Renormalizations of circle hoemomorphisms with a single break point

Conjugations between circle maps with a single break point

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.

Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As a consequence of this, we show that the space of real C∞-coboundaries of such a diffeomorphism is closed in C∞(T) if and only if its rotation number is Diophantine.

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.

In this paper, we prove pure point spectrum for a large class of Schrödinger operators over circle maps with conditions on the rotation number going beyond the Diophantine. More specifically, we develop the scheme to obtain pure point spectrum for Schrödinger operators with monotone bi-Lipschitz potentials over orientation-preserving circle homeomorphisms with Diophantine or weakly Liouville rotation number. The localization is uniform when the coupling constant is large enough.

Reverse Transcription Polymerase Chain Reaction (RT-PCR) using new designed primers pair for Heat Shock Protein70 homologue (HSP70h) of Olive leaf yellowing-associated virus revealed 667 amplified product of 10 olive accessions collected from various olive-growing regions in Tunisia. Amplicons were cloned and sequenced. The sequences were deposited in the international databases. Pairwise sequence comparisons among 10 Tunisian isolates along with a reference sequence (AJ440010) extracted from GenBank revealed a nucleotide identity of 86.06-99.40 and an amino acid similarity of 91.89-99.55. Sequence multiple alignments were searched for evidence of recombination using three methods, ie. Differences of Sums of Squares (DSS) implemented in TOPALi v2.5 software and Single Breakpoint (SBP) along with GARD, a genetic algorithm, both incorporated in HyPhy package. All used methods pointed out the presence of putative breaking points in partially sequenced HSP70h-coding gene. Since failing to account for recombination can mislead the phylogeny inference and can elevate the false positive error rate in positive selection assessment, the use of GARD resulted in the reconstruction of different phylogenies on the left as well as on the right sides of putative recombination breaking points, and the 11 accessions were distributed into at least three clusters compared to MEGA6 software which delineated only two clades. Nonetheless, by dividing the aligned sequences at breakpoints into separate sequence sets, MEGA6 delineated a clustering pattern different from the former two. As a result, recombination reshuffled the affiliation of the different accessions to the clusters. Analysis of selection pressures exerted on HSP70h encoded protein using different models (SLAC, IFEL, FEL, REL, PARRIS, FUBAR, MEME, GA Branch, and PRIME) taking into account recombination, and implemented in HyPhy package, revealed that it underwent predominantly purifying selection as confirmed by Tajima’s D, Fu and Li’s D and F tests, and SNAP algorithm. However, a few sites were also under positive selection as assessed by various models such as FEL, IFEL, REL, MEME, and PRIME.

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.

In this work, we consider two class $P$-homeomorphisms, $f$ and $g$, of the circle with break point singularities, that are differentiable maps except at some singular points where the derivative has a jump. Assume that they have the same irrational rotation number of bounded type and that the derivatives $\text{Df}$ and $\text{Dg}$ are absolutely continuous on every continuity interval of $\text{Df}$ and $\text{Dg}$, respectively. We show that if $f$ and $g$ are not break-equivalent, then any topological conjugating $h$ between $f$ and $g$ is a singular function, i.e., it is continuous on the circle, but $\text{Dh}(x)=0$ almost everywhere (a.e.) with respect to the Lebesgue measure. In particular, this result holds under some combinatorial assumptions on the jumps at break points. It also generalizes previous results obtained for one and two break points and complements that of Cunha–Smania which was established for break equivalence.

As was known to H. Poincare, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms; that is, torus homeomorphisms isotopic to the identity for which the rotation set is a point with rationally independent irrational coordinates.

Let $T \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, be a circle homeomorphism with one break point $x_{b}$, at which $ T'(x) $ has a discontinuity of the first kind and both one-sided derivatives at the point $x_{b} $ are strictly positive. Assume that the rotation number $\rho_{T}$ is irrational and its decomposition into a continued fraction beginning from a certain place coincides with the golden mean, i.e., $\rho_{T}=[m_{1}, m_{2}, \ldots, m_{l}, \, m_{l + 1}, \ldots] $, $ m_{s} = 1$, $s> l> 0$. Since the rotation number is irrational, the map $ T $ is strictly ergodic, that is, possesses a unique probability invariant measure $\mu_{T}$. A.A. Dzhalilov and K.M. Khanin proved that the probability invariant measure $ \mu_{G} $ of any circle homeomorphism $ G \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0$, with one break point $ x_{b} $ and the irrational rotation number $ \rho_{G} $ is singular with respect to the Lebesgue measure $ \lambda $ on the circle, i.e., there is a measurable subset of $ A \subset S^{1} $ such that $ \mu_ {G} (A) = 1 $ and $ \lambda (A) = 0$. We will construct a thermodynamic formalism for homeomorphisms $ T_{b} \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, with one break at the point $ x_{b} $ and rotation number equal to the golden mean, i.e., $ \rho_{T}:= \frac {\sqrt{5} -1}{2} $. Using the constructed thermodynamic formalism, we study the exponents of singularity of the invariant measure $ \mu_{T} $ of homeomorphism $ T $.

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.

In this paper we investigate maps of the two-torus $\mathbb{T}^{2}$ of the form $T(x,y)=(x+\unicode[STIX]{x1D714},g(x)+f(y))$ for Diophantine $\unicode[STIX]{x1D714}\in \mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\rightarrow \mathbb{T}$, where each $g$ is strictly monotone and of degree 2 and each $f$ is an orientation-preserving circle homeomorphism. For our class of $f$ and $g$, we show that $T$ is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two $T$-invariant graphs. One of the graphs is a strange non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in $\mathbb{T}^{2}$. Only a low-regularity assumption (Lipschitz) is needed on the maps $f$ and $g$, and the results are robust with respect to Lipschitz-small perturbations of $f$ and $g$.

We define anoscillating sequence, an important example of which is generated by the Möbius function in number theory. We also define aminimally mean attractable (MMA) flowand aminimally mean-L-stable (MMLS) flow. One of the main results is that any oscillating sequence is linearly disjoint from all MMA and MMLS flows. In particular, this confirms Sarnak’s conjecture for all MMA and MMLS flows. We provide several examples of flows that are MMA and MMLS. These examples include flows defined by all$p$-adic polynomials of integral coefficients, all$p$-adic rational maps with good reduction, all automorphisms of the$2$-torus with zero topological entropy, all diagonalizable affine maps of the$2$-torus with zero topological entropy, all Feigenbaum maps, and all orientation-preserving circle homeomorphisms.