Vadim Kaloshin
Publications:
Koudjinan C. E., Kaloshin V.
On Some Invariants of Birkhoff Billiards Under Conjugacy
2022, vol. 27, no. 5, pp. 525537
Abstract
In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the ``normalized'' Mather's $\beta$function
are invariant under $C^\infty$conjugacies.
In contrast, we prove that any two elliptic billiard maps are $C^0$conjugate near their respective boundaries, and $C^\infty$conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar.

Buhovsky L., Kaloshin V.
Nonisometric Domains with the Same Marvizi–Melrose Invariants
2018, vol. 23, no. 1, pp. 5459
Abstract
For any strictly convex planar domain
$\Omega \subset \mathbb R^2$ with a $C^\infty$ boundary
one can associate an infinite sequence of spectral
invariants introduced by Marvizi–Merlose~\cite{MM}.
These invariants can generically be determined using
the spectrum of the Dirichlet problem of the Laplace operator.
A natural question asks if this collection is sufficient to determine
$\Omega$ up to isometry. In this paper we give
a counterexample, namely, we present two nonisometric
domains $\Omega$ and $\bar \Omega$ with the same collection
of Marvizi–Melrose invariants. Moreover, each domain
has countably many periodic orbits $\{S^n\}_{n \geqslant 1}$ (resp.
$\{ \bar S^n\}_{n \geqslant 1}$) of period going to infinity such that
$ S^n $ and $ \bar S^n $ have the same period and perimeter for each $ n $.
