On calibrated and separating sub-actions

We consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Hölder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability measures. If ā denotes the minimizing ergodic value of A, a sub-action u for A is by definition a continuous function such that A ≥ u ○ σ ? u + ā. We call contact locus of u with respect to A the subset of Σ where A = u ○ σ ? u + ā. A calibrated sub-action u gives the possibility to construct, for any point x ε Σ, backward orbits in the contact locus of u. In the opposite direction, a separating sub-action gives the smallest contact locus of A, that we call Ω(A), the set of non-wandering points with respect to A.We prove that separating sub-actions are generic among Hölder sub-actions. We also prove that, under certain conditions on Ω(A), any calibrated sub-action is of the form u(x) = u(x _i ) + h _A (x _i , x) for some x _i ∈ Ω(A), where h _A (x, y) denotes the Peierls barrier of A. We present the proofs in the holonomic optimization model, a formalism which allows to take into account a two-sided transitive subshift of finite type \((\hat \Sigma , \hat \sigma )\).     

A note about topologically transitive cylindrical cascades

Letσ: Σ → Σ be a topologically mixing shift of finite type. Letβ: Σ → ? be a continuous function, and σ _β be the skew-product ofσ byβ. Assume that σ _β has a positive semi-orbit that reaches any upper height, and any lower height. Then, arbitrarilyC ⁰-close toβ there exists a Hölder mapβ′: Σ → ? such that the skew-product $\sigma _{\beta ^\prime } $ ofσ byβ′ is topologically transitive.     

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

We give sufficient conditions for a shift space (Σ, σ) to be intrinsically ergodic, along with sufficient conditions for every subshift factor of Σ to be intrinsically ergodic. As an application, we show that every subshift factor of a β-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.     

Dimension theory of iterated function systems

Let {S_i} be an iterated function system (IFS) on ?^d with attractor K. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, ??}. We define the projection entropy function h_π on the space of invariant measures on Σ associated with the coding map π : Σ → K and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (respectively, the direct product of finitely many conformal IFSs), without any separation condition, the projection of an ergodic measure under π is always exactly dimensional and its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (respectively, the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFSs, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. © 2008 Wiley Periodicals, Inc.     

A representation theorem for maxitive measures

A maxitive measure is a nonnegative function η on a σ-algebra Σ and such that η(U_j A_j ) = sup_j η(A_j) for all countable disjoint families of sets (A_j) in Σ. A representation theorem for such measures is established, and next applied to represent Köthe function M-spaces as L_∞-spaces.     

Negative entropy,zero temperature and Markov chains on the interval

We consider ergodic optimization for the shift map on the modified Bernoulli space σ: [0, 1]^? → [0, 1]^?, where [0, 1] is the unit closed interval, and the potential A: [0, 1]^? → ? considered depends on the two first coordinates of [0, 1]^?. We are interested in finding stationary Markov probabilities µ_∞ on [0, 1]^? that maximize the value ∫ Adµ, among all stationary (i.e. σ-invariant) probabilities µ on [0, 1]^?. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential A. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities µ _β which weakly converges to µ_∞. The probabilities µ _β are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure in Thermodynamic Formalism. As the potential A depends only on the first two coordinates, instead of the probability µ on [0, 1]^?, we can consider its projection ν on [0, 1]². We look at the problem in both ways. If µ_∞ is the maximizing probability on [0, 1]^?, we also have that its projection ν _∞ is maximizing for A. The hypothesis about stationarity on the maximization problem can also be seen as a transhipment problem. Under the hypothesis of A being C ² and the twist condition, that is,

$\frac{{\partial ^2 A}}{{\partial x\partial y}}(x,y) \ne 0, for all (x,y) \in [0,1]^2 ,$

we show the graph property of the maximizing probability ν on [0, 1]². Moreover, the graph is monotonous. An important result we get is: the maximizing probability is unique generically in Mañé’s sense. Finally, we exhibit a separating sub-action for A.

     

On marginal distributions and isomorphisms of stationary processes

LetT be an invertible ergodic aperiodic measure preserving transformation of a Lebesgue space, letA be a finite alphabet, and let π be a probability measure onA ⁿ which admits a mixing shift-invariant measureμ _π onΩ=A ^? such that the marginals of anyn successive coordinates are π and the entropyh(T) ofT is smaller than the entropy of the shift in (Ω,μ _π). Then there exists a shift invariant measure ν_π in Ω which also has marginals π and for whichT is isomorphic to the shift in (Ω, ν_π). This contains Krieger's finite generator theorem and strengthens the measure theoretic part of his approximation theorem for shift-invariant measures by showing that the preassigned marginal π can not only be achieved up to an ε>0 but exactly. Our result also contains an as yet unpublished theorem of Krieger, which says thatT can be embedded in an arbitrary mixing subshift of finite type, as long as the entropy of the subshift under the measure with maximal entropy exceeds that ofT. In the final section we show that the method can be extended to yield also exact marginals for the generator in the Jewett-Krieger theorem, i.e.T is shown to be isomorphic to a shift in (Ω, ν_π) where ν_π has exact marginals π and the shift is uniquely ergodic on the support of ν_π.     

Operateurs essentiellement reguliers dans les espaces de Banach

Given a closed operatorA acting in a Banach spaceX, we define the regular (respectively the essentialy regular) spectrum σ _r (A) (respectively σ _e,r (A)) ofA. We prove that σ _r (A) and σ _e,r (A) are a closed subsets of the classical spectrum σ(A) ofA. Morever ifA is bounded we prove that σ _r (A) and σ _e,r (A)) satisfies the spectral mapping theorem.     

Topologically inseparable functions I: Finitary case

Given a finite set A and a distinguished function f: A → A, we study the set of all functions g: A → A that are continuous for all topologies for which f is continuous. The main result is a characterization of the functions f such that this set is trivial, that is, contains only the constant functions and the iterates of f.     

Topological dynamics for multidimensional perturbations of maps with covering relations and Liapunov condition

In this paper, we study topological dynamics of high-dimensional systems which are perturbed from a continuous map on Rm×Rk of the form (f(x),g(x,y)). Assume that f has covering relations determined by a transition matrix A. If g is locally trapping, we show that any small C⁰ perturbed system has a compact positively invariant set restricted to which the system is topologically semi-conjugate to the one-sided subshift of finite type induced by A. In addition, if the covering relations satisfy a strong Liapunov condition and g is a contraction, we show that any small C¹ perturbed homeomorphism has a compact invariant set restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by A. Some other results about multidimensional perturbations of f are also obtained. The strong Liapunov condition for covering relations is adapted with modification from the cone condition in Zgliczyński (2009) [11]. Our results extend those in Juang et al. (2008) [1], Li et al. (2008) [2], Li and Malkin (2006) [3], Misiurewicz and Zgliczyński (2001) [4] by considering a larger class of maps f and their multidimensional perturbations, and by concluding conjugacy rather than entropy. Our results are applicable to both the logistic and Hénon families.     

Construction of Fuzzy σ-algebras using triangular norms

Generalizing the definitions given by the author [Fuzzy Sets and Systems4 (1980), 83–93] we introduce and study T-fuzzy σ-algebras, T being any triangular norm. The main result is that for a large class of triangular norms each T-fuzzy σ-algebra is generated, i.e., consists of all functions μ:X → [0, 1] being measurable with respect to some σ-algebra on X.     

Topologically inseparable functions II: Infinitary case

Given a set A and a function A: A → A, we study the set of all functions g: A → A that are continuous for all topologies for which f continuous. We prove that in a sense to be made precise in the text, for any essentially infinitary function f, any non-constant such g equals f ⁿ , for some n∈ ?. We also prove a similar result for the clone of n-ary functions from A ⁿ → A.     

On C*-Algebras from Interval Maps

Given a unimodal interval map f, we construct partial isometries acting on Hilbert spaces associated to the orbit of each point. Then we prove that such partial isometries give rise to representations of a C*-algebra associated to the subshift encoding the kneading sequence of the critical point. This construction has the advantage of incorporating maps with a non necessarily Markov partition (e.g. Fibonacci unimodal map). If we are indeed in the presence of a finite Markov partition, then we prove that these new representations coincide with the (previously considered by the authors) representations arising from the Cuntz–Krieger algebra of the underlying (finite) transition matrix.     

Simple equations on real intervals

If A is an absolute retract in the class of metric spaces, and if Σ is a consistent set of simple equations, then A is compatible with Σ, i.e., there are continuous operations on A that model Σ.     

Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type

In this article we study how a subshift can simulate another one, where the notion of simulation is given by operations on subshifts inspired by the dynamical systems theory (factor, projective subaction …). There exists a correspondence between the notion of simulation and the set of forbidden patterns. The main result of this paper states that any effective subshift of dimension d—that is a subshift whose set of forbidden patterns can be generated by a Turing machine—can be obtained by applying dynamical operations on a subshift of finite type of dimension d+1—a subshift that can be defined by a finite set of forbidden patterns. This result improves Hochman’s (Invent. Math. 176(1):131–167, 2009).     

An existence theorem for probability measures invariant under a Markov kernel

LetP be a Markov kernel defined on a measurable space (X,A). A probability measure μ onA is said to beP-invariant if μ(A=∫P(x,A)dμ(x) for allA ∈A∈A. In this note we prove a criterion for the existence ofP-invariant probabilities which is, in particular, a substantial generalization of a classical theorem due to Oxtoby and Ulam ([5]). As another consequence of our main result, it is shown that every pseudocompact topological space admits aP-invariant Baire probability measure for any Feller kernelP.     

The Myhill property for strongly irreducible subshifts over amenable groups

Let G be an amenable group and let A be a finite set. We prove that if X ? A ^G is a strongly irreducible subshift then X has the Myhill property, that is, every pre-injective cellular automaton ?? : X ?? X is surjective.     

On a dual submanifold homomorphism

In this paper we develop a technique to study the homomorphisma: MU _* (B U₁)→M U_*?2 (B U₁) defined by assigning to the class off: M→B U ₁ the class off _oi: N→B U₁, wherei: N→M is the submanifold dual tof*(γ₁)?f*(γ₁), and γ₁→B U is the 3 universal line boundle. So that we can present a (σ_n), where σ_nis the class of the classifying map of the canonical line boundle overC P ⁿ, in terms of the σ_i’s and chosen generators of Π_★(M U).     

Hyperbolicity of the invariant sets for the real polynomial maps

In this paper, the conditions under which there exits a uniformly hyperbolic invariant set for the map f_a(x) = ag(x) are studied, where a is a real parameter, and g(x) is a monic real-coefficient polynomial. It is shown that for certain parameter regions, the map has a uniformly hyperbolic invariant set on which it is topologically conjugate to the one-sided subshift of finite type for A, where ∣a∣ is sufficiently large, A is an eventually positive transition matrix, and g has at least two different real zeros or only one real zero. Further, it is proved that there exists an invariant set on which the map is topologically semiconjugate to the one-sided subshift of finite type for a particular irreducible transition matrix under certain conditions, and one type of these maps is not hyperbolic on the invariant set.     

Random walks on the triangular prism and other vertex-transitive graphs

In this paper we consider Markov chains of the following type: the state space is the set of vertices of a connected, regular graph, and for each vertex transitions are to the adjacent vertices, which equal probabilities. The proof is given that the mean first-passage matrix F of such a Markov chain is symmetric, when the underlying graph is vertex-transitive. Hence, we can apply results from a previous paper, in which we investigated general, finite, ergodic Markov chains, with the property F= F^T.