Let $\beta >1$ be a non-integer. We consider expansions of the form ${\sum }_{i=1}^{\infty }{d}_i/{\beta }^i$, where the digits ${\left(d_i\right)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor \left(\beta -1\right)]$. We show existence and uniqueness of a $K_{\beta }$-invariant probability measure, absolutely continuous with respect to $m_p\otimes \lambda$, where $m_p$ is the Bernoulli measure on ${\{0,1\}}^{\mathbb{N}}$ with parameter $p$ ($0<p<1$) and $\lambda$ is the normalized Lebesgue measure on $[0,\lfloor \beta \rfloor (\beta -1\left)\right]$. Furthermore, this measure is of the form $m_p\otimes {\mu }_{\beta ,p}$, where ${\mu }_{\beta ,p}$ is equivalent to $\lambda$. We prove that the measure of maximal entropy and $m_p\otimes \lambda$ are mutually...

The properties of topological dynamical systems $(X,T)$ which are disjoint from all minimal systems of zero entropy, ${ℳ}_0$, are investigated. Unlike the measurable case, it is known that topological $K$-systems make up a proper subset of the systems which are disjoint from ${ℳ}_0$. We show that $(X,T)$ has an invariant measure with full support, and if in addition $(X,T)$ is transitive, then $(X,T)$ is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists...

We prove the existence of entropy solutions to unilateral problems associated to equations of the type $Au-div\left(\varphi \left(u\right)\right)=\mu \in L¹\left(\Omega \right)+W^{-1,p^{\text{'}}\left(·\right)}\left(\Omega \right)$, where A is a Leray-Lions operator acting from $W{₀}^{1,p\left(·\right)}\left(\Omega \right)$ into its dual $W^{-1,p\left(·\right)}\left(\Omega \right)$ and $\varphi \in C⁰(ℝ,{ℝ}^N)$.

In the following text for a discrete finite nonempty set $K$ and a self-map $\varphi :X\to X$ we investigate interaction between different entropies of generalized shift ${\sigma }_{\varphi }:K^X\to K^X$, ${\left(x_{\alpha }\right)}_{\alpha \in X}\mapsto {\left(x_{\varphi \left(\alpha \right)}\right)}_{\alpha \in X}$ and cellularities of some Alexandroff topologies on $X$.

Let $\beta >1$ be a non-integer. We consider $\beta$-expansions of the form ${\sum }_{i=1}^{\infty }{d}_i/{\beta }^i$, where the digits ${\left(d_i\right)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor /\left(\beta -1\right)]$. We show that $K_{\beta }$ has a unique mixing measure ${\nu }_{\beta }$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ${\nu }_{\beta }$ the digits ${\left(d_i\right)}_{i\ge 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness...

We study II${}_1$ factors $M$ and $N$ associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result : every finite index $M$-$N$-bimodule (in particular, every isomorphism between $M$ and $N$) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index $M$-$M$-bimodules is identified with an extended Hecke fusion algebra,...

Let $F=(f_1,...,f_q)$ be a polynomial dominating map from ${ℂ}^n$ to ${ℂ}^q$. We study the quotient ${𝒯}^1\left(F\right)$ of polynomial 1-forms that are exact along the generic fibres of $F$, by 1-forms of type $\mathrm{d}R+\sum a_i\mathrm{d}{f}_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${𝒯}^1\left(F\right)$ is always a torsion $ℂ[t_1,...,t_q]$-module. Then we determine under which conditions on $F$ we have ${𝒯}^1\left(F\right)=0$. As an application, we study the behaviour of a class of algebraic $({ℂ}^p,+)$-actions on ${ℂ}^n$, and determine in particular when these actions are trivial.

Let $f$ be a dominant rational map of ${\mathbb{P}}^k$ such that there exists $s\&lt;k$ with ${\lambda }_s\left(f\right)\&gt;{\lambda }_l\left(f\right)$ for all $l$. Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $\mathrm{Aut}\left({\mathbb{P}}^k\right)$, the map $f\circ A$ admits a hyperbolic measure of maximal entropy $log{\lambda }_s\left(f\right)$ with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of $f$ to ${\mathbb{P}}^{k+1}$. This provides many examples where non uniform hyperbolic dynamics is established. One of the key tools is to approximate...

This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs $x_i$ and outputs $y_i$, i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator $f_z$ on the base of given data $z:=((x₁,y₁),...,(x_m,y_m))$ that approximates well the regression function...

Let ${\alpha }_i,{\beta }_i>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=(t^{\alpha ₁}x₁,...,t^{\alpha ₙ}xₙ)$, $t\circ x=(t^{\beta ₁}x₁,...,t^{\beta ₙ}xₙ)$ and $\left|\right|x\left|\right|={\sum }_{i=1}^n{\left|x_i\right|}^{1/{\alpha }_i}$. Let φ₁,...,φₙ be real functions in $C^{\infty }(ℝⁿ-0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_E{(x,\phi \left(x\right))\left|\right|x\left|\right|}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^n{\alpha }_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{\mu }f=\mu \ast f$ and let $\left|\right|T_{\mu }{\left|\right|}_{p,q}$ be the operator norm of $T_{\mu }$ from $L^p\left({ℝ}^{2n}\right)$ into $L^q\left({ℝ}^{2n}\right)$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_{\mu }$ is defined by $E_{\mu }=(1/p,1/q):\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_i\ne {\beta }_k$ for 1 ≤ i,k ≤ n we characterize the...

This paper discusses $n$-island finite automata whose transition graphs can be expressed as $n$-member sequences of islands $i_1,i_2,\cdots ,i_n$, where there is a bridge leaving $i_j$ and entering $i_{j+1}$ for each $1\le j\le n-1$. It concentrates its attention on even computation defined as any sequence of moves during which these automata make the same number of moves in each of the islands. Under the assumption that these automata work only in an evenly computational way, the paper proves its main result stating that $n$-island finite...

Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${\left({ϵ}_i\right)}_{i=1}^{\infty }\in {0,1}^{\mathbb{N}}$ a β-expansion for x if $x={\sum }_{i=1}^{\infty }{ϵ}_i{\beta }^{-i}$. We call a finite sequence ${\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n$ an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given $\Psi :\mathbb{N}\to {ℝ}_{\ge 0}$, we introduce the following subset of ℝ: $W_{\beta }\left(\Psi \right):={\bigcap }_{m=1}^{\infty }{\bigcup }_{n=m}^{\infty }{\bigcup }_{{\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n}[{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right),{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)+\Psi \left(n\right)]$In other words, $W_{\beta }\left(\Psi \right)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0\le x-{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)\le \Psi \left(n\right)$. When ${\sum }_{n=1}^{\infty }{2}^n\Psi \left(n\right)<\infty$, the Borel-Cantelli lemma tells us that the Lebesgue measure...

Let $\mu$ be a probability measure on $[0,1]$ which is invariant and ergodic for $T_a\left(x\right)=ax\mathrm{𝚖𝚘𝚍}1$, and $0<\mathrm{𝚍𝚒𝚖}\mu <1$. Let $f$ be a local diffeomorphism on some open set. We show that if $E\subseteq ℝ$ and $\left(f\mu \right){\left|{}_E\sim \mu \right|}_E$, then $f^{\text{'}}\left(x\right)\in \left\{\pm a^r:r\in \mathbb{Q}\right\}$ at $\mu$-a.e. point $x\in f^{-1}E$. In particular, if $g$ is a piecewise-analytic map preserving $\mu$ then there is an open $g$-invariant set $U$ containing supp $\mu$ such that $g\left|{}_U\right.$ is piecewise-linear with slopes which are rational powers of $a$. In a similar vein, for $\mu$ as above, if $b$ is another integer and $a,b$ are not powers of a common integer, and if $\nu$ is...

It is known that $ℬ\left({\ell }^p\right)$ is not amenable for p = 1,2,∞, but whether or not $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞), then so are ${\ell }^{\infty }\left(ℬ\left({\ell }^p\right)\right)$ and ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$. Moreover, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable so is ${\ell }^{\infty }(,\left(E\right))$ for any index set and for any infinite-dimensional ${ℒ}^p$-space E; in particular, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable for p ∈ (1,∞), then so is ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$. We show that ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over...

We investigate the recovery of $k$-sparse signals using the ${\ell }_1$-${\ell }_2$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$-sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...

For a Banach space $E$ and a probability space $(X,𝒜,\lambda )$, a new proof is given that a measure $\mu :𝒜\to E$, with $\mu \ll \lambda$, has RN derivative with respect to $\lambda$ iff there is a compact or a weakly compact $C\subset E$ such that ${\left|\mu \right|}_C:𝒜\to [0,\infty ]$ is a finite valued countably additive measure. Here we define ${\left|\mu \right|}_C\left(A\right)=sup\{{\sum }_k|\langle \mu \left(A_k\right),f_k\rangle \left|\right\}$ where $\left\{A_k\right\}$ is a finite disjoint collection of elements from $𝒜$, each contained in $A$, and $\left\{f_k\right\}\subset E^{\text{'}}$ satisfies ${sup}_k\left|f_k\left(C\right)\right|\le 1$. Then the result is extended to the case when $E$ is a Frechet space.

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers. Then a sequence $x\in A^{\mathbb{N}}$, where $A$ is a finite alphabet, is both $\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimately...