Preprints and published papers , Habilitation thesis

**Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing**(.pdf)

F. Pène, B. Saussol, preprint.**Abstract:**We consider some nonuniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs-Markov-Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball B(x,r) converges to a Poisson distribution as the radius r -> 0 and after suitable normalization.**Return- and Hitting-time limits for rare events of null-recurrent Markov maps**, (.pdf)

F. Pène, B.Saussol, R. Zweimüller, preprint.**Abstract:**We determine limit distributions for return- and hitting-time functions of certain asymptotically rare events for conservative ergodic infinite measure preserving transformations with regularly varying asymptotic type. Our abstract result applies, in particular, to shrinking cylinders around typical points of null-recurrent renewal shifts and infinite measure preserving interval maps with neutral fixed points.**Exponential law for random subshifts of finite type**(.pdf)

Jérôme Rousseau, Benoît Saussol, Paulo Varandas, preprint.**Abstract:**In this paper we study the distribution of hitting times for a class of random dynamical systems. We prove that for invariant measures with super-polynomial decay of correlations hitting times to dynamically defined cylinders satisfy exponential distribution. Similar results are obtained for random expanding maps. We emphasize that what we establish is a quenched exponential law for hitting times.**Skew products, quantitative recurrence, shrinking targets and decay of correlations**, (.ps, .pdf)

Stefano Galatolo, Jérôme Rousseau, Benoît Saussol, To appear in*Ergodic Theory and Dynamical Systems***Abstract:**We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recurrence (also with respect to given observables) and hitting time scale behavior depend on the arithmetical properties of the extension. By this we show that those systems have a polynomial decay of correlations with respect to C^{r}observables, and give estimations for its exponent, which depend on r and on the arithmetical properties of the system. We also show examples of systems of this kind having not the shrinking target property, and having a trivial limit distribution of return time statistics.**Recurrence rates and hitting-time distributions for random walks on the line**, (.ps / .pdf)

F. Pène, B.Saussol, R. Zweimüller, accepted for publication in*Annals of Probability*.**Abstract:**We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as r goes to zero, of the return time to (-r,r). We then refine this result by establishing a limit theorem for the hitting-time distributions of (x-r,x+r) with arbitrary real x.**Central limit theorem for dimension of Gibbs measures in hyperbolic dynamics**, (.ps / .pdf)

R. Leplaideur, B.Saussol,*Stochastic and Dynamics***12**-2 (2012).**Abstract:**We consider a class of non-conformal expanding maps on the d-dimensional torus. For an equilibrium measure of an Hoelder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. An unexpected consequence is that when the measure is not absolutely continuous, then half of the balls of radius ε have a measure smaller than ε^{δ}and half of them have a measure larger than ε^{δ}, where δ is the Hausdorff dimension of the measure. We first show that the problem is equivalent to the study of the fluctuations of some Birkhoff sums. Then we use general results from probability theory as the weak invariance principle and random change of time to get our main theorem. Our method also applies to conformal repellers and Axiom A surface diffeomorphisms and possibly to a class of one-dimensional non uniformly expanding maps. These generalizations are presented at the end of the paper.**Hitting and returning into rare events for all alpha-mixing processes**, (.ps / .pdf)

M. Abadi, B.Saussol,*Stochastic Processes and their Applications***121-2**(2011) 314-323.**Abstract:**We prove that for any α-mixing process the hitting time of any n-string A_{n}converges, when suitably normalized, to an exponential law. We identify the normalization constant λ(A_{n}). A similar statement holds also for the return time.**An introduction to quantitative Poincaré recurrence in dynamical systems**, (.ps / .pdf)

B.Saussol,*Reviews in Mathematical Physics***21**-8 (2009) 949-979.**Abstract:**We present some recurrence results in the context of ergodic theory and dynamical systems. The main focus will be on smooth dynamical systems, in particular those with some chaotic/hyperbolic behavior. The aim is to compute recurrence rates, limiting distributions of return times, and short returns. We choose to give the full proofs of the results directly related to recurrence, avoiding as possible to hide the ideas behind technical details. This drove us to consider as our basic dynamical system a one-dimensional expanding map of the interval. We note however that most of the arguments still apply to higher dimensional or less uniform situations, so that most of the statements continue to hold. Some basic notions from the thermodynamic formalism and the dimension theory of dynamical systems will be recalled.**Back to balls in billiards**, (.ps / .pdf)

F. Pène, B.Saussol,*Communications in mathematical physics***293**-3 (2010) 837-866; présentation grand public.**Abstract:**We consider a billiard in the plane with periodic configuration of convex scatterers. This system is recurrent, in the sense that almost every orbit comes back arbitrarily close to the initial point. In this paper we study the time needed to get back in an r-ball about the initial point, in the phase space and also for the position, in the limit when r->0. We establish the existence of an almost sure convergence rate, and prove a convergence in distribution for the rescaled return times.**Poincaré recurrence for observations**, (.ps/ .pdf)

J. Rousseau, B.Saussol,*Transactions A.M.S.***362**-11 (2010) 5845-5859**Abstract:**A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of di fferent quantities, called an observation. Following this idea and in the continuity of Boshernitzan's work, for a measure preserving system, we study Poincaré recurrence for the observation. The link between the return time for the observation and the Hausdorff dimension of the image of the invariant measure is considered. We prove that when the decay of correlations is super polynomial, the recurrence rates for the observations and the pointwise dimensions relatively to the push-forward are equal.**Quantitative recurrence in two dimensional extended processes**, (.ps / .ps.gz / .pdf / .dvi )

F. Pène, B.Saussol,*Ann. Inst. H. Poincaré proba-stat***45**-4 (2009) 1065-1084.**Abstract:**Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighborhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including Z^{2}-extension of hyperbolic dynamics. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a convergence in distribution of the rescaled return times near the origin.**Large deviations for return times in non-rectangle sets for Axiom A diffeomorphisms**, (.ps / .ps.gz / .pdf / .dvi )

R. Leplaideur, B.Saussol,*Discrete and Continuous Dynamical Systems A***22**(2008) 327-344**Abstract:**For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition.**Recurrence rate in rapidly mixing dynamical systems**, (.ps / .ps.gz / .pdf / .dvi )

B.Saussol,*Discrete and Continuous Dynamical Systems A***15**(2006) 259-267**Abstract:**For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return close to its starting point. We prove that when the decay of correlation is super-polynomial the recurrence rates and the pointwise dimensions are equal. This gives a broad class of systems for which the recurrence rate equals the Hausdorff dimension of the invariant measure.**Products of non-stationary random matrices and Multiperiodic equations of several scaling factors**, (.ps / .ps.gz / .pdf / .dvi )

A.H.Fan, B.Saussol, J.Schmeling,*Pacific Journal of Mathematics***214**(2004) 31-54**Abstract:**Let*b>1*be a real number and*M: R -> GL(C*^{d}*)*be a uniformly almost periodic matrix-valued function. We study the asymptotic behavior of the product*P*_{n}*(x) =M(b*^{n-1}*x) ... M(bx) M(x)*. Under some condition we prove a theorem of Furstenberg-Kesten type for such products of non-stationary random matrices. Theorems of Kingman and Oseledec type are also proved. The obtained results are applied to multiplicative functions defined by commensurable scaling factors. We get a positive answer to a Strichartz conjecture on the asymptotic behavior of such multiperiodic functions. The case where*b*is a Pisot-Vijayaraghavan number is well studied.**Recurrence spectrum in smooth dynamical system**, (.ps / .ps.gz / .pdf / .dvi )

B.Saussol, J.Wu,*Nonlinearity***16**(2003) 1991-2001**Abstract:**We prove that for conformal expanding maps the return time does have constant multifractal spectrum. This is the counterpart of the result by Feng and Wu in the symbolic setting.**Recurrence and Lyapunov exponents for diffeomorphisms**, (.ps / .ps.gz / .pdf / .dvi )

B.Saussol, S.Troubetzkoy, S.Vaienti,*Moscow Mathematical Journal***3**(2003) 189-203**Abstract:**We prove two inequalities between the Lyapunov exponents of a diffeomorphism and its local recurrence properties. We give examples showing that each of the inequalities is optimal.**Distribution of frequencies of digits via multifractal analysis**, (.ps / .ps.gz / .pdf / .dvi )

L.Barreira, B.Saussol, J.Schmeling,*Journal of Number Theory***97/2**(2002) 413-442**Abstract:**We study the Hausdorff dimension of a large class of sets in the real line defined in terms of the distribution of frequencies of digits for the representation in some integer base. In particular, our results unify and extend classical work of Borel, Besicovitch, Eggleston, and Billingsley in several directions. Our methods are based on recent results concerning the multifractal analysis of dynamical systems and often allow us to obtain explicit expressions for the Hausdorff dimension. This work is still another illustration of the role that the theory of dynamical systems can play in number theory.**On the uniform hyperbolicity of certain hyperbolic systems**, (.ps / .ps.gz / .pdf / .dvi )

J.F.Alves, V.Araújo, B.Saussol,*Proc. Amer. Math. Soc.***131**(2003) 1303-1309**Abstract:**We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.**Recurrence, dimensions and Lyapunov exponents**, (.ps / .ps.gz / .pdf / .dvi )

B.Saussol, S.Troubetzkoy, S.Vaienti,*Journal of Statistical Physics***106**(2002) 623-634**Abstract:**We show that the Poincaré return time of a typical cylinder is at least its length. For one dimensional maps we express the Lyapunov exponent and dimension via return times.**On pointwise dimensions and spectra of measures**, (.ps / .ps.gz / .pdf / .dvi )

J.-R. Chazottes, B.Saussol,*C. R. Acad. Sci. Paris Sér I Math.***333**(2001) 719-723**Abstract:**We give a new definition of the lower pointwise dimension associated with a Borel probability measure with respect to a general Caratheodory-Pesin structure. Then we show that the spectrum of the measure coincides with the essential supremum of the lower pointwise dimension. We provide an example coming from dynamical systems.**Variational principles for hyperbolic flows**, (.ps / .ps.gz / .pdf / .dvi )

L.Barreira, B.Saussol,*Fields Institute Communications***31**(2002) 43-63**Abstract:**We establish a conditional variational principle for hyperbolic flows. In particular we provide an explicit expression for the topological entropy of the level sets of Birkhoff averages, and obtain a very simple new proof of the corresponding multifractal analysis. One application is that for a geodesic flow*F*on a compact Riemannian manifold of negative sectional curvature, if there exists a geodesic_{t}*F*_{t}*x*with ``average'' scalar curvature K, then there exist uncountably many geodesics with the same ``average'' scalar curvature K. The variational principle can also be used to establish the analyticity of several new classes of multifractal spectra for hyperbolic flows.**Pointwise dimensions for Poincaré recurrence associated with maps and special flows**, (.ps / .ps.gz / .pdf / .dvi )

V.Afraimovich, J.-R.Chazottes, B.Saussol,*Discrete and Continuous Dynamical Systems A***9**(2003) 263-280**Abstract:**We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subsh ift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a "typical" cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.**Statistics of return time via inducing**, (.ps / .ps.gz / .pdf / .dvi )

H.Bruin, B.Saussol, S.Troubetzkoy, S.Vaienti,*Ergodic Theory and Dynamical Systems***23**(2003) 991-1013**Abstract:**We prove that return time statistics of a dynamical system do not change if one passes to an induced (i.e. first return) map. We apply this to show exponential return time statistics in i) smooth interval maps with nowhere--dense critical orbits and ii) certain interval maps with neutral fixed points. The method also applies to iii) certain quadratic maps of the complex plane.**Product structure of Poincaré recurrence**. (.ps / .ps.gz / .pdf / .dvi )

L.Barreira, B.Saussol,*Ergodic Theory and Dynamical Systems***22**(2002) 33-61**Abstract:**We provide new non-trivial quantitative information on the behavior of Poincaré recurrence. In particular we establish the almost everywhere coincidence of the recurrence rate and of the pointwise dimension for a large class of repellers, including repellers without finite Markov partitions.

Using this information, we are able to show that for locally maximal hyperbolic sets the recurrence rate possesses a certain local product structure, which closely imitates the product structure provided by the families of local stable and unstable manifolds, as well as the almost product structure of hyperbolic measures.**Higher dimensional multifractal analysis**. (.ps / .ps.gz / .pdf / .dvi )

L.Barreira, J.Schmeling, B.Saussol,*Journal de Mathématiques pures et appliquées***81**(2002) 67-91**Abstract:**We establish a higher-dimensional version of multifractal analysis for several classes of hyperbolic dynamical systems. This means that we consider multifractal decompositions which are associated to multi-dimensional parameters. In particular, we obtain a conditional variational principle, which shows that the topological entropy of the level sets of pointwise dimensions, local entropies, and Lyapunov exponents can be*simultaneously*approximated by the entropy of ergodic measures. A similar result holds for the Hausdorff dimension.

This study allows us to exhibit new nontrivial phenomena absent in the one-dimensional multifractal analysis. In particular, while the domain of definition of a one-dimensional spectrum is always an interval, we show that for higher-dimensional spectra the domain need not be convex and may even have empty interior, while still containing an uncountable number of points. Furthermore, the interior of the domain of a higher-dimensional spectrum has in general more than one connected component.**Local dimension for Poincaré recurrence**.

V.Afraimovich, J.-R. Chazottes, B.Saussol.*Electron. Res. Announc. Amer. Math. Soc.***6**(2000), 64-74**Abstract:**Pointwise dimensions and spectra for measures associated with Poincaré recurrences are calculated for arbitrary weakly specified subshifts with positive entropy. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. It is also proved that the Poincaré recurrence for a "typical" cylinders is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy.**Hausdorff dimension of measures via Poincaré recurrence**. (.ps / .ps.gz / .pdf / .dvi )

L.Barreira, B.Saussol.*Communication in mathematical physics***219**(2001) 443-463**Abstract:**We study the quantitative behavior of Poincaré recurrence. In particular, for an equilibrium measure on a locally maximal hyperbolic set of a diffeomorphism*f*with Hoelder derivative, we show that the recurrence rate to each point coincides almost everywhere with the Hausdorff dimension*d*of the measure, that is, inf {*k>0*: dist(*f*^{k}*x, x*) <*r*} ~*r*^{-d}. This result is a non-trivial generalization of work of Boshernitzan concerning the quantitative behavior of recurrence, and is a dimensional version of work of Ornstein and Weiss for the entropy. We stress that our approach uses different techniques. Furthermore, our results motivate the introduction of a new method to compute the Hausdorff dimension of measures.**On fluctuations and the exponential statistics of return times**. ( .ps / .ps.gz / .pdf / .dvi )

B.Saussol.*Nonlinearity***14**(2001) 179-191**Abstract:**This paper presents some facts related to the exponential statistic of return time. First of all we show that this behavior is valid in a large class of dynamical system.

Second, we investigate the question of computing the speed of convergence to this limiting law. We show that this speed carries informations about the system under consideration, while via a local analysis we can relate it to some combinatorial property of some orbits.

Finally, we prove that for an arbitrary dynamical systems, the existence of an exponential statistic for the return time implies the equivalence between the fluctuations of empirical entropies and repetition times.**Variational principles and mixed multifractal spectra**. ( .ps / .ps.gz / .pdf / .dvi )

L.Barreira, B.Saussol.*Transactions AMS***353**(2001) 3919-3944**Abstract:**We establish a "conditional" variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy.

Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the "mixed" multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the "non-mixed" multifractal spectra.**Multifractal analysis of hyperbolic flows**. (.ps / .ps.gz / .pdf / .dvi )

L.Barreira, B.Saussol.*Communication in Mathematical Physics***214**(2000) 339-371**Abstract:**We establish the multifractal analysis of hyperbolic flows and of suspension flows over subshifts of finite type. A non-trivial consequence of our results is that for every Hoelder continuous function non-cohomologous to a constant, the set of points without Birkhoff average has full topological entropy.**Dimensions for recurrence times: topological and dynamical properties**. ( .ps / .ps.gz / .pdf / .dvi )

V.Penné, B. Saussol, S. Vaienti.*Discrete and Continuous Dynamical Systems***5**(1999) 783-798**Abstract:**In this paper we give new properties of the dimension introduced by Afraimovich to characterize Poincaré recurrence and which we proposed to call Afraimovich-Pesin's (AP's) dimension. We will show in particular that AP's dimension is a topological invariant and that it often coincides with the asymptotic distribution of periodic points: deviations from this behavior could suggest that the AP's dimension is sensible to some "non-typical"' points.**Statistics of return times: a general framework and new applications**. ( .ps / .ps.gz / .pdf / .dvi )

M.Hirata, B.Saussol, S.Vaienti.*Communication in Mathematical Physics***206**(1999) 33-55**Abstract:**In this paper we provide general estimates for the errors between the distribution of the first, and more generally, the Kth return time (suitably rescaled) and the Poisson law for measurable dynamical systems. In the case that the system exhibits strong mixing properties, these bounds are explicitly expressed in terms of the speed of mixing. Using these approximations, the Poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval.**Fractal and statistical characteristics of recurrence times**. ( .ps / .ps.gz / .pdf / .dvi )

V. Penné, B. Saussol, S. Vaienti.*Journal de Physique*(Paris), proceding of the conference*Disorders and Chaos*(Rome) in honor of Giovanni Paladin**Abstract:**In this paper we introduce and discuss two proprieties related to recurrences in dynamical systems. The first gives the asymptotic law for the return time in a neighborhood, while the second gives a topological index of fractal type to characterize the system or some regions of the system.**Absolutely continuous invariant measures for multidimensional expanding maps**. ( .ps / .ps.gz / .pdf / .dvi )

B. Saussol.*Israel Journal of Mathematics***116**(2000) 223-248**Abstract:**We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps with singularities. We allow maps that are discontinuous on some extremely wild sets, the shape of the discontinuities being completely ignored with our approach.**A probabilistic approach to intermittency**. ( .ps / .ps.gz / .dvi )

C. Liverani, B. Saussol, S. Vaienti.*Ergodic Theory and Dynamical Systems***19**(1999) 671-685**Abstract:**We present an original approach which allows to investigate the statistical properties of a non-uniform hyperbolic map of the interval. Based on a stochastic approximation of the deterministic map, this method gives essentially the optimal polynomial bound for the decay of correlations, the degree depending on the order of the tangency at the neutral fixed point.**Conformal measure and decay of correlations for covering weighted systems**. ( .ps / .ps.gz / .pdf / .dvi )

C. Liverani, B. Saussol, S. Vaienti.*Ergodic Theory and Dynamical Systems***18**(1998) 1399-1420**Abstract:**We show that for a large class of piecewise monotonic transformations on a totally ordered, compact set one can construct conformal measures and obtain exponential mixing rate for the associated equilibrium state. The method is based on the study of the Perron-Frobenius operator. The conformal measure, the density of the invariant measure and the rate of mixing are deduced by using an appropriate Hilbert metric, without any compactness arguments, even in the case of a countable to one transformation.

**Récurrence de Poincaré dans les systèmes dynamiques hyperboliques.**( .ps / .ps.gz / .pdf / .dvi )

Habilitation à diriger des recherches.*Université de Picardie Jules Verne, Amiens*, 2003**Abstract:**Le théorème de récurrence de Poincaré est fondamental dans la théorie des systèmes dynamiques. Il dit que l'existence d'une mesure invariante finie entraîne une récurrence non-triviale dans les ensembles de mesure positive. Toutefois il n'amène qu'une information de nature qualitative. L'objet de la récurrence quantitative que nous étudions dans ce mémoire est précisément de mesurer la façon dont les retours s'opèrent.

Nous nous proposons ici de mener cette étude, principalement dans le cadre des systèmes dynamiques hyperboliques, selon trois axes complémentaires.

- On peut considérer les temps de retour comme des variables aléatoires et s'intéresser alors à leur loi. Nous verrons que l'on a souvent des convergences vers des lois exponentielles.

- Combien de temps faut-il attendre pour qu'un système revienne dans un état proche de son état initial, disons avec une erreur r ? Nous verrons que ce temps est en puissance de r, et nous relierons ces vitesses de divergence aux dimensions des mesures invariantes.

- Quel est le temps minimal de retour dans une boule de rayon r ? Nous verrons que ce temps n'est pas nécessairement "typique" et on obtiendra cette fois des temps logarithmiques en r, les vitesses auront alors plutôt des relations avec les exposants de Lyapunov du système.**Etude statistique de systèmes dynamiques dilatants.**( .ps / .ps.gz / .pdf / .dvi )

Thèse de Doctorat, directed by Prof. Sandro Vaienti.*Université de Toulon et du Var*, 1998**Abstract:**This thesis treats two fundamental aspects of dynamical systems: Ergodic Theory and Differentiable Dynamic, the latter being more concrete from a certain point of view, closer to "real life" dynamic. We tried, as far as it was possible, to give not only asymptotic results but also finite time (or size) estimations.

In a first step, we review some statistical properties and metric characteristics of dynamical systems. We focus on dynamical translation of usual probility theorems, like central limit theorem. In particular, we show that in several cases, the time needed by the system to return in a neighborhood of its initial state closely follows a Poisson law, the error being a function of the size of the neighborhood and chaotic properties of the system.

In a second part, we study three kind of dynamical systems: piecewise monotonic transformation with an infinite number of branches, intermittent map of the interval (with an indifferent fixed point) and finally multidimensional piecewise expanding maps with singularities. We will prove for these systems the existence of SRB measures (and other equilibrium states in the first case), and we will find constructive estimates on the rate of decay of correlations.