Preprints and published papers , Habilitation thesis
Spatio-temporal Poisson processes for visits to small sets
(.pdf)
F. Pène, B. Saussol, To appear in Israel Journal of Mathematics
Abstract: For many measure preserving dynamical systems (Ω; T;m) the successive hitting times to a small set is well approximated by a Poisson process on the real line. In this work we define a new process obtained from recording not only the successive times n of visits to a set A, but also the position Tn(x) in A of the orbit, in the limit where m(A) -> 0. We obtain a convergence of this process, suitably normalized, to a Poisson point process in time and space under some decorrelation condition. We present several new applications to hyperbolic maps and SRB measures, including the case of a neighborhood of a periodic point, and some billiards such as Sinai billiards, Bunimovich stadium and diamond billiard.
Large deviations for return times
(.pdf)
A. Coutinho, J. Rousseau, B. Saussol, Nonlinearity 31 (2018), no. 11, 5162-5179
Abstract: We prove a large deviation result for the return times of the orbits of a dynamical system in a r-neighborhood of an initial point x. Our result is a differentiable version of the work by Jain and Bansal who considered the return time of a stationnary and ergodic process defined in a space of infinite measure.
Linear response for random dynamical systems
(.pdf)
W. Bahsoun, M. Ruziboev, B. Saussol, To appear in Advances in Mathematics
Abstract:We study for the first time linear response for random compositions of maps, chosen independently according to a distribution P. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when P changes smoothly to Pε For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to ε ; moreover, we obtain a linear response formula. We apply our results to iid compositions, with respect to various distributions Pε, of uniformly expanding circle maps, Gauss-Renyi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations.
Linear response in the intermittent family: differentiation in a weighted C0-norm
(.pdf)
W. Bahsoun, B. Saussol, Discrete Contin. Dyn. Syst. 36 (2016), no. 12, 6657-6668.
Abstract: We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infnite SRB measure. In particular, we obtain a linear response formula for both finite and infnite SRB measures. To the best of our knowledge, this is the first work that contains a linear response result for infinite measure preserving systems.
Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing
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F. Pène, B. Saussol, Ergodic Theory Dynam. Systems 36 (2016), no. 8, 2602-2626.
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,
Ergodic Theory Dynam. Systems 37 (2017), no. 1, 244-276.
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.
Almost sure convergence of the clustering factor in a-mixing processes.
(.pdf)
M. Abadi, B. Saussol, Stoch. Dyn. 16 (2016), no. 3, 1660016, 11 pp.
Abstract: Abadi and Saussol (2011) have proved that the first time a dynamical system, starting from its equilibrium measure, hits a target set A has approximately an exponential law. These results holds for systems satisfying the α-mixing condition with rate function α decreasing to zero at any rate. The parameter of the exponential law is the product λ(A)μ(A), where the later is the measure of the set A; only bounds for λ(A) were given. In this note we prove that, if the rate function α decreases algebraically and if the target set is a sequence of nested cylinders sets An(x) around a point x, then λ(An) converges to one for almost every point x. As a byproduct, we obtain the corresponding result for return times.
An elementary way to rigorously estimate convergence to equilibrium and escape rates.
(.pdf)
S. Galatolo, I. Nisoli, B. Saussol, J. Comput. Dyn. 2 (2015), no. 1, 51-64.
Abstract: We show an elementary method to obtain (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rate for systems satisfying a Lasota Yorke inequality. The bounds are deduced by the ones of suitable approximations of the system's transfer operator. We also present some rigorous experiment on some nontrivial example.
Exponential law for random subshifts of finite type
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Jérôme Rousseau, Benoît Saussol, Paulo Varandas.
Stochastic processes and their Applications. 124 (2014), no. 10, 3260-3276.
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,
Ergodic Theory and Dynamical Systems 35 (2015), no. 6, 1814-1845.
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 Cr 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,
Annals of Probability 41-2 (2013) 619-635.
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 εδ and half of them have a measure larger than εδ, 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 An converges, when suitably normalized, to an exponential law. We identify the normalization constant λ(An). 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 Z2-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(Cd) be a uniformly almost periodic matrix-valued function. We study the asymptotic behavior of the product Pn(x) =M(bn-1x) ... 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,
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.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 Ft on a compact Riemannian manifold of negative sectional curvature, if there exists a geodesic Ftx 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,
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/ .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.
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/ .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(fkx, 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.
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.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 /
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/ .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
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/ .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
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/ .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
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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
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/ .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
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/ .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.