endomorphism f : S1 → S1, whose f-invariant probability measures maximize . We prove that the set of endomorphisms having a ϕ maximizing invariant measure supported on a periodic orbit is dense.

Circle homeomorphisms with two break points

Let f be a circle class P homeomorphism with two break points 0 and c. If the rotation number of f is of bounded type and f is C2(S1∖{0, c}) then the unique f-invariant probability measure is absolutely continuous with respect to the Lebesgue measure if and only if 0 and c are on the same orbit and the product of their f-jumps is 1. We indicate how this result extends to class P homeomorphisms of rotation number of bounded type and with a finite number of break points such that f admits at least two break points 0 and c not on the same orbit and that the jump of f at c is not the product of some f-jumps at breaks points not belonging to the orbits of 0 and c.

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.

The Perron-Frobenius operators, invariant measures and representations of the Cuntz-Krieger algebras

For a transformation F on a measure space (X,μ), we show that the Perron-Frobenius operator of F can be written by a representation (L2(X,μ),π) of the Cuntz-Krieger algebra OA associated with F when F satisfies some assumption. Especially, when OA is the Cuntz algebra ON and (L2(X,μ),π) in the above is some irreducible representation of ON, then there is an F-invariant measure on X which is absolutely continuous with respect to μ.