Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$ ) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$ . We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$ .

Eigentime identity of the weighted (m,n)-flower networks

The eigentime identity for random walks on the weighted networks is the expected time for a walker going from a node to another node. Eigentime identity can be studied by the sum of reciprocals of all nonzero Laplacian eigenvalues on the weighted networks. In this paper, we study the weighted [Formula: see text]-flower networks with the weight factor [Formula: see text]. We divide the set of the nonzero Laplacian eigenvalues into three subsets according to the obtained characteristic polynomial. Then we obtain the analytic expression of the eigentime identity [Formula: see text] of the weighted [Formula: see text]-flower networks by using the characteristic polynomial of Laplacian and recurrent structure of Markov spectrum. We take [Formula: see text], [Formula: see text] as example, and show that the leading term of the eigentime identity on the weighted [Formula: see text]-flower networks obey superlinearly, linearly with the network size.

M∖L is Not Closed

We show that $1+3/\sqrt{2}$ is a point of the Lagrange spectrum $L$ that is accumulated by a sequence of elements of the complement $M\!\setminus\! L$ of the Lagrange spectrum in the Markov spectrum $M$. In particular, $M\!\setminus\! L$ is not a closed subset of $\mathbb{R}$, so that a question by T. Bousch has a negative answer.

EIGENTIME IDENTITIES OF FRACTAL FLOWER NETWORKS

The eigentime identity for random walks on networks is the expected time for a walker going from a node to another node. In this paper, our purpose is to calculate the eigentime identities of flower networks by using the characteristic polynomials of normalized Laplacian and recurrent structure of Markov spectrum.

A special case of effective equidistribution with explicit constants

An effective equidistribution with explicit constants for the isometry group of rational forms with signature (2,1) is proved. As an application we get an effective discreteness of the Markov spectrum.

HERMITIAN POINTS IN MARKOV SPECTRA

Let Hn be the upper half-space model of the n-dimensional hyperbolic space. For n=3, Hermitian points in the Markov spectrum of the extended Bianchi group Bd are introduced for any d. If ν is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H3 with diameter 1/ν, which depends on one continuous parameter. It is shown that ν2 ≤ |D|/24 for any imaginary quadratic field with discriminant D, whose ideal-class group contains no cyclic subgroup of order 4, and in many other cases. Similarly, in the case of n = 4, if ν is a Hermitian point in the Markov spectrum for SV(Z4), some discrete group of isometries of H4, then the corresponding set of extremal geodesics depends on two continuous parameters.