Related papers: Maximizing dimension for Bernoulli measures and th…
We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems with overlaps $\mathcal S$. We prove exact dimensionality for these image measures, and find a…
We describe a general method of arithmetic coding of geodesics on the modular surface based on a two parameter family of continued fraction transformations studied previously by the authors. The finite rectangular structure of the…
In this survey paper, I first review the history of Bernoulli numbers, then examine the modern definition of Bernoulli numbers and the appearance of Bernoulli numbers in expansion of functions. I revisit some properties of Bernoulli numbers…
We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain…
We establish new results on the dimension of density of states for the Anderson-Bernoulli model at small disorder
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…
The presence of large partial quotients can invalidate many classical limit theorems in the metric theory of continued fractions. A commonly employed strategy to overcome this problem is to discard the largest partial quotient when…
Measures generated by Iterated Function Systems composed of uncountably many one--dimensional affine maps are studied. We present numerical techniques as well as rigorous results that establish whether these measures are absolutely or…
Using optimal transport we study some dynamical properties of expanding circle maps acting on measures by push-forward. Using the definition of the tangent space to the space of measures introduced by Gigli, their derivative at the unique…
On a one-sided shift of finite type we prove that for a generic Holder continuous function there is a unique maximizing measure. We show that b-Holder continuous functions can be approximated in the a-Holder topology, a<b, by a function…
We consider the problem of discriminating between states of a specified set with maximum confidence. For a set of linearly independent states unambiguous discrimination is possible if we allow for the possibility of an inconclusive result.…
We derive the class of covariant measurements which are optimal according to the maximum likelihood criterion. The optimization problem is fully resolved in the case of pure input states, under the physically meaningful hypotheses of…
We study measures on $[0,1]$ which are driven by a finite Markov chain and which generalize the famous Bernoulli products. We propose a hands-on approach to determine the structure function $\tau$ and to prove that the multifractal…
We construct surface measures associated to Gaussian measures in separable Banach spaces, and we prove several properties including an integration by parts formula.
We show that for $0<\gamma, \gamma' <1$ and for measurable subsets of the unit square with Lebesgue measure $\gamma$ there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the…
We provide a complete picture of the upper quantization dimension in terms of the R\'enyi dimension by proving that the upper quantization dimension $\bar{D}_{r}(\nu)$ of order $r>0$ for an arbitrary compactly supported Borel probability…
Under the assumption of a natural subadditive potential, the so called cylinder function, working on the symbol space we prove the existence of the ergodic invariant probability measure satisfying the equilibrium state. As an application we…
We characterize the situation of having many normal measures on a measurable cardinal. We show the plausibility of having many normal measures on each compact cardinal.
We study the one-dimensional expanding Lorenz maps and show the existence of dense subset D of Lorens maps such that each f in D has an uncountable set of ergodic invariant probabilities with infinite Lyapunov exponent and positive entropy.…
We study the statistical properties of piecewise expanding maps in the general setting of metric measure spaces. We provide sufficient conditions for exponential mixing of such systems with explicit estimates on the constants. We also…