Hitting functions for mixed partitions

We introduce a renormalization procedure which allows us to study in a unified and concise way different properties of the irrational rotations on the unit circle $$\beta \mapsto \{\alpha + \beta\}$$ , $$\alpha \in {\mathbb{R}}\!\setminus\! {\mathbb{Q}}$$ . In particular, we obtain sharp results for the diffusion of the walk on $${\mathbb{Z}}$$ generated by the location of points of the sequence {n α + β} on a binary partition of the unit interval. Finally, we give some applications of our method.

We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by $α$ where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $β$. When $α$ is badly approximable and $β$ is badly approximable with respect to $α$, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D.Dolgopyat and O.Sarig), namely we show that for any fixed initial point, the occupancy random variables, suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when $α$ is quadratic irrational, $β$ is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig (Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use renormalization, but in order to treat irrational values of $β$, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}|<+\infty ,\end{eqnarray}$$ where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.

Recently Li and Yorke introduced the notion of chaos for mappings from the class C 0 ( I , I ) {C^0}(I,I) , where I I is a compact real interval. In the present paper we give a characterization of the class M ⊂ C 0 ( I , I ) M \subset {C^0}(I,I) of mappings chaotic in this sense. As is well known, M M contains the mappings of positive topological entropy. We show that M M contains also certain (but not all) mappings that have both zero topological entropy and infinite attractors. Moreover, we show that the complement of M M consists of maps that have only trajectories approximate by cycles. Finally, it turns out that the original Li and Yorke notion of chaos can be replaced by (an equivalent notion of) δ \delta -chaos, distinguishable on a certain level δ > 0 \delta > 0 .

The critical behavior of the effective conductivity σ* of the random resistor network in Zd, near its percolation threshold, is considered. The network has bonds assigned the conductivities 1 and ε ≥ 0 in the volume fractions p and 1−p. Motivated by the statistical mechanics of an Ising ferromagnet at temperature T in a field H, we introduce a partition function and free energy for the resistor network, which establishes a direct correspondence between the two problems. In particular, we show that the free energies for the resistor network and the Ising model both have the same type of integral representation, which has the interpretation of the complex potential due to a charge distribution on [0, 1] in the s=1/(1−ε) plane for the resistor network, and on the unit circle in the z=exp(−2βH) plane for the ferromagnet. Based on this correspondence, we develop a Yang–Lee picture of the onset of nonanalytic behavior of the effective conductivity σ*, so that the percolation threshold p=pc is characterized as an accumulation point of zeros of the partition function in the complex p-plane as ε → 0. A scheme is developed to find the locations of a certain sequence of zeros in the p-plane, which is based on Padé approximation of a perturbation expansion of σ*(p,ε) around a homogeneous medium (ε=1). Furthermore, for ε ≳ 0, we construct a domain 𝒟ε containing [0, 1] in the p-plane in which σ*(p,ε) is analytic, and which collapses as ε → 0. The explicit construction of this domain allows us to obtain a lower bound on the size of the gap in zeros of the partition function around the percolation threshold p=pc, which leads to the gap exponent inequality Δ≤1.

Let n be a probability measure on R. We say that a a-finite measure \ is irreducible with respect to a if there does not exist a Borel set A with a(A).u(Ac) > 0 such that fAn(A' -x) \(dx) = 0.It is well known that the Lebesgue measure m(dx) is irreducible with respect to any discrete measure whose support is R. We prove that every absolutely continuous measure is irreducible with respect to any probability measure whose support is R and give an application of this fact to random walks on the unit circle.

Bounded functions of vanishing mean oscillation on compact metric spaces

We present both analytical and numerical results on the position of partition function zeros on the complex magnetic field plane of the q=2 state (Ising) and the q=3 state Potts model defined on phi(3) Feynman diagrams (thin random graphs). Our analytic results are based on the ideas of destructive interference of coexisting phases and low temperature expansions. For the case of the Ising model, an argument based on a symmetry of the saddle point equations leads us to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs. For the q=3 state Potts model, our perturbative results indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic results are confirmed by finite lattice numerical calculations.

Consider a uniformly chosen element $X_n$ of the $n$-fold wreath product $\Gamma_n = G \wr G \wr \cdots \wr G$, where $G$ is a finite permutation group acting transitively on some set of size $s$. The eigenvalues of $X_n$ in the natural $s^n$-dimensional permutation representation (the composition representation) are investigated by considering the random measure $\Xi_n$ on the unit circle that assigns mass $1$ to each eigenvalue. It is shown that if $f$ is a trigonometric polynomial, then $\lim_{n \rightarrow \infty} P\{\int f d\Xi_n \ne s^n \int f d\lambda\}=0$, where $\lambda$ is normalised Lebesgue measure on the unit circle. In particular, $s^{-n} \Xi_n$ converges weakly in probability to $\lambda$ as $n \rightarrow \infty$. For a large class of test functions $f$ with non-terminating Fourier expansions, it is shown that there exists a constant $c$ and a non-zero random variable $W$ (both depending on $f$) such that $c^{-n} \int f d\Xi_n$ converges in distribution as $n \rightarrow \infty$ to $W$. These results have applications to Sylow $p$-groups of symmetric groups and autmorphism groups of regular rooted trees.

We examine dynamical systems which are ‘nonchaotic’ on a big (in the sense of Lebesgue measure) set in each neighbourhood of a fixed point $x_{0}$, that is, the entropy of this system is zero on a set for which $x_{0}$ is a density point. Considerations connected with this family of functions are linked with functions attracting positive entropy at $x_{0}$, that is, each mapping sufficiently close to the function has positive entropy on each neighbourhood of $x_{0}$.

Spontaneous supersymmetry breaking in matrix models from the viewpoints of localization and Nicolai mapping

A Chaotic Function with a Scrambled Set of Positive Lebesgue Measure

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).

Let W = U n I W, be an F0 subset of the unit circle of measure O and let {qn), n > 1, be a decreasing sequence with q, 1, there is such an H of norm 1 for which lim,rH(re' w) = I /n. The proof of the theorem depends on the existence of a special collection of closed sets {SA), A > 1, real, for which the function h, defined by h(x) = an + [(inf{AIx E S)) n](a+a), a= In qn, is such that the function H(w) = exp(1/2r) f [(w + eau)/(eiu w)]h(u) du has the required properties. Some of the techniques used are similar to those developed in an earlier paper [1]. NOTATION. Let I be a fixed interval on the real line. For each Lebesgue measurable subset E of I, JEl denotes its Lebesgue measure. E' denotes the complement of E in I. The letters k, m, n and s when they occur as symbols denote natural numbers. C denotes the unit circle and D the unit disc. Purpose. The aim of this work is to prove the following theorem. THEOREM. Let W = U n= I W, be an F,J subset of the unit circle of measure 0 and let {q }, n > 1 be a strictly decreasing sequence with q1 1, there is such an H of norm 1 for which limr 1 H(rei'.) = l/n. Preliminary lemmas. Three lemmas precede the proof of the theorem. In addition, frequent reference is made to a lemma of Lusin-Menchoff. For convenience this lemma is stated first. LUSIN-MENCHOFF LEMMA. Let M1 be an arbitrary nonempty measurable subset of (0, 1). Let M2 be a closed subset of M1 consisting only of points of Received by the editors February 2, 1977. AMS (MOS) subject classifications (1970). Primary 26A24, 26A30, 30A88.

Almost periodic Verblunsky coefficients and reproducing kernels on Riemann surfaces