Expanding maps of solenoids

We prove that compact metric groups which admit expanding maps must be solenoidal groups, and that every expanding map on a solenoidal group is topologically conjugate to a positively expansive group endomorphism. This first was studied by Shub for expanding differentiable maps of tori and by Manning for Anosov diffeomorphisms of tori.     

First-order differentiability of the flow of a system withL p controls

This paper considers the study of the regularity of the flow of a nonautonomous nonlinear control process when the set of control maps is endowed with theL ^p -topology. Roughly speaking, it is proved that, if the norm of the mapf(t, x, u) defining the process together with its first derivatives goes to infinity, with the norm ofu not faster thanu ^p ,p>1, then the flow isC ¹ in theL ^p -topology. This property implies that, if the control maps are bounded, then the flow is differentiable in anyL ^p ,p>1. Moreover, it is proved that the only systems for which the flow is differentiable inL ¹ are the affine ones.This research was supported by a grant from Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica, Rome, Italy.     

Intrinsic ergodicity of smooth interval maps

We generalize the technique of Markov Extension, introduced by F. Hofbauer [10] for piecewise monotonic maps, to arbitrary smooth interval maps. We also use A. M. Blokh’s [1] Spectral Decomposition, and a strengthened version of Y. Yomdin’s [23] and S. E. Newhouse’s [14] results on differentiable mappings and local entropy. In this way, we reduce the study ofC ^r interval maps to the consideration of a finite number of irreducible topological Markov chains, after discarding a small entropy set. For example, we show thatC ^∞ maps have the same properties, with respect to intrinsic ergodicity, as have piecewise monotonic maps.     

Repellers for non-uniformly expanding maps with singular or critical points

GivenanergodicmeasurewithpositiveentropyandonlypositiveLyapunov exponents, its dynamical quantifiers can be approximated by means of quantifiers of some family of uniformly expanding repellers. Here non-uniformly expanding maps are studied that are C ^1+β smooth outside a set of possibly critical or singular points.     

Regularity of the Level Set Flow

We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative and satisfies the equation classically. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed C¹ manifold with cylindrical singularities. © 2017 Wiley Periodicals, Inc.     

Bernoulli maps of the interval

The ergodic properties of expanding piecewiseC ² maps of the interval are studied. It is shown that such a map is Bernoulli if it is weak-mixing. Conditions are given that imply weak-mixing (and hence Bernoulliness). Partially supported by NSF Grant MCS74-19388 and the Sloan Foundation.     

Smoothness Properties of Semiflows for Differential Equations with State-Dependent Delays

Differential equations with state-dependent delay can often be written as (t)=f(x_t) with a continuously differentiable map f from an open subset of the space C¹=C¹([-h,0], {}^n), {h>0}, into {}^n. In a previous paper we proved that under two mild additional conditions the set is a continuously differentiable n-codimensional submanifold of C ¹, on which the solutions define a continuous semiflow F with continuously differentiable solution operators F_t=F(t,·), t 0. Here we show that under slightly stronger conditions the semiflow F is continuously differentiable on the subset of its domain given by {t> h}. This yields, among others, Poincaré return maps on transversals to periodic orbits. All hypotheses hold for an example which is based on Newton's law and models automatic position control by echo.     

The Singularities of the Distance Function Near Convex Boundary Points

In an open bounded set Ω, we consider the distance function from ∂Ω associated to a Riemannian metric with C ^1,1 coefficients. Assuming that Ω is convex near a boundary point x ₀, we show that the distance function is differentiable at x ₀ if and only if there exists the tangent space to ∂Ω at x ₀. Furthermore, if the distance function is not differentiable at x ₀ then there exists a Lipschitz continuous curve, with initial point at x ₀, such that the distance function is not differentiable along such a curve.      

AC 1 expanding map of the circle which is not weak-mixing

In this paper, we construct an example of aC ¹ expanding map of the circle which preserves Lebesgue measure such that the system is ergodic, but not weak-mixing. This contrasts with the case ofC ^1+ε maps, where any such map preserving Lebesgue measure has a Bernoulli natural extension and hence is weak-mixing.     

Periodic orbits and expansiveness

We prove that if a local diffeomorphism has expanding periodic points robustly then it is an expanding map. Using this, we reobtain a result due to Sakai: generic positively expansive maps are expanding. Our methods also show a global version of a result by Gan and Yang: generic expansive diffeomorphisms are Axiom A without cycles.     

Expanding measures

We prove that any C1+α transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller (1990) [33] for maps of the interval with negative Schwarzian derivative.Given a non-uniformly expanding set, we also show how to construct a Markov structure such that any invariant measure defined on this set can be lifted. We used these structure to study decay of correlations and others statistical properties for general expanding measures.     

On the relationship between differentiability and absolute continuity of measures on ℝn

Summary We give an elementary proof of the fact that a finite Borel measure on ⁿ is absolutely continuous with a C ¹ density if and only if it has directional derivatives which are continuous almost everywhere. The Radon-Nikodym derivative of a differentiable measure is given in terms of the directional derivatives.     

Lineability of sets of nowhere analytic functions

Although the set of nowhere analytic functions on [0,1] is clearly not a linear space, we show that the family of such functions in the space of C^∞-smooth functions contains, except for zero, a dense linear submanifold. The result is even obtained for the smaller class of functions having Pringsheim singularities everywhere. Moreover, in spite of the fact that the space of differentiable functions on [0,1] contains no closed infinite-dimensional manifold in C([0,1]), we prove that the space of real C^∞-smooth functions on (0,1) does contain such a manifold in C((0,1)).     

Good Banach spaces for piecewise hyperbolic maps via interpolation

We introduce a weak transversality condition for piecewise C^1+α and piecewise hyperbolic maps which admit a C^1+α stable distribution. We show bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel–Lizorkin type which are better than previously known estimates (when our assumption on the stable distribution holds). In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) but gives a new result on bounded multipliers. Our method applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results on a simpler scale of Banach spaces.     

Hermitian and Positive Integrated C-cosine Functions on Banach Spaces

Peculiar properties of hermitian and positive n-times integrated C-cosine functions on Banach spaces are investigated. Among them are: (1) Any nondegenerate positiven -times integrated C-cosine function is infinitely differentiable in operator norm; (2) An exponentially bounded, nondegenerateC -cosine function on L ^p () (1L ¹(), C₀ , in case C has dense range) is positive if and only if its generator is bounded, positive, and commutes with C.     

On the regularity of the roots of hyperbolic polynomials

We prove that a hyperbolic monic polynomial whose coefficients are functions of class C ^r of a parameter t admits roots of class C ¹ in t, if r is the maximal multiplicity of the roots as t varies. Moreover, if the coefficients are functions of t of class C ^2r , then the roots may be chosen two times differentiable at every point in t. This improves, among others, previous results of Bronšteĭn, Mandai, Wakabayashi and Kriegl, Losik and Michor.     

Rigidity of critical circle mappings I

We prove that two C ³ critical circle maps with the same rotation number in a special set ? are C ^1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C ⁰ sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ^∞ critical circle maps with the same rotation number that are not C ^1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers. Received November 1, 1998 / final version received July 7, 1999     

Necessary conditions for efficiency in terms of the Michel–Penot subdifferentials

In this article, we study a multiobjective optimization problem involving inequality and equality cone constraints and a set constraint in which the functions are either locally Lipschitz or Fréchet differentiable (not necessarily C ¹-functions). Under various constraint qualifications, Kuhn–Tucker necessary conditions for efficiency in terms of the Michel–Penot subdifferentials are established.     

Convergence of best approximations in a smooth Banach space

Let X be a reflexive, strictly convex Banach space such that both X and X^* have Fréchet differentiable norms, and let {C_n} be a sequence of non-empty closed convex subsets of X. We prove that the sequence of best approximations {p(x ¦ C_n)} of any x ε X converges if and only if lim C_n exists and is not empty. We also discuss measurability of closed convex set valued functions.     

On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces

Given a rational homology classh in a two dimensional torusT ², we show that the set of Riemannian metrics inT ² with no geodesic foliations having rotation numberh isC ^k dense for everyk N. We also show that, generically in theC ² topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.Partially supported by CNPq-GMD, FAPERJ, and the University of Freiburg.