In this short note, we investigate noninvertible stochastic dynamical systems on the unit interval [ 0 , 1 ] . We provide a handy condition for unique ergodicity for systems that are injective in mean. On the other hand, we give concrete examples where unique ergodicity fails.
For any unbranched double covering of a compact Riemann surface, we study the associated character varieties that are unitary in the global sense, which we call GL n ⋊ 〈 σ 〉 -character varieties. We introduce k > 0 punctures on the surface, restrict the monodromies around the punctures to generic semi-simple conjugacy classes in GL n , and compute the E-polynomials of these character varieties using the character table of GL n ( q ) . The result is expressed as the inner product of certain symmetric functions. We are then led to a conjectural formula for the mixed Hodge polynomial, which is built out of (modified) Macdonald polynomials, their self-pairings, and self-pairings of wreath Macdonald polynomials.
In this article, we investigate the Gevrey regularity and the summability of the formal solutions in time of some inhomogeneous linear systems of partial differential equations with analytic coefficients at the origin of ℂ 2 . Given such a system, we first prove that the Gevrey regularity of its formal solutions follows a noteworthy dichotomy with respect to the s -Gevrey regularity of the inhomogeneity: for s ≥ s 0 with s 0 a nonnegative rational number entirely determined by a convenient Newton polygon attached to the given system, the solutions inherit the Gevrey regularity of the inhomogeneity; for s < s 0 , if any exists, the solutions keep the s 0 -Gevrey regularity defined by the structure of the initial system. Then, assuming s 0 > 0 and some convenient additional assumptions on the system, we give a necessary and sufficient condition for the 1 / s 0 -summability of the formal solutions in a given direction. Several examples illustrate these results.
Peral–Miyachi’s celebrated theorem states that the operator ( I - Δ ) - α 2 exp ( i - Δ ) is bounded on L p ( ℝ d ) if and only if α ≥ s p : = ( d - 1 ) 1 p - 1 2 . We extend this result to operators of the form ℒ = - ∑ j = 1 d a j + d ∂ j a j ∂ j , such that, for j = 1 , ⋯ , d , the functions a j and a j + d only depend on x j , are bounded above and below, but are merely Lipschitz continuous. This is below the C 1 , 1 regularity that is required in general situations. We construct spaces on which exp ( i ℒ ) is bounded by lifting L p functions to tent spaces, using wave packets adapted to the coefficients. The result then follows from Sobolev embedding properties of these spaces.
We study the sheaf of the locally square integrable holomorphic section of a vector bundle with semi-positive curved singular Hermitian metric. We confirm the coherence when its induced determinant metric has analytic singularities.
