Georgia Institute of Technology College of Sciences

Proving positivity of the top Lyapunov exponent ($\lambda_1$​) and obtaining parameter-dependent lower bounds is an interesting and challenging problem for SDEs (stochastic differential equations). We outline methods to obtain lower bounds and establish positivity of $\lambda_1$​ for certain SDEs, combining the coordinate rescaling framework of Pinsky–Wihstutz (1988) for nilpotent linear It\^{o} systems with Fisher information formulas for Lyapunov exponents introduced by J. Bedrossian, A. Blumenthal, and S. Punshon-Smith (2022). This approach uses hypoellipticity and regularity of 2nd order linear PDEs.

We apply these techniques to a 2-D toy SDE to obtain positive lower bounds and small-noise scaling (in terms of noise parameter $\sigma$) for $\lambda_1$​ as $\sigma \to 0$. These techniques avoid computing the stationary density explicitly, using only qualitative regularity of the limiting stationary density coming from hypoellipticity. We also present how a similar approach yields shear-induced chaos for a stochastically driven limit cycle closely related to the Hopf normal form with additive noise, by proving $\lambda_1 > 0$. Finally, we briefly discuss additional SDEs where we believe variants of these ideas may yield positive lower bounds on $\lambda_1$. This work is part of ongoing joint work with Samuel Punshon-Smith.
 

In a recent joint work with J. Buzzi, Y. Shi, and J. Yang, given a diffeomorphism preserving a one-dimensional expanding foliation $\mathcal F$ with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with controlled Jacobian and a Gibbs property.

We then prove that for any measure of maximal $\mathcal F$-entropy, its conditional measures on each leaf must be equivalent to the reference measures.

When the reference measures are equivalent to the leafwise Lebesgue measure, we prove that the log-determinant of $f$ must be cohomologous to a constant.

We will consider several applications, including the strong and center foliations of Anosov diffeomorphisms, factor over Anosov diffeomorphisms, and perturbations of the time-one map of geodesic flows on surfaces with negative curvature. We will also discuss several conjectures on the unique ergodicity and (exponential) equidistribution for the strong unstable foliations of Anosov systems. 

Zoom link: https://gatech.zoom.us/j/92005780980?pwd=ptlx7KdBAbHI3DTvv6V7fjFn27LDaE.1

Meeting ID: 920 0578 0980
Passcode: 604975

Consider a sequence of independent and identically distributed SL(2, R) matrices. There are several classical results by Le Page, Tutubalin, Benoist, Quint, and others that establish various forms of the central limit theorem for the products of such matrices. I will talk about a recent joint work with Anton Gorodetski and Victor Kleptsyn, where we generalize these results to the non-stationary case. Specifically, we prove that the properly shifted and normalized logarithm of the norm of a product of independent (but not necessarily identically distributed) SL(2, R) matrices converges to the standard normal distribution under natural assumptions. A key component of our proof is the regularity of the distribution of the unstable vector associated with these products.

The classical Wiener-Wintner Theorem says that for all measure preserving systems, and bounded functions f, there is a set of full measure so that the averages below converge for all continuous functions  g from the circle (R/Z)  to the complex numbers.

N^{-1} \sum_{n=1}^N  g( \pi n) f(T^n). 

We extend this result to averages over the prime integers. The proof uses structure of measure preserving systems, higher order Fourier analysis, and the Heath-Brown approximate to the von Mangoldt function.  A key result is a surprisingly small  Gowers norm estimate for the Heath-Brown approximate with fixed height.  

Joint work with  Y. Chen, A. Fragkos,  J. Fornal, B. Krause, and H. Mousavi.