Ergodic Invariant Measure Research Articles

An adic dynamical system related to the Delannoy numbers

AbstractWe introduce an adic (Bratteli–Vershik) dynamical system based on a diagram whose path counts from the root are the Delannoy numbers. We identify the ergodic invariant measures, prove total ergodicity for each of them, and initiate the study of the dimension group and other dynamical properties.

Nonuniform measure rigidity

We consider an ergodic invariant measure µ for a smooth actionof Z k , k � 2, on a (k+1)-dimensional manifold or for a locally free smooth action of R k , k � 2 on a (2k + 1)-dimensional manifold. We prove that if µ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z k has positive entropy, then µ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpinski gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.

SRB measures for certain Markov processes

We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $\delta_0$ and $\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.

The Euler Adic Dynamical System and Path Counts in the Euler Graph

We give a formula for generalized Eulerian numbers, prove monotonicity of sequences of certain ratios of the Eulerian numbers, and apply these results to obtain a new proof that the natural symmetric measure for the Bratteli-Vershik {dynamical} system based on the Euler graph is the unique fully supported invariant ergodic Borel probability measure. Key ingredients of the proof are a two-dimensional induction argument and a one-to-one correspondence between most paths from two vertices at the same level to another vertex.

Q-Distributions on boxed plane partitions

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.

Existence of a measurable saturated compensation function between subshifts and its applications

AbstractWe show the existence of a bounded Borel measurable saturated compensation function for any factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. These problems were studied in [23] for a compact invariant set whose symbolic representation is a shift of finite type under the condition of the existence of a saturated compensation function. By using the ergodic equilibrium states of a constant multiple of a Borel measurable compensation function, we extend the results to the general case where this condition might not hold, presenting a formula for the Hausdorff dimension for a compact invariant set whose symbolic representation is a subshift and studying invariant ergodic measures of full dimension. We study uniqueness and properties of such measures for a compact invariant set whose symbolic representation is a topologically mixing shift of finite type.

The local variational principle of topological pressure for sub-additive potentials

Let ( X , T ) be a topological dynamical system and F = { f n } n = 1 ∞ be a sub-additive potential on C ( X , R ) . Let U be an open cover of X . Then for any T -invariant measure μ , let F ∗ ( μ ) = lim n → ∞ 1 n ∫ f n d μ . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle: P ( T , F , U ) = sup μ { h μ ( T , U ) + F ∗ ( μ ) : μ ∈ M ( X , T ) } where h μ ( T , U ) denotes the measure-theoretic entropy of μ relative to U and the supremum can be attained by a T -invariant ergodic measure. The main purpose of this paper is to generalize a result of Huang and Yi (2007) [17]. In the paper [17], they proved the local variational principle of pressure for additive potentials. Furthermore, we prove the result P ( T , F ) = lim d i a m ( U ) → 0 P ( T , F ; U ) . Moreover, we obtained P ( T , F ) = sup μ { h μ ( T ) + F ∗ ( μ ) : μ ∈ M ( X , T ) } , which gives another proof of the topological pressure variational principle for sub-additive potentials from Cao et al. (2008) [14].

An example of a non-uniquely ergodic lamination

AbstractThis paper presents an example of Riemann surface lamination with at least two ergodic invariant measures. The generic leaves for those measures are of different growth and have different numbers of ends.

Equilibrium states for partially hyperbolic horseshoes

AbstractWe study ergodic properties of invariant measures for the partially hyperbolic horseshoes, introduced in Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys. 29 (2009), 433–474]. These maps have a one-dimensional center direction Ec, and are at the boundary of the (uniformly) hyperbolic diffeomorphisms (they are constructed bifurcating hyperbolic horseshoes via heterodimensional cycles). We prove that every ergodic measure is hyperbolic, but the set of Lyapunov exponents in the central direction has gap: all ergodic invariant measures have negative exponent, with the exception of one ergodic measure with positive exponent. As a consequence, we obtain the existence of equilibrium states for any continuous potential. We also prove that there exists a phase transition for the smooth family of potentials given by ϕt=t log ∣DF∣Ec∣.

Multifractal analysis of weak Gibbs measures for non-uniformly expanding C1 maps

AbstractWe will consider the local dimension spectrum of a weak Gibbs measure on a C1 non-uniformly hyperbolic system of Manneville–Pomeau type. We will present the spectrum in three ways: using invariant measures, ergodic invariant measures supported on hyperbolic sets and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.

The recurrence dimension for piecewise monotonic maps of the interval

We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval [0, 1], giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure function. Mathematics Subject Classification (2000): 37E05 (primary); 37C45, 28A80, 37B40, 28A78 (secondary).

On the distribution of periodic orbits

Let f : $M\to M$ be a $C^{1+\varepsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of theperiodic orbits of $f|_\Lambda$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure Ptop$(\varphi)$ can be computed by the values of the potential $\varphi$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we provethat certain equilibrium states are Bowen measures. Finally, we derive a large deviation resultfor the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.

The dynamics of the NAIRU model with two switching regimes

We consider a model of inflation and unemployment proposed in Ferri et al. (JEBO, 2001), in which the dynamics are described by a discontinuous piecewise linear map, made up of two branches. We shall show that the bounded dynamics may be classified in two cases: we may have either regular dynamics with stable cycles of any period or quasiperiodic trajectories, or only chaotic dynamics ( pure chaos in which a unique absolutely continuous invariant ergodic measure exists, and structurally stable), in a rich variety of cyclical chaotic intervals. The main results are the analytical formulation of the border collision bifurcation curves, through which we give a complete picture of the possible outcomes of the model.

Measure of full dimension for some nonconformal repellers

Given $(X,T)$ and $(Y,S)$ mixing subshifts of finite type such that $(Y,S)$ is a factor of $(X,T)$ with factor map $\pi$:$\ X\to Y$, and positive Hölder continuous functions $\varphi$:$\ X\to \mathbb{R}$ and $\psi$:$\ Y\to \mathbb{R}$, we prove that the maximum of <p align="center"> $\frac{h_{\mu\circ \pi^{-1}}(S)}{\int \psi\circ\pi\d\mu}+ \frac{h_\mu(T)-h_{\mu\circ \pi^{-1}}(S)}{\int \varphi\d\mu}$ <p align="left" class="times"> over all $T$-invariant Borel probability measures $\mu$ on $X$ is attained on the subset of ergodic measures. Here $h_\mu(T)$ stands for the metric entropy of $T$ with respect to $\mu$. As an application, we prove the existence of an ergodic invariant measure with full dimension for a class of transformations treated in [11], and also for the transformations treated in [17], where the author considers nonlinear skew-product perturbations of <i>general Sierpinski carpets</i>. In order to do so we establish a variational principle for the topological pressure of certain noncompact sets.

A self-dual induction for three-interval exchange transformations

We describe completely a new induction process on three-interval exchange transformations, under its additive and multiplicative forms; we describe the natural extension of the induction map, show that it is self-dual for the notion of duality which links the Rauzy/Zorich and da Rocha inductions, and compute a finite ergodic invariant measure (for the multiplicative form). Then we explain the connection with the published negative slope algorithm of Ferenczi, Holton and Zamboni [S. Ferenczi, C. Holton, and L. Zamboni, The structure of three-interval exchange transformations I: An arithmetic study, Ann. Inst. Fourier (Grenoble), 51 (2001), pp. 861–901], and use our measure-theoretic study to get new results on three-interval exchange transformations.

Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds

Extending earlier results for analytic curve segments, in this article we describe the asymptotic behavior of evolution of a finite segment of a Cn-smooth curve under the geodesic flow on the unit tangent bundle of a hyperbolic n-manifold of finite volume. In particular, we show that if the curve satisfies certain natural geometric conditions, then the pushforward of the parameter measure on the curve under the geodesic flow converges to the normalized canonical Riemannian measure on the tangent bundle in the limit. We also study the limits of geodesic evolution of shrinking segments. We use Ratner's classification of ergodic invariant measures for unipotent flows on homogeneous spaces of SO(n,1) and an observation relating local growth properties of smooth curves and dynamics of linear SL(2,R)-actions

Almost additive multifractal analysis

For the class of almost additive sequences, we establish a conditional variational principle for the dimension spectra in the context of the nonadditive thermodynamic formalism. This generalizes the classical thermodynamic formalism, by replacing the topological pressure of a single function by the topological pressure of a sequence of functions. In particular, we show that each level set of the multifractal decomposition carries a full measure, that is, an ergodic invariant measure with dimension equal to the dimension of the level set. We also show that the spectra are continuous and that the irregular sets have full dimension.

Invariant measures for interval maps with critical points and singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.

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.