We define a $C^1$ distance between submanifolds of a riemannian manifold $M$ and show that, if a compact submanifold $N$ is not moved too much under the isometric action of a compact group $G$, there is a $G$-invariant submanifold $C^1$-close to $N$. The proof involves a procedure of averaging nearby submanifolds of riemannian manifolds in a symmetric way. The procedure combines averaging techniques of Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney’s idea of realizing submanifolds as zeros...

Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure $m_G$. We consider the map ${\tau }_{\phi }$ defined on X × G by ${\tau }_{\phi }:(x,g)\mapsto (\tau x,\phi \left(x\right)g)$ and the cocycle ${\left(\phi ₙ\right)}_{n\in \mathbb{Z}}$ generated by φ. Using a characterization of the ergodic invariant measures for ${\tau }_{\phi }$, we give the form of the ergodic decomposition of $\mu \left(dx\right)\otimes m_G\left(dg\right)$ or more generally of the ${\tau }_{\phi }$-invariant measures ${\mu }_{\chi }\left(dx\right)\otimes \chi \left(g\right)m_G\left(dg\right)$, where ${\mu }_{\chi }\left(dx\right)$ is χ∘φ-conformal for an exponential χ on G.

A submanifold $M^m$ of a generalized Sasakian-space-form ${\overline{M}}^{2n+1}(f_1,f_2,f_3)$ is said to be $C$-totally real submanifold if $\xi \in \Gamma \left(T^{\perp }M\right)$ and $\phi X\in \Gamma \left(T^{\perp }M\right)$ for all $X\in \Gamma \left(TM\right)$. In particular, if $m=n$, then $M^n$ is called Legendrian submanifold. Here, we derive Wintgen inequalities on Legendrian submanifolds of generalized Sasakian-space-forms with respect to different connections; namely, quarter symmetric metric connection, Schouten-van Kampen connection and Tanaka-Webster connection.

We show that for any irrational number α and a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $li{m}_{l\to \infty }\left|\right||m_l\alpha \left|\right||=0$, there exists a continuous measure μ on the circle such that $li{m}_{l\to \infty }{\int }_{}\left|\right||m_l\theta \left|\right||d\mu \left(\theta \right)=0$. This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $\left|\right||m_l\alpha \left|\right||\to 0$ and such that $m_l\theta \left[1\right]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.

Let $M$ be an $n$-dimensional submanifold in the unit sphere $S^{n+p}$, we call $M$ a $k$-extremal submanifold if it is a critical point of the functional ${\int }_M{\rho }^{2k}\mathrm{d}v$. In this paper, we can study gap phenomenon for these submanifolds.

Let X be a closed subspace of $L^p\left(\mu \right)$, where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $su{p}_{n\in \mathbb{Z}}\left|\right|Uⁿ\left|\right|<\infty$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like ${\sum }_{n\ge 1}\left(Uⁿf\right)/n^{1-\alpha }$, 0 ≤ α < 1, in terms of $\left|\right|f+\cdots +U^{n-1}f{\left|\right|}_p$, generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...

We give the definition of $f$-biminimal submanifolds and derive the equation for $f$-biminimal submanifolds. As an application, we give some examples of $f$-biminimal manifolds. Finally, we consider $f$-minimal hypersurfaces in the product space ${ℝ}^n\times {𝕊}^1\left(a\right)$ and derive two rigidity theorems.

Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on $L^p\left(\nu \right)$, $T₁,...,T_k$, $n̅=(n₁,...,n_k)\in {\mathbb{N}}^k$ and $\alpha ̅=(\alpha ₁,...,{\alpha }_k)$ with $0<{\alpha }_j\le 1$, we define the ergodic Cesàro-α̅ averages ${}_{n̅,\alpha ̅}f=1/\left({\prod }_{j=1}^k{A}_{n_j}^{{\alpha }_j}\right){\sum }_{i_k=0}^{n_k}\cdots {\sum }_{i₁=0}^{n₁}{\prod }_{j=1}^k{A}_{n_j-i_j}^{{\alpha }_j-1}{T}_k^{i_k}\cdots T{₁}^{i₁}f$. For these averages we prove the almost everywhere convergence on X and the convergence in the $L^p\left(\nu \right)$ norm, when $n₁,...,n_k\to \infty$ independently, for all $f\in L^p\left(d\nu \right)$ with p > 1/α⁎ where $\alpha ⁎=mi{n}_{1\le j\le k}{\alpha }_j$. In the limit case p = 1/α⁎, we prove that the averages ${}_{n̅,\alpha ̅}f$ converge almost everywhere on X for all f in the Orlicz-Lorentz space $\Lambda (1/\alpha ⁎,{\phi }_{m-1})$ with $\phi ₘ\left(t\right)=t{(1+log⁺t)}^m$. To obtain the result in the limit case we need...

Let $U₁,...,U_d$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $\mathbb{Z}{₊}^d$ and the associated collection of rectangular parallelepipeds in ${ℝ}^d$ with sides parallel to the axes and dimensions of the form $n₁\times \cdots \times n_d$ with $(n₁,...,n_d)\in \Gamma .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{\Gamma }$ are defined respectively on $L¹\left({ℝ}^d\right)$ and L¹(Ω) by $M_{}g\left(x\right)=su{p}_{x\in R\in }1/\left|R\right|{\int }_R\left|g\left(y\right)\right|dy$ and $M_{\Gamma }f\left(\omega \right)=su{p}_{(n₁,...,n_d)\in \Gamma }1/n₁\cdots n_d{\sum }_{j₁=0}^{n₁-1}\cdots {\sum }_{j_d=0}^{n_d-1}\left|f\left(U{₁}^{j₁}\cdots U_d^{j_d}\omega \right)\right|$. Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate ...

In ergodic theory, certain sequences of averages $A_{k}f$ may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence $A_{m_k}f$ of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are $A_{k}f\left(x\right)=1/\left(2^k\right){\sum }_{j=4^k+1}^{4^k+2^k}f\left(T^{j}x\right)$, then the subsequence $A_{k²}f$ will not be pointwise good even on $L^{\infty }$, but the subsequence $A_{2^k}f$ will be pointwise good on L¹. Understanding when the hyperexponential...

We obtain estimates for derivative and cross-ratio distortion for $C^{2+\eta }$ (any η > 0) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition. For two intervals J ⊂ T, the cross-ratio is defined as the value B(T,J): = (|T| |J|)/(|L| |R|) where L,R are the left and right connected components of T∖J respectively. For an interval map g such that $g_T:T\to ℝ$ is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J): = B(g(T),g(J))/B(T,J). We prove...

Let $\Omega$ be a bounded domain in ${ℝ}^n$ with smooth boundary $\partial \Omega$. We consider the equation $d^2\Delta u-u+u^{\frac{n-k+2}{n-k-2}}=0\text{in}\Omega$, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial \Omega$, which is non-degenerate, and certain weighted average of sectional curvatures of $\partial \Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that $d^2{\left|\nabla u_d\right|}^2\rightharpoonup S{\delta }_K\mathrm{as}d\to 0$ in the sense of measures,...

We consider iterated function systems on the interval with random perturbation. Let $Y_{\epsilon }$ be uniformly distributed in [1-ε,1+ ε] and let $f_i\in C^{1+\alpha }$ be contractions with fixpoints $a_i$. We consider the iterated function system $Y_{\epsilon }{f}_i+a_i(1-Y_{\epsilon }){ⁿ}_{i=1}$, where each of the maps is chosen with probability $p_i$. It is shown that the invariant density is in L² and its L² norm does not grow faster than 1/√ε as ε vanishes. The proof relies on defining a piecewise hyperbolic dynamical system on the cube with an SRB-measure whose projection...

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...

A stable deformation $f^t$ of a real map-germ $f:ℝⁿ,0\to {ℝ}^p,0$ is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification $f_{ℂ}^t$ are real. A related notion is that of a good real perturbation $f^t$ of f (studied e.g. by Mond and his coworkers) for which the homology of the image (for n < p) or discriminant (for n ≥ p) of $f^t$ coincides with that of $f_C^t$. The class of map germs having an M-deformation is, in some sense, much larger than the one having a good...

Given any compact manifold $M$, we construct a non-empty open subset $𝒪$ of the space ${\mathrm{Diff}}^1\left(M\right)$ of $C^1$-diffeomorphisms and a dense subset $𝒟\subset 𝒪$ such that the centralizer of every diffeomorphism in $𝒟$ is uncountable, hence non-trivial.

Let $f:{\mathbb{P}}^k\to {\mathbb{P}}^k$ be a dominating rational mapping of first algebraic degree $\lambda \ge 2$. If $S$ is a positive closed current of bidegree $(1,1)$ on ${\mathbb{P}}^k$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks ${\lambda }^{-n}{\left(f^n\right)}^{*}S$ converge to the Green current $T_f$. For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$-invariant currents.