{{C^m}} Positive Eigenvectors for Linear Operators Arising in the Computation of Hausdorff Dimension

We consider a broad class of linear Perron–Frobenius operators \({\Lambda:X \rightarrow X}\), where \({X}\) is a real Banach space of \({C^m}\) functions. We prove the existence of a strictly positive \({C^m}\) eigenvector \({v}\) with eigenvalue \({r=r(\Lambda) =}\) the spectral radius of \({\Lambda}\). We prove (see Theorem 6.5 in Sect. 6 of this paper) that \({r(\Lambda)}\) is an algebraically simple eigenvalue and that, if \({\sigma(\Lambda)}\) denotes the spectrum of the complexification of \({\Lambda,\sigma(\Lambda) \backslash \{r(\Lambda)\}\subseteq \{\zeta \in \mathbb{C} \big| |\zeta| \le r_*\}}\), where \({r_* < r(\Lambda)}\). Furthermore, if \({u \in X}\) is any strictly positive function, \({(\frac 1r \Lambda)^k(u) \rightarrow s_u v}\) as \({k \rightarrow \infty}\), where \({s_u > 0}\) and convergence is in the norm topology on \({X}\). In applications to the computation of Hausdorff dimension, one is given a parametrized family \({\Lambda_s,s > s_*}\), of such operators and one wants to determine the (unique) value \({s_0}\) such that \({r(\Lambda_{s_0})=1}\). In another paper (Falk and Nussbaum in C\({^{\rm m}}\) Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension, submitted) we prove that explicit estimates on the partial derivatives of the positive eigenvector \({v_s}\) of \({\Lambda_s}\) can be obtained and that this information can be used to give rigorous, sharp upper and lower bounds for \({s_0}\).     

A remark on hereditarily nonparadoxical sets

Call a set \({A \subseteq \mathbb {R}}\)paradoxical if there are disjoint \({A_0, A_1 \subseteq A}\) such that both \({A_0}\) and \({A_1}\) are equidecomposable with \({A}\) via countabbly many translations. \({X \subseteq \mathbb {R}}\) is hereditarily nonparadoxical if no uncountable subset of \({X}\) is paradoxical. Penconek raised the question if every hereditarily nonparadoxical set \({X \subseteq \mathbb {R}}\) is the union of countably many sets, each omitting nontrivial solutions of \({x - y = z - t}\). Nowik showed that the answer is ‘yes’, as long as \({|X| \leq \aleph_\omega}\). Here we show that consistently there exists a counterexample of cardinality \({\aleph_{\omega+1}}\) and it is also consistent that the continuum is arbitrarily large and Penconek’s statement holds for any \({X}\).     

Random approximation and the vertex index of convex bodies

We prove that there exists an absolute constant \({\alpha > 1}\) with the following property: if K is a convex body in \({{\mathbb R}^n}\) whose center of mass is at the origin, then a random subset \({X\subset K}\) of cardinality \({{\rm card}(X)=\lceil\alphan\rceil }\) satisfies with probability greater than \({1-e^{-c_1n}}\)

$$K\subseteq c_2n\, {\rm conv}(X),$$

where \({c_1, c_2 > 0}\) are absolute constants. As an application we show that the vertex index of any convex body K in \({{\mathbb R}^n}\) is bounded by \({c_3n^2}\), where \({c_3 > 0}\) is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.

     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

Cutoff on all Ramanujan graphs

We show that on every Ramanujan graph \({G}\), the simple random walk exhibits cutoff: when \({G}\) has \({n}\) vertices and degree \({d}\), the total-variation distance of the walk from the uniform distribution at time \({t=\frac{d}{d-2} \log_{d-1} n + s\sqrt{\log n}}\) is asymptotically \({{\mathbb{P}}(Z > c \, s)}\) where \({Z}\) is a standard normal variable and \({c=c(d)}\) is an explicit constant. Furthermore, for all \({1 \leq p \leq \infty}\), \({d}\)-regular Ramanujan graphs minimize the asymptotic \({L^p}\)-mixing time for SRW among all \({d}\)-regular graphs. Our proof also shows that, for every vertex \({x}\) in \({G}\) as above, its distance from \({n-o(n)}\) of the vertices is asymptotically \({\log_{d-1} n}\).     

Additive iterative roots of identity and Hamel bases

We describe a class of discontinuous additive functions \({a:X\to X}\) on a real topological vector space X such that \({a^n={\rm id}_X}\) and \({a({\mathcal{H}}){\setminus} {\mathcal{H}}\neq\emptyset}\) for every infinite set \({{\mathcal{H}}\subset X}\) of vectors linearly independent over \({\mathbb{Q}}\). We prove the density of the family of all such functions in the linear topological space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced on \({{\mathcal{A}}_X}\) by the Tychonoff topology of the space X^X. Moreover, we consider additive functions \({a\in{\mathcal{A}}_X}\) satisfying \({a^n={\rm id}_X}\) and \({a({\mathcal{H}})= {\mathcal{H}}}\) for some Hamel basis \({{\mathcal{H}}}\) of X. We show that the class of all such functions is also dense in \({{\mathcal{A}}_X}\). The method is based on decomposition theorems for linear endomorphisms.     

On the Convergence to Equilibrium of Unbounded Observables Under a Family of Intermittent Interval Maps

We consider a family \({\{T_{r}: [0, 1] \circlearrowleft \}_{r\in[0, 1]}}\) of Markov interval maps interpolating between the tent map \({T_{0}}\) and the Farey map \({T_{1}}\). Letting \({\mathcal{P}_{r}}\) denote the Perron–Frobenius operator of \({T_{r}}\), we show, for \({\beta \in [0, 1]}\) and \({\alpha \in (0, 1)}\), that the asymptotic behaviour of the iterates of \({\mathcal{P}_{r}}\) applied to observables with a singularity at \({\beta}\) of order \({\alpha}\) is dependent on the structure of the \({\omega}\)-limit set of \({\beta}\) with respect to \({T_{r}}\). The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.     

Cantor’s intersection theorem for <Emphasis Type="Italic">K</Emphasis>-metric spaces with a solid cone and a contraction principle

We establish an extension of Cantor’s intersection theorem for a \({K}\)-metric space (\({X, d}\)), where \({d}\) is a generalized metric taking values in a solid cone \({K}\) in a Banach space \({E}\). This generalizes a recent result of Alnafei, Radenovi? and Shahzad (2011) obtained for a \({K}\)-metric space over a solid strongly minihedral cone. Next we show that our Cantor’s theorem yields a special case of a generalization of Banach’s contraction principle given very recently by Cvetkovi? and Rako?evi? (2014): we assume that a mapping \({T}\) satisfies the condition “\({d(Tx, Ty) \preceq \Lambda (d(x, y))}\)” for \({x, y \in X}\), where \({\preceq}\) is a partial order induced by \({K}\), and \({\Lambda : E \rightarrow E}\) is a linear positive operator with the spectral radius less than one. We also obtain new characterizations of convergence in the sense of Huang and Zhang in a \({K}\)-metric space.     

Symmetry for a general class of overdetermined elliptic problems

Let \({\Omega}\) a bounded domain in \({\mathbb{R} ^N }\), and let \({u\in C^1 (\overline{\Omega})}\) a weak solution of the following overdetermined BVP: \({-\nabla (g(|\nabla u|)|\nabla u|^{-1}\nabla u)=f(|x|,u)}\), \({ u > 0 }\) in \({\Omega }\) and \({u=0, \ |\nabla u(x)|=\lambda (|x|)}\) on \({\partial \Omega }\), where \({g\in C([0,+\infty)\cap C^1 ((0,+\infty ) ) }\) with \({g(0)=0}\), \({g'(t) > 0}\) for \({t > 0}\), \({f\in C([0,+\infty ) \times [0, +\infty ) )}\), f is nonincreasing in \({|x|}\), \({\lambda \in C([0, +\infty )) }\) and \({\lambda }\) is positive and nondecreasing. We show that \({\Omega }\) is a ball and u satisfies some “local” kind of symmetry. The proof is based on the method of continuous Steiner symmetrization.     

Orthogonal projections of cubes and regular simplices into a plane

We show that for every \({k\ge 2}\) and \({n\ge k}\), there is an \({n}\)-dimensional unit cube in \({\mathbb{R}^n}\) which is mapped to a regular \({2k}\)-gon by an orthogonal projection in \({\mathbb{R}^n}\) onto a \({2}\)-dimensional subspace. Moreover, by increasing dimension \({n}\), arbitrary large regular \({2k}\)-gon can be obtained in such a way. On the other hand, for every \({m\ge 3}\) and \({n\ge m-1}\), there is an \({n}\)-dimensional regular simplex of unit edge in \({\mathbb{R}^n}\) which is mapped to a regular \({m}\)-gon by an orthogonal projection onto a plane. Moreover, contrary to the cube case, arbitrary small regular \({m}\)-gon can be obtained in such a way, by increasing dimension \({n}\).     

Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space

We show that if \({f\colon X\to Y}\) is a quasisymmetric mapping between Ahlfors regular spaces, then \({dim_H f(E)\leq dim_H E}\) for “almost every” bounded Ahlfors regular set \({E\subseteq X}\). If additionally, \({X}\) and \({Y}\) are Loewner spaces then \({dim_H f(E)=dim_H E}\) for “almost every" Ahlfors regular set \({E\subset X}\). The precise statements of these results are given in terms of Fuglede’s modulus of measures. As a corollary of these general theorems we show that if \({f}\) is a quasiconformal map of \({\mathbb{R}^N}\), \({N\geq 2}\), then for Lebesgue a.e. \({y\in\mathbb{R}^N}\) we have \({dim_H f(y+E) = dim_H E}\). A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if \({E \subset {\mathbb{R}}}\) is Ahlfors \({d}\)-regular, \({d < 1}\), then some component of \({f(E \times {\mathbb{R}})}\) has dimension at most \({2/(d+1)}\), and we construct examples to show this bound is sharp. In addition, we show there is a \({1}\)-dimensional set \({S\subseteq \mathbb R}\) and planar quasiconformal map \({f}\) such that \({f({\mathbb{R}} \times S)}\) contains no rectifiable sub-arcs. These results generalize work of Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and answer questions posed in Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and Capogna et al. (Mapping theory in metric spaces. http://aimpl.org/mappingmetric, 2016).     

On the angle sum of lines

What is the maximum of the sum of the pairwise (non-obtuse) angles formed by n lines in the Euclidean 3-space? This question was posed by Fejes Tóth in (Acta Math Acad Sci Hung 10:13–19, 1959). Fejes Tóth solved the problem for \({n \leq 6}\), and proved the asymptotic upper bound \({n^{2} \pi /5}\) as \({n \to \infty}\). He conjectured that the maximum is asymptotically equal to \({n^{2} \pi /6}\) as \({n \to \infty}\). The main result of this paper is an upper bound on the sum of the angles of n lines in the Euclidean 3-space that is asymptotically equal to \({3n^{2} \pi /16}\) as \({n \to \infty}\).     

Some homological properties of {T}-Lau product algebras

Let \({T}\) be a homomorphism from a Banach algebra \({B}\) to a Banach algebra \({A}\). The Cartesian product space \({A\times B}\) with \({T}\)-Lau multiplication and \({\ell^1}\)-norm becomes a new Banach algebra \({A\times _T B}\). We investigate the notions such as approximate amenability, pseudo amenability, \({\phi}\)-pseudo amenability, \({\phi}\)-biflatness and \({\phi}\)-biprojectivity for Banach algebra \({A\times_T B}\). We also present an example to show that approximate amenability of \({A}\) and \({B}\) is not stable for \({A\times _TB}\). Finally we characterize the double centralizer algebra of \({A\times _T B}\) and present an application of this characterization.     

Continued fraction normality is not preserved along arithmetic progressions

It is well known that if \({0.a_1a_2a_3\ldots}\) is the base-\({b}\) expansion of a number normal to base-\({b}\), then the numbers \({0.a_ka_{m+k}a_{2m+k}\ldots}\) for \({m\ge 2}\), \({k\ge 1}\) are all normal to base-\({b}\) as well. In contrast, given a continued fraction expansion \({\langle a_1,a_2,a_3,\ldots\rangle}\) that is normal (now with respect to the continued fraction expansion), we show that for any integers \({m\ge 2}\), \({k\ge 1}\), the continued fraction \({\langle a_k, a_{m+k},a_{2m+k},a_{3m+k},\ldots\rangle}\) will never be normal.     

On pluri-canonical systems of arithmetic surfaces

Let \({S}\) be a Dedekind scheme with perfect residue fields at closed points. Let \({f: X\rightarrow S}\) be a minimal regular arithmetic surface of fibre genus at least 2 and let \({f': X'\rightarrow S}\) be the canonical model of \({f}\). It is well known that \({\omega_{X'/S}}\) is relatively ample. In this paper we prove that \({\omega_{X'/S}^{\otimes n}}\) is relatively very ample for all \({n\geq 3}\).     

On Euler characteristics of Selmer groups for abelian varieties over global function fields

Let F be a global function field of characteristic \({p > 0}\), \({K/F}\) an \({\ell}\)-adic Lie extension (\({ \ell \neq p}\)), and \({A/F}\) an abelian variety. We provide Euler characteristic formulas for the Gal\({(K/F)}\)-module \({Sel_A(K)_\ell}\).     

The Holditch-Type Theorem for the Polar Moment of Inertia of the Orbit Curve in the Generalized Complex Plane

In this study, we first calculate the polar moment of inertia of orbit curves under one-parameter planar motion in the generalized complex plane \({{\mathbb{C}_p}}\) and then give the Holditch-type theorem for \({{\mathbb{C}_p}}\): When the fixed points \({X}\) and \({Y}\) on the moving plane \({{\mathbb{K}_p} \subset {\mathbb{C}_p}}\) trace the same curve \({k}\) with the polar moment of inertia \({{T_X}}\), the different point \({Z}\) on this line segment \({XY}\) traces another curve \({{k_Z}}\) with the polar moment of inertia \({{T_Z}}\) during the one-parameter planar motion in the fixed plane \({{\mathbb{K}'_p} \subset {\mathbb{C}_p}}\). Thus, we obtain that the difference between the polar moments of inertia of these curves \({( {{T_Z} - {T_X}} )}\) depends on the only the \({p}\)-distances of this points and \({p}\)-rotation angle of the motion, \({{T_X} - {T_Z} = {\delta _p}ab.}\)     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are \({H(\zeta) = U + U^{-1} + V + \zeta V^{-1}}\) and \({H_{m,n} = U + V + q^{-mn}U^{-m}V^{-n}}\), where \({U}\) and \({V}\) are self-adjoint Weyl operators satisfying \({UV = q^{2}VU}\) with \({q = {\rm e}^{{\rm i}\pi b^{2}}}\), \({b > 0}\) and \({\zeta > 0}\), \({m, n \in \mathbb{N}}\). We prove that \({H(\zeta)}\) and \({H_{m,n}}\) are self-adjoint operators with purely discrete spectrum on \({L^{2}(\mathbb{R})}\). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean \({\sum_{j\ge 1}(\lambda - \lambda_{j})_{+}}\) as \({\lambda \to \infty}\) and prove the Weyl law for the eigenvalue counting function \({N(\lambda)}\) for these operators, which imply that their inverses are of trace class.     

On the k-free values of the polynomial {xy^k + C}

Consider the polynomial \({f(x, y) = xy^k + C}\) for \({k \geq 2}\) and any nonzero integer constant C. We derive an asymptotic formula for the k-free values of \({f(x, y)}\) when \({x, y \leq H}\). We also prove a similar result for the k-free values of \({f(p, q)}\) when \({p, q \leq H}\) are primes, thus extending Erd?s’ conjecture for our specific polynomial. The strongest tool we use is a recent generalization of the determinant method due to Reuss.