We extend the definition of involutivity to non-Lipschitz tangent subbundles using generalized functions. prove Frobenius Theorem with sharp regularity estimate when subbundle is log-Lipschitz: if $${\mathcal {V}}$$ a log-Lipschitz involutive rank r, then for any $$\varepsilon >0$$ , locally there homeomorphism $$\Phi (u,v)$$ such that ,\frac{\partial \Phi }{\partial u^1},\dots u^r}\in C^{0,1-\...

We consider continuous SL(2, R)-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an SO(2, R)-cocycle. Using this, we show that if a cocycle’s homotopy class does not display a certain obstruction to uniform hy...

We prove a number of homeomorphism results for nonlinear Fredholm maps of index zero and their perturbations. Moreover, we show that k-ball and k-set perturbations of these homeomorphisms are again homeomorphisms or that the corresponding equations are finitely solvable. Various generalized first Fredholm theorems are given and finite solvability of general ( odd ) Fredholm maps is also studied...

This work aims to present the concept of infra soft $b$-open sets(IS-b-open sets) as a generalized new class open sets(IS-open sets). We first investigate their basic properties and study behaviours under homeomorphism maps finite product spaces. Then, we establish some operators such interior, closure, limit boundary using IS-b-open sets IS-$b$-closed sets. The relationships between them are i...

We construct a family of involutions on the space $\mathfrak{gl}_n'(\mathbb C)$ $n\times n$ matrices with real eigenvalues interpolating complex conjugation and transpose. deduce from it stratified homeomorphism between symmetric eigenvalues, which restricts to analytic isomorphism individual $\mathrm{GL}_n(\mathbb R)$-adjoint orbits $\mathrm{O}_n(\mathbb C)$-adjoint orbits. also establish simi...

The notation and terminology used in this paper have been introduced in the following articles: [20], [9], [25], [7], [8], [4], [16], [24], [21], [19], [13], [18], [23], [1], [2], [10], [5], [17], [11], [3], [22], [14], [12], [6], [26], and [15]. We use the following convention: i, j, n denote natural numbers, f , g, h, k denote finite sequences of elements of R, and M , N denote non empty metr...