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...

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 $\varphi \in C⁰(ℝ,{ℝ}^N)$.

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...

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...

We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule $F=F_{[l,r]}$, l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy $h_{v⃗}\left(\Phi \right)$, v⃗= (x,y) ∈ ℝ², is bounded above by $max\left(\right|z_l|,|z_r\left|\right)logA$ if $z_l{z}_r\ge 0$ and by $|z_r-z_l|$ in the opposite case, where $z_l=x+ly$, $z_r=x+ry$. We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.

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...

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...

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 $X$.

Suppose that $\mu$ is an absolutely continuous probability measure on ${}^R$n, for large $n$. Then $\mu$ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if $n\ge {(C/\epsilon )}^{Cd}$, then there exist $d$-dimensional marginals of $\mu$ that are $\epsilon$-far from being sphericallysymmetric, in an appropriate sense. Here $C>0$ is a universal constant.

We consider $C^2$ families $t\mapsto f_t$ of $C^4$ unimodal maps $f_t$ whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure ${\mu }_t$ of $f_t$ depends differentiably on $t$, as a distribution of order $1$. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of ${\mu }_t$ for a Benedicks-Carleson map $f_t$, in terms of a single smooth function and the...

For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^2$, let $AC(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\left\{0\right\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly...

Let $X$ be a $\mu$-symmetric Hunt process on a LCCB space $𝙴$. For an open set $𝙶\subseteq 𝙴$, let ${\tau }_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$. Let $r:[0,\infty [\to [0,\infty [$ and $R\left(t\right)={\int }_0^{t}r\left(s\right)\mathrm{d}s$. We give necessary and sufficient conditions for ${𝔼}_{\mu }R\left({\tau }_{𝙶}\right)lt;\infty$ in terms of the behavior near the origin of the spectral measure of $-A^{𝙶}$. When $r\left(t\right)=t^l$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for ${\tau }_{𝙶}$...

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

Let ${}_s$ be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis ${}_s$ differentiates the integral of f if s ∉ S, and $D{̅}_{s}f\left(x\right)=limsu{p}_{diam\left(R\right)\to 0,x\in R{\in }_s}{\left|R\right|}^{-1}{\int }_{R}f=\infty$ almost everywhere if s ∈ S. If the condition $D{̅}_{s}f\left(x\right)=\infty$ holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a $G_{\delta }$ (resp. a $G_{\delta \sigma }$).

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and...

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...

In this work we study the problem $u_t{-div\left(\right|x|}^{-2\gamma }\nabla u)=\lambda \frac{u^{\alpha }}{{\left|x\right|}^{2(\gamma +1)}}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0\left(x\right)$ in $\Omega$, $\Omega \subset {ℝ}^N$ $(N\ge 2)$ is a bounded regular domain such that $0\in \Omega$, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <\left(N-2\right)/2$, $f$ and $u_0$ are positive functions such that $f\in L^1(\Omega \times (0,T))$ and $u_0\in L^1\left(\Omega \right)$. The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $\alpha =1$, $1+\gamma >0$; (ii) the nonexistence of solutions if $\alpha >1$, $1+\gamma >0$; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f\left(x\right)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....