Cohomological equations and invariant distributions for minimal circle diffeomorphisms

We introduce the Mass Migration Process (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\ge 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence in particular zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We obtain necessary and sufficient conditions for the existence of translation-invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation-invariant, invariant probability measures. We also consider attractiveness properties (through couplings) and condensation phenomena for MMP. We illustrate our results on many examples; in particular, we prove the coexistence of both condensation and attractiveness in one of them.

In this paper, we show two related results on circle diffeomorphisms. The first result is on quasi-reducibility: for a Baire-dense set of α, for any diffeomorphism f of rotation number α, it is possible to accumulate Rα with a sequence , hn being a diffeomorphism. The second result is: for a Baire-dense set of α, given two commuting diffeomorphisms f and g, such that f has α for rotation number, it is possible to approach each of them by commuting diffeomorphisms fn and gn that are differentiably conjugated to rotations.In particular, it implies that if α is in this Baire-dense set, and if β is an irrational number such that (α, β) are not simultaneously Diophantine, then the set of commuting diffeomorphisms (f, g) with singular conjugacy, and with rotation numbers (α, β), respectively, is C∞-dense in the set of commuting diffeomorphisms with rotation numbers (α, β).

A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper we establish this principle for a large class of bicritical circle maps, which are $C^3$ circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo for the case of a single critical point. When combined with some recent papers, our main theorem implies $C^{1+\alpha}$ rigidity for real-analytic bicritical circle maps with rotation number of bounded type.

This paper continues, and in some sense completes, the study begun in [2]. We consider diffeomorphisms f of T = R/Z with irrational rotation number. A theorem of Denjoy states that if f ∈ C, it is conjugate to a rotation. The conjugating homeomorphims may be non-absolutely-continuous in which case the unique f-invariant measure on T is not equivalent to the Lebesgue measure μ. On the other hand, it is clear that f maps μ onto a measure equivalentto it, and it is this non-singular transformation that we proposie to study here. The natural classification theory for non-singular transformations on a Lebesgue space, which parallels the classification of von Neumann factors, is done modulo orbit-equivalence (also know as Dye-equivalence or weakequivalence). We list the main definitions and results concerning orbitequivalence, which we use, in section 1. The main contributors to the theory of orbit-equivalence are Hopf, Dye, Krieger, Connes and Woods, and we refer the reader to their works or to the forthcoming exposition [3] for the proofs. In section 2 we give a characterization of those transformationswhich are orbitequivalent to smooth diffeomorphisms of the circle. We show that every Cdiffeomorphism with irrational rotation number is of product type, and that every transformation of product type is orbit-equivalent to some C∞-diffeomorphism. Using the same ideas, one can easily show that any non-singular ergodic system is orbit-equivalent to some homeomorphism of T acting on the Legbesgue measure. Thus our results show the precise limitations imposed on the ergodic properties of mapping on T by smoothness conditions.

Consider a one-parameter family of circle diffeomorphisms which unfolds a saddle-node periodic orbit at the edge of an ‘Arnold tongue’. Recently it has been shown that homoclinic orbits of the saddle-node periodic points induce a ‘transition map’ which completely describes the smooth conjugacy classes of such maps and determines the universalities of the bifurcations resulting from the disappearance of the saddle-node periodic points. We show that after the bifurcation the relative density (measure) of parameter values corresponding to irrational rotation numbers is completely determined by the transition map and give a formula for this density. It turns out that this density is always less than 1 and generically greater than 0, with the exceptional cases having infinite co-dimension.

Let k ≥ 1 k \ge 1 be an integer and f f a holomorphic endomorphism of P k ( C ) \mathbb {P}^k (\mathbb {C}) of algebraic degree d ≥ 2 d\geq 2 . We introduce a dynamical volume dimension for ergodic f f -invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than ( k − 1 ) log ⁡ d (k-1)\log d , a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k = 1 k=1 , but depends on the dynamics of f f to incorporate the absence of an analogue of Koebe’s theorem and the non-conformality of holomorphic endomorphisms for k ≥ 2 k\geq 2 . If ν \nu is an ergodic f f -invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of ν \nu . As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k ≥ 1 k\geq 1 . Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

Let \mathcal{G}_{n} denote the space of n -generated marked groups. We prove that, for every n\ge 2 , there exist 2^{\aleph_0} non-atomic, \operatorname{Out}(F_{n}) -invariant, mixing probability measures on \mathcal{G}_{n} . On the other hand, there are non-empty closed subsets of \mathcal{G}_{n} that admit no \operatorname{Out}(F_{n}) -invariant probability measure. Acylindrical hyperbolicity of the group \operatorname{Aut}(F_{n}) plays a crucial role in the proof of both results. We also discuss model-theoretic implications of the existence of \operatorname{Out}(F_{n}) -invariant, ergodic probability measures on \mathcal{G}_{n} .

Let $f$ be an orientation preserving circle homeomorphism with asingle break point $x_b,$ i.e. with a jump in the first derivative$f'$ at the point $x_b,$ and with irrational rotation number$\rho=\rho_{f}.$ Suppose that $f$ satisfies the Katznelson andOrnstein smoothness conditions, i.e. $f'$ is absolutely continuouson $[x_b,x_b+1]$ and $f''(x)\in \mathbb{L}^{p}([0,1), d\ell)$ forsome $p>1$, where $\ell$ is Lebesque measure. We prove, that therenormalizations of $f$ are approximated by linear-fractionalfunctions in $\mathbb{C}^{1+L^{1}}$, that means, $f$ is approximatedin $C^{1}-$ norm and $f''$ is appoximated in $L^{1}-$ norm.Also it is shown, thatrenormalizations of circle diffeomorphisms with irrational rotationnumber satisfying the Katznelson and Ornstein smoothness conditionsare close to linear functions in $\mathbb{C}^{1+L^{1}}$- norm.

For a chain transitive flow, the averages produced by the invariant probability measures are approximated by multi-valued averages of an inflated flow. The results are applied to singularly perturbed differential equations with slow and fast trajectories. Here, the multi-valued character of the averaged system is due to the presence of many invariant probability measures, often caused by resonances. Hence, the inflation of a chain transitive flow gives information on the relevance of resonances. The Takens equation modelling strong elastic constraints is treated in detail.

Let $T_{f}$ : S1 → S1 be a circle homeomorphism with two break points ab, cb that means the derivative $Df$ of its lift $f\ :\ \mathbb{R}\rightarrow\mathbb{R}$ has discontinuities at the points ã b, ĉb, which are the representative points ofab, cb in the interval $[0,1)$, and irrationalrotation number ρf. Suppose that $Df$ is absolutely continuous on every connected intervalof the set [0,1]\{ãb, ĉb}, that DlogDf ∈ L1([0,1]) and the product of the jump ratios of $ Df $ at thebreak points is nontrivial, i.e.$\frac{Df_{-}(\tilde{a}_{b})}{Df_{+}(\tilde{a}_{b})}\frac{Df_{-}(\tilde{c}_{b})}{Df_{+}(\tilde{c}_{b})}\ne1$.We prove, that the unique Tf - invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure onS1.

We consider the spaceN ofC 2 twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps inN with nonlinearityk large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem).In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.

Letf be a “flat spot” circle map with irrational rotation number. Located at the edges of the flat spot are non-flat critical points (S: x→Axv,v≧1). First, we define scalings associated with the closest returns of the orbit of the critical point. Under the assumption that these scalings go to zero, we prove that the derivative of long iterates of the critical value can be expressed in the scalings. The asymptotic behavior of the derivatives and the scalings can then be calculated. We concentrate on the cases for which one can prove the above assumption. In particular, let one of the singularities be linear. These maps arise for example as the lower bound of the non-decreasing truncations of non-invertible bimodal circle maps. It follows that the derivatives grow at a sub-exponential rate.

It is shown that the generators of two discrete Heisenberg-Weyl groups with irrational rotation numbers θ and −1/θ generate the whole algebraB of operators onL 2(R). The natural action of the modular group inB is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed.

Approximation of invariant measures of a class of backward Euler-Maruyama scheme for stochastic functional differential equations

In this paper we prove under a general condition on the relation between metric and partial order of the underlying space an equivalence to stochastic order of probability measures. Secondly we show that the set of all probability measures which are placed stochastically between two fixed probability measures is compact in the topology of weak convergence. Using weak-convergence arguments we apply these two theorems to Markov processes, getting results on bounds and existence of their invariant probability measures. Finally, we give a transience criterion for some semi-Markov processes.