ALEXANDER TEPLYAEV

LIST OF PUBLICATIONS (ABSTRACTS)

[9] Gradients on fractals, to appear in the Journal of Functional Analysis.

In this paper I defined and studied a gradient on p.c.f. (post critically finite, or finitely ramified) fractals. I used Dirichlet (energy) form analysis developed for such fractals by Kigami. I considered both nondegenerate and degenerate harmonic structures (where a nonzero harmonic function can be identically zero on an open set). I showed that for any function of finite energy a weak gradient exists almost everywhere with respect to a certain measure $\ \nu$. Moreover, the energy itself is equal to the integral of a certain semi-norm of the gradient if the harmonic structure is weakly nondegenerate. This result was proved by Kusuoka in a different form. I proved that for a $\ C^1$-function on the Sierpinski gasket the gradient considered here and Kusuoka's gradient essentially coincide with a gradient considered by Kigami. The gradient at a junction point was studied by Strichartz in relation to the Taylor approximation on fractals. He also proved the existence of the gradient almost everywhere with respect to the Hausdorff (Bernoulli) measure for a function in the domain of the Laplacian. In this paper I obtained certain continuity properties of the gradient for a function in the domain of the Laplacian. As an appendix, I proved an estimate of the local energy of harmonic functions which was stated by Strichartz as a hypothesis.

[8] What is not in the domain of the Laplacian on a Sierpinski gasket type fractal (with O. Ben-Bassat and R. Strichartz). Journal of Functional Analysis, 166 (1999), 197-217

We considered the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of $\Delta$ if f is continuous and $\Delta f$ is defined as a continuous function. We showed that if f is a nonconstant function in the domain of $\Delta$, then f² is not in the domain of $\Delta$. We gave two proofs of this fact. The first is based on the analog of the pointwise identity $\Delta f^2 - 2 f \Delta f = \vert\nabla f\vert^2$, where we showed that $\vert\nabla f\vert^2$ does not exist as a continuous function. In fact we showed that a correct interpretation of $\Delta f^2$ is as a singular measure. The second proof is based on a dichotomy for the local behavior of a function in the domain of $\Delta$, at a junction point x₀ of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for |f(x)-f(x₀)| in terms of $d(x,x_0)^\beta$ for a certain value $\beta<1$, and in the nontypical case (vanishing the normal derivative) we have an upper bound with an exponent greater than 2. These results involved the study of products of random matrices and are related to the theory of abstract Dirichlet forms.

[7] Spectral analysis on infinite Sierpinski gaskets. Journal of Functional Analysis 159 (1998), 537-567.

In this paper I studied the spectral properties of the Laplacian on infinite Sierpinski gaskets. I proved that the Laplacian with the Neumann boundary condition has pure point spectrum. Moreover, the set of eigenfunctions with compact support is complete. The same is true if the infinite Sierpinski gasket has no boundary, but is false for the Laplacian with the Dirichlet boundary condition. In all these cases I described the spectrum of the Laplacian and all the eigenfunctions with compact support. To obtain these results, first I proved certain new formulae for eigenprojectors of the Laplacian on finite Sierpinski pre-gaskets. Then I proved that the spectrum of the discrete Laplacian on a Sierpinski lattice is pure point, and the eigenfunctions are localized.

[6] Pure point spectrum of the Laplacians on fractal graphs (with L. Malozemov). Journal of Functional Analysis 129 (1995), 390-405.

In this article we studied spectral properties of the Laplacians on self-similar infinite graphs (pre-fractals). Such graphs have been extensively investigated by physicists in connection with percolation clusters, quasicrystals, fractal electrical networks and porous materials. We defined a class of two point self-similar graphs and showed the Laplacian on such graphs has pure point spectrum with localized eigenfunctions. We also showed that these results hold for a Schrödinger operator with potential consisting of independent identically distributed random variables, that gives an Anderson localization result.

[5] A difference equation arising from the trigonometric moment problem having random reflection coefficients - an operator-theoretic approach (with J.S. Geronimo). Journal of Functional Analysis 123 (1994), 12-45.

In this work we applied the operator theoretic approach, developed in my paper [2], for doubly infinite (two-sided) analogs of equations which determine the orthogonal polynomials on the unit circle. Therefore we were able to use many results from the general theory of random operators in a situation where more classical methods of difference equations had been usually applied. We assumed that the elements of the operators involved are obtained from a stationary process and investigated the properties of the spectrum of these operators. In particular, we examined the relation between the Lyapunov exponent and the absolutely continuous component of the spectral measure.

[4] Continuous analogues of random polynomials that are orthogonal on the circle. Teoria Veroyatnostey i Primeneniya 39 (1994), 588-604; English translation in Theory Probab. Appl. 39 (1995), 476-489.

M. G. Krein described continuous analogs of polynomials, orthogonal on the unit circle. They are solutions of certain canonical system of differential equations and provide, in a sense, a generalization of the Fourier transform from $L^2(\Bbb R, \lambda)$ to $L^2(\Bbb R, \mu)$ where $\lambda$ is the Lebesgue measure on $\Bbb R$ and $\mu$ is a measure uniquely defined by these differential equations. Such equations are related to study of the one dimensional continuous Schrödinger equation. Also they can be used to solve an important factorization problem in the theory of analytic functions. I considered Krein's differential equations with random coefficients and obtained conditions that ensure the absolute continuity of the spectral measure $\mu$ with probability one. I gave estimations of the density of this measure and indicated consequences related to the deterministic case. Also I defined a notion of the spectral measure for a system of canonical stochastic differential equations. In particular, I constructed stochastic differential analogs of orthogonal polynomials on the circle and proved Krein's theorem and the almost sure absolute continuity of the corresponding spectral measure under some mild assumptions. In the appendix, I proved some new results on Krein's equations with nonrandom coefficients.

[3] Absolute continuity of the spectrum of random polynomials that are orthogonal on the circle and their continual analogues. Zapiski Nauchnyh Seminarov LOMI, 194 (1991), 170-173; English translation in Journal of Math. Sciences 75 (1995), 1982-1985.

In this article I generalized the results of my first paper for a situation when the random parameters are dependent. More presizely, I considered parameters which form a semimartingale difference sequence. Also I indicated how a similar construction can be done in the continuous setting, which was studied in my paper [4] in detail.

[2] The pure point spectrum of random orthogonal polynomials on the circle. Doklady Akad. Nauk SSSR 320 (1991), 49-53; English translation in Soviet Math. Dokl. 44 (1992), 407-411.

Here I studied a situation when the random parameters $\{a_n\}_{n\geqslant 0}$ considered in [1] are independent identically distributed. This random spectral problem is closely related to one dimensional Anderson localization. In particular, I was able to modify a method invented by B. Simon and T. Wolff in the case of an one-dimensional discrete Schrödinger operator for a certain unitary difference operator. The main result is the following theorem. Let the parameters $\{a_n\}_{n\geqslant 0}$ be i.i.d. random variables with absolutely continuous distribution, and let $\text{\bf E}\,{ \log}\,(1-\vert a\sb 0\vert )>-\infty$. Then with probability one the measure $\sigma$ corresponding to this sequence of parameters is discrete. Also I proved that with probability one the sequence $\vert\phi_n(z)\vert$, where $\phi_n(z)$ is the orthogonal polynomial of degree n, decreases exponentially fast if $\sigma\{z\} > 0$.

[1] Properties of polynomials that are orthogonal on the circle with random parameters. Zapiski Nauchnyh Seminarov LOMI 177 (1989), 157-162; English translation in Journal of Soviet Mathematics 61 (1992), 1931-1934.

In this article I considered polynomials that are orthogonal with respect to a measure $\sigma$ on the unit circle. Such polynomials play an important role in Approximation Theory, Statistics, Probability and Analysis. There exists a sequence of parameters $\{a_n\}_{n\geqslant 0}$ that generate these polynomials and uniquely determine the spectral measure $\sigma$. I considered the case when the parameters $\{a_n\}_{n\geqslant 0}$ are independent random variables and proved the following theorem. Let $\sum \sp {\infty }\sb {n=0}\text{\bf E}\,\vert a_n \vert \sp 2<\infty$, $\sum \sp {\infty }\sb {n=0}\vert \text{\bf E}\, a_n \vert <\infty$. Then with probability one: (a) the measure $\sigma$ is absolutely continuous, and (b) $\exp \{\gamma \vert \log ( \sigma ')\vert \, \log (\vert \log ( \sigma ')\vert )\}$ is integrable for some $\gamma >0$. Here $\sigma '$ is the density of $\sigma$ with respect to the Lebesgue measure on the circle. This is a generalization of a result of E. M. Nikishin.