Intrinsic Ergodicity Research Articles

Similarities and differences between specification and non-uniform specification

AbstractPavlov [Adv. Math.295 (2016), 250–270; Nonlinearity32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

Intrinsic Ergodicity, Generators, and Symbolic Representations of Algebraic Group Actions

Intrinsic Ergodicity, Generators, and Symbolic Representations of Algebraic Group Actions

Внутренняя эргодичность, порождающие разбиения и символические представления алгебраических действий групп

Строятся естественные символические представления внутренне эргодических, но не обязательно расширительных главных алгебраических действий счетных бесконечных аменабельных групп. С помощью этих представлений получены порождающие разбиения (с точностью до множеств меры нуль) для таких действий.

Effective intrinsic ergodicity for countable state Markov shifts

For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and the measure of maximal entropy in terms of the difference of their entropies. This extends an earlier result for subshifts of finite type, due to Kadyrov. We provide a similar bound for equilibrium measures of strongly positively recurrent potentials, in terms of the pressure difference. For measures with nearly maximal entropy, we have new, and sharp, bounds. The strong positive recurrence condition is necessary.

On entropy and intrinsic ergodicity of coded subshifts

Any coded subshift X C X_C defined by a set C C of code words contains a subshift, which we call L C L_C , consisting of limits of single code words. We show that when C C satisfies the unique decipherability property, the topological entropy h ( X C ) h(X_C) of X C X_C is determined completely by h ( L C ) h(L_C) and the number of code words of each length. More specifically, we show that h ( X C ) = h ( L C ) h(X_C) = h(L_C) exactly when a certain infinite series is less than or equal to 1 1 , and when that series is greater than 1 1 , we give a formula for h ( X C ) h(X_C) . In the latter case, an immediate corollary (using a result from [Israel J. Math. 192 (2012), pp. 785–817] is that X C X_C has a unique measure of maximal entropy.

Intrinsic ergodicity for factors of -shifts

We show that every subshift factor of the ()-shift is intrinsically ergodic, when and the ()-expansion of 1 is not periodic with odd period. Moreover, the unique measure of maximal entropy satisfies a certain Gibbs property. This is an application of the technique established by Climenhaga and Thompson to prove intrinsic ergodicity for symbolic systems beyond specification. We also prove that there exists a subshift factor of the ()-shift which is not intrinsically ergodic in the cases other than the above.

One-sided almost specification and intrinsic ergodicity

We define a new property called one-sided almost specification, which lies between the properties of specification and almost specification, and prove that it guarantees intrinsic ergodicity (i.e. uniqueness of the measure of maximal entropy) if the corresponding mistake function $g$ is bounded. We also show that uniqueness may fail for unbounded $g$ such as $\log \log n$. Our results have consequences for almost specification: we prove that almost specification with $g\equiv 1$ implies one-sided almost specification (with $g\equiv 1$) and hence uniqueness. On the other hand, the second author showed recently that almost specification with $g\equiv 4$ does not imply uniqueness.

On entropy of dynamical systems with almost specification

We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), 785–817]. Elaborating on our examples we prove that sufficient conditions for every shift factor of a shift space to be intrinsically ergodic given by Climenhaga and Thompson are in some sense best possible; moreover, the weak specification property neither implies intrinsic ergodicity, nor follows from almost specification. We also construct a dynamical system with the weak specification property, which does not have the almost specification property. We prove that the minimal points are dense in the support of any invariant measure of a system with the almost specification property. Furthermore, if a system with almost specification has an invariant measure with non-trivial support, then it also has uniform positive entropy over the support of any invariant measure and cannot be minimal.

On intrinsic ergodicity and weakenings of the specification property

Since seminal work of Bowen [2], it has been known that the specification property implies various useful properties about an expansive topological dynamical system, among them uniqueness of the measure of maximal entropy (often referred to as intrinsic ergodicity). Weakenings of the specification property have been defined and profitably applied in various works such as [6,9,11,16,17].It has been an open question (see p. 798 of [4]) whether two of these properties, which we here call almost specification and non-uniform specification, imply intrinsic ergodicity for expansive topological systems. We answer this question negatively by exhibiting examples of subshifts with multiple measures of maximal entropy with disjoint support which have non-uniform specification with any gap function f(n)=O(ln⁡n) or almost specification with any mistake function g(n)≥4. We also show some results in the opposite direction, showing that subshifts with non-uniform specification with gap function f(n)=o(ln⁡n) or almost specification with mistake function g(n)=1 cannot have multiple measures of maximal entropy with disjoint support.

On intrinsic ergodicity of factors of subshifts

It is well known that any $\mathbb{Z}$ subshift with the specification property has the property that every factor is intrinsically ergodic, i.e. every factor has a unique factor of maximal entropy. In recent work, other $\mathbb{Z}$ subshifts have been shown to possess this property as well, including $\unicode[STIX]{x1D6FD}$-shifts and a class of $S$-gap shifts. We give two results that show that the situation for $\mathbb{Z}^{d}$ subshifts with $d>1$ is quite different. First, for any $d>1$, we show that any $\mathbb{Z}^{d}$ subshift possessing a certain mixing property must have a factor with positive entropy which is not intrinsically ergodic. In particular, this shows that for $d>1$, $\mathbb{Z}^{d}$ subshifts with specification cannot have all factors intrinsically ergodic. We also give an example of a $\mathbb{Z}^{2}$ shift of finite type, introduced by Hochman, which is not even topologically mixing, but for which every positive-entropy subshift factor is intrinsically ergodic.

Magnetic three states of matter: A quantum Monte Carlo study of spin liquids

We present thermodynamic phase diagrams showing magnetic analog of "three states of matter," namely, spin liquid, paramagnetic, and magnetically ordered phases, obtained by unbiased quantum Monte Carlo simulations. Our simulations are carried out for Kitaev's toric codes in two and three dimensions, i.e., the simplest realizations of gapped topological $Z_2$ spin liquids, extended by a nearest-neighbor ferromagnetic Ising coupling. We find that the ordered phase borders on the spin liquid by a discontinuous transition line in three dimensions, while it grows continuously from the quantum critical point in two dimensions. In both cases, our results elucidate peculiar proximity effects to the nearby spin liquids in the high-temperature paramagnetic phase, even when the ground state is magnetically ordered. The thorough study of magnetic three states of matter is achieved by introducing the "fictitious vertex" method into the directed loop algorithm. This provides a generic prescription to simulate models with off-diagonal multispin interactions, in which the conventional scheme may suffer from intrinsic ergodicity breakdown as in the present case.

Ergodic Properties of Visible Lattice Points

Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as ℬ-free numbers with Sarnak’s conjecture on the “randomness” of the Mobius function; another is the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.

Ergodic properties of visible lattice points

Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as ℬ-free numbers with Sarnak’s conjecture on the “randomness” of the Möbius function; another is the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.

Topological and ergodic properties of symmetric sub-shifts

The family of symmetric one sided sub-shifts in two symbols given by a sequence $a$ is studied. We analyse some of their topological properties such as transitivity, the specification property and intrinsic ergodicity. It is shown that almost every member of this family admits only one measure of maximal entropy. It is shown that the same results hold for attractors of the family of open dynamical systems arising from the doubling map with a centred symmetric hole depending on one parameter, and for the set of points that have unique $\beta$-expansion for $\beta \in (\varphi,2)$ where $\varphi$ is the Golden Ratio.

Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

We show that partially hyperbolic diffeomorphisms of $$d$$ -dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a global stability result, i.e. every partially hyperbolic diffeomorphism as above is leaf-conjugate to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.

Intrinsic ergodicity beyond specification: β-shifts, S-gap shifts, and their factors

We give sufficient conditions for a shift space $(\Sigma,\sigma)$ to be intrinsically ergodic, along with sufficient conditions for every subshift factor of $\Sigma$ to be intrinsically ergodic. As an application, we show that every subshift factor of a $\beta$-shift is intrinsically ergodic, which answers an open question included in Mike Boyle's article "Open problems in symbolic dynamics". We obtain the same result for $S$-gap shifts, and describe an application of our conditions to more general coded systems. One novelty of our approach is the introduction of a new version of the specification property that is well adapted to the study of symbolic spaces with a non-uniform structure.

Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part

We prove that any (absolutely) partially hyperbolic diffeomorphism f f of T 3 \mathbb {T}^3 homotopic to a hyperbolic automorphism A A is intrinsically ergodic; that is, it has a unique entropy maximizing measure μ \mu .

Fuzzy controller based on chaos optimal design and its application

In order to overcome difficulty of tuning parameters of fuzzy controller, a chaos optimal design method based on annealing strategy is proposed. First, apply the chaotic variables to search for parameters of fuzzy controller, and transform the optimal variables into chaotic variables by carrier-wave method. Making use of the intrinsic stochastic property and ergodicity of chaos movement to escape from the local minimum and direct optimization searching within global range, an approximate global optimal solution is obtained. Then, the chaos local searching and optimization based on annealing strategy are cited, the parameters are optimized again within the limits of the approximate global optimal solution, the optimization is realized by means of combination of global and partial chaos searching, which can converge quickly to global optimal value. Finally, the third order system and discrete nonlinear system are simulated and compared with traditional method of fuzzy control. The results show that the new chaos optimal design method is superior to fuzzy control method, and that the control results are of high precision, with no overshoot and fast response.

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II

We consider a class of piecewise monotonically increasing functionsf on the unit intervalI. We want to determine the measures with maximal entropy for these transformations. In part I we construct a shift-space Σ + isomorphic to (I, f) generalizing the \-shift and another shift Σ M over an infinite alphabet, which is of finite type given by an infinite transition matrixM. Σ M has the same set of maximal measures as (I, f) and we are able to compute the maximal measures of maximal measures of. In part II we try to bring these results back to (I, f). There are only finitely many ergodic maximal measures for (I, f). The supports of two of them have at most finitely many points in common. If (I, f) is topologically transitive it has unique maximal measure.

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy

On intrinsic ergodicity of piecewise monotonic transformations with positive entropy