We study families of Axiom A skew products with the transversality condition and in particular, the Hausdorff dimension of their fibers, by using thermodynamical formalism. The maps we consider can be non-invertible, and the study of their dynamics is influenced greatly by this fact. We introduce and employ probability measures (constructed from equilibrium measures on the natural extension), w...

2 The Riordan skew algebra K[[x]] ⋊ M of formal power series under multiplication and substitution 3 2.1 Basics on formal power series . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Skew algebra of formal power series under substitution . . . . . . . . . . 4 2.2.1 Right-distributive algebras . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Substitution of formal power series . . . . ....

Nisan (STOC 1991) exhibited a polynomial which is computable by linear-size non-commutative circuits but requires exponential-size non-commutative algebraic branching programs. Nisan’s hard polynomial is in fact computable by linear-size “skew circuits.” Skew circuits are circuits where every multiplication gate has the property that all but one of its children is an input variable or a scalar....

We propose an approach to the attractors of skew products that tries to avoid unnecessary structures on the base space and rejects the assumption on the invariance of an attractor. When nonivertible maps in the base are allowed, one can encounter the mystery of the vanishing attractor. In the second part of the paper, we show that if the fiber maps are concave interval maps then contraction in ...

We study two register arithmetic computation and skew arithmetic circuits. Our main results are the following: • For commutative computations, we show that an exponential circuit size lower bound for a model of 2-register straight-line programs (SLPs) which is a universal model of computation (unlike width-2 algebraic branching programs that are not universal [AW11]). • For noncommutative compu...

We use Hopf algebras to prove a version of the Littlewood–Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood–Richardson rules for Schur P and Q-functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of...

Chaotic multimodal skew tent maps are constructed to design an efficient chaos-based image encryption scheme with an efficient permutation-diffusion mechanism, in which permuting the positions of image pixels incorporates with changing the grey values of image pixels to confuse the relationship between cipher-image and plain-image. In the permutation process, a multimodal skew tent map is utili...