 Okay, so thank you very much. Thank you for the invitation. It's really a pleasure to be here again. I think this is my fourth time in Trieste each time for a conference on dynamical systems. So what I will speak about concerns would join works with mostly with the run and clepsin. There are also some works with the run clepsin and filimonov and there is this soccer team with Alvarez, filimonov, clepsin, malice, menino and Michele Triestino. And this concerns some questions arising from the community of dimension one fallations. But let me start with something very classical and elementary that appears in all books on dynamical systems which is the most basic tool in one-dimensional dynamics which is control of distortion. So for me I will, for this talk I will only consider control of distortion for maps without singularities because I will deal with diffeomorphisms. And by this I mean well you take a map from a one-dimensional manifold to itself and you try to get a bound of the derivative of the iterates of this map and you try to compare these derivatives for two different points and you try to get some uniform bound for this portion here which is independent on the number of iterates. And everybody knows that if you get something like this then you should get a theorem because this gives you a kind of control of the geometry when passing from one scale to another. So in general when you pass from a small scale to another one. So this is the general principle if you get something like this then you should get some theorem. So in my case since I will deal with fallations and group actions I will not only consider a single diffeomorphism but I will consider finitely many diffeomorphisms and I will do compositions instead of iteration of a single diffeomorphism. So in my case I will consider instead of the nth iterate of f I will consider a map like fn which is going to be the composition of certain maps GIN, GIN-1 and so on where this Gij, these maps here belong to a finite family of diffeomorphisms. So what is important here is to have a finite family say of generators of your dynamics. And again instead of getting something like this the idea will be to get a control of distortion for the nth composition of these maps. Not for every kind of composition but for certain compositions that allows to say something about the dynamics. And just to complete this discussion let me just remind you the most famous and simplest the simplest argument in this direction which is due to actually to the Danjua but it was perished by some other people. So Danjua did it for the case of a single diffeomorphism Trots realized that it works in a different manner but it was a successor that introduced this technique in the world of fallations. And it says something which is very easy to state so it says that you take two points leaving in interval i which is wandering, sorry let me change it, leaving in some interval j which is wandering for the dynamics which means that the image of the along the compositions are two by two disjoint. So you take j then in this setting you take the first image of j the second image or the image up to the second level and so on. So these are two by two disjoint then you get control of distortion provided that you impose some regularity on the generators and the regularities that they are all C2 diffeomorphisms or at least C2 local diffeomorphism. Actually C1 plus bounded variation is enough but anyway so then in this case you get this uniform bound for the distortion and the constant is very explicit so this is less or equal to L times C where L is the length of your manifold or your one dimensional manifold i and C is the elliptic constant for the derivative of the generator. So it is the maximum elliptic constant for the derivative of the generator. And well I think everybody knows the argument so you just take the logarithm of this and then you do the computation and I think everybody knows. The important point here is that since the intervals are two by two disjoint and this sum and the logarithm of this is controlled by the total sum of the length of the intervals and since they are disjoint then this total sum is bounded by the length of the manifold. This is the Danjoua trial success estimate. And there is another estimate that in general it is also attribute to Danjoua but I think this is wrong because Danjoua didn't think about this. In the case where you don't have this property but you know that the dynamics is somehow contracting in a hyperbolic way. So this is the hyperbolic case in some sense. This is referred to as being the folklore argument in the book of Paulis and Takens so I don't know who really stated for the first time but the idea is that if you take two points and if you know that the derivatives are larger or equal than certain constant which is larger than one then again you get control of distortion. And in this case you get this control of distortion for points that remain that are bounded distance along the direction. And the idea is that well here you have the total sum of the lengths is finite because the intervals are disjoint and here the total sum of the lengths is finite because when you take the dynamics in the inverse in the inverse sense then the length of the interval is decreasing in an exponential way so the total sum is controlled by a geometric series so it is finite. And with this very simple tool and some arguments in each case one can prove the classical theorems in this setting so the first is Danjoua theorem which says that well if you take as I said before this still work for this control of distortion is still work for C1 plus bounded variation diffeomorphisms so if you take a C1 plus bounded variation diffeomorphism with irrational rotation number C1 plus bounded variation diffeomorphism of the circle then the theorem says that f is minimum. So there is no invariant counter set for the dynamics so it is minimal and actually because of Fonkari theorem says that because of Fonkari theorem it is conjugate topologically conjugate to the rotation. And there is also the Jean-Claire's top version of this theorem for f which is homomorphism of the circle of course with irrational rotation number but which is real analytic which is somehow different because here you allow many some critical points but since you are real analytic you only have finitely many of those and they are non-degenerate and then there is some extra argument here. And of course this applies this version of the control of distortion applies to the situation because if you have a diffeomorphism which is not minimal then there is an invariant counter set and then you take the gap of the invariant counter set so which means the connected component of a connected component of the complement of the counter set and then you start iterating your diffeomorphism and this gap is a boundary interval for the dynamics so you get control of distortion and at the end there is an extra argument in order to get a contradiction. There is always an extra argument to get a contradiction but the point is that the main tool is there in the control of distortion technique. Sorry? Ah, then, sorry. F is minimal. Sorry. Okay. So, well, here there is a very big difference is from 32 and this is from, sorry, 53. Okay. And there is another, well, here you know that the diffeomorphism is minimal so the next question is to ask whether it is ergodic with respect to the natural measure which is the Lebesgue measure in this case and this is also a theorem which was independently proved by Michel Hermann and Katok. So again, if you take a C1 plus bounded variation diffeomorphism with the rational rotation number then, well, according to Najrath theorem it is minimal but this theorem that says that F is ergodic with respect to the Lebesgue measure. Okay. Which is not at all a trivial stuff because there is an invariant measure for diffeomorphism but in general the invariant measure of the diffeomorphism is singular with respect to the Lebesgue measure. So this is not obvious at all. And again for this theorem there is, well, both, the proofs are very different. So, but the proof by Katok uses this principle in a very tricky way and it uses the fact that if you look at the dynamic of the composition which means that you take a point and then you take the nth interval of the dynamic of the composition which means that you take the image of this point by the iterate to the corresponding to the power qn where qn is the denominator in the rational, in the rational approximation of the, of the rotation number then this interval here which I will denote by jn, well, it is not wandering for the dynamics but it is almost wandering up to certain time so that the whole circle is covered by the iterates of this interval and the multiplicity of the covering is bounded, actually it is bounded by two. So this allows to, although these intervals are not, are not joined, they are almost joined in a certain sense. So you can still apply the, you still get the control of distortion and having control of distortion, you can prove something and in this case this is the the rigidity of the action. And well, this is for the case of a single diffeomorphism but for the case we are interested now we have certain results especially in the, in the, somehow in the hyperbolic case. So there is another theorem that I will just state, I will not recall all the words here but well the theorem says that if you take a c1 plus alpha dynamically defined cantor set, well the Levegue measure of this cantor set is equal to zero and actually the whole dimension is smaller than one, okay. This means more or less that this cantor set is given by a Markov partition. And there is another theorem that I think it goes back to Sullivan but it is not clear, maybe it was already known is that well if you take say a group gamma acting on a, or a pseudo group acting on a one dimensional manifold by c1 plus alpha local diffeomorphisms and if you assume that it is hyperbolic in the sense that for example for every x in the manifold there exists some element for which the derivative at this point is larger than one then the action is ergodic, well this is in the case of minimality. So if you have a minimal action which is expanding which means that for every point there is an expanding element then the action is ergodic and this uses this principle here again control of distortion going from a small scale to a large one and so on. So there is also a version for the case where there invariant cantor sets in that case what you have is that if every point of the cantor set is expanded by a group element then the Levegue measure of the cantor set is zero. And motivated by this kind of results there were some conjectures by many people so I will write some names here but actually I should say that the first time I learned about this condition was reading the book of the Meloam by String which is the following. So I will state this conjecture in the word of co-dimension one-folliation so assume that you have a co-dimension one-folliation on a compact manifold. So a co-dimension one-folliation can be seen as a dynamical object where the dynamics is given by the along the maps. This is the issue here. Not really, if you want to do something like this, you have to impose some regularity condition is that this foleyation is of class C2 transversal because the dynamics is given by the allonomous along transversals. And well the conjectures are the following. If f is minimal then which means that if the only saturated closed sets are the anti-set and the whole manifold then f is ergodic again with respect to the Levegue measure which means that the only measurable saturated sets are those of zero measure of total measure. And if there is a what is called an exceptional leaf which is a saturated set such that transversally looks as a contour set. So I will state in this way if there is a transversal contour set then the Levegue measure of this contour set should be equal to zero. And the gaps should be finitely many up to the equivalence relation given by the dynamics. And so let me just say that more or less the transversal minus the contour set is made up of finitely many orbits. And well these are still open but what we did with the rank lepsin and the rank lepsin and filament up was to deal with this question in the simplest case, I mean not in the simplest case I mean in the case where the foliation is given by a suspension of a group action. So let me remind you that if you have a group action I will consider group actions on the circle so I will take a group gamma acting on the circle say finitely generated essentially the condition here corresponds to the condition here that the manifold is compact. And when you have this then you can produce a foliation on a three manifold by the suspension procedure. So you take a very for instance you take a very big surface with a very large genus you map this into a free group of a large number of generators so that this maps into the into gamma and then you get some group of homomorphism here and then you take well you take the Poincare disk time s1 then you get an action on this product space where a map g say gamma here maps px into gamma p phi of gamma x where this is the deck transformation and this is the action provided by this composition here and delta is the okay this is the unit disk hypervalid disk okay so I am looking at this as the covering of the surface okay and then you question by this by this action and the question is a foliation on sg times the circle okay the question is a co-dimension one foliation on the surface times the circle that retains all the dynamics of the group action okay no no no but yeah yeah yeah yeah yeah yeah yeah yeah okay so by this I mean that I take gamma is generated by k elements so I consider these elements as free elements and here I take the g equals to 2k so I take the k element that maps into the generator and the other one I send to the identity and in this way I get exactly what I said okay okay what is important is that if you take a group action then you produce a foliation so if you want to prove something for foliation in this world the first thing you should try to do is prove something for group actions okay and so the conjecture so the conjecture so become a question about group action so take a gamma acting on the circle by gamma finite generated and gamma acting by c2 different morphisms and the question is two-fold or actually three-fold so in the in the minimal case the question is if the action is is minimal whether it is ergodic with respect to the Lebesgue measure and the second thing is if there is a cantor set an invariant cantor set actually here this case is easier to stay because if there is an invariant cantor set then this cantor set turns to be unique so say k then the Lebesgue measure should be equal to zero and the circle and the number of connected components of the complement should be finite up to so it should be it should consist of finitely many orbits okay this is somehow G san saliva and this is Hector's conjecture so let me just say that for instance for the case of fuchsian groups acting on the circle this is not obvious this is a theorem this is a theorem by alphors which actually also holds in dimension two so it was for fuchsian groups and clenion groups this is called the alphors finiteness theorem okay let me just give an example a very simple example of what should be what should we have in mind in this theorem so take the circle and take two maps you can take it take them inside pressure to R say f by this I mean that f maps the complement of this interval into the interior of this interval okay and then take G doing this this is the ping-pong dynamics so G is taking the complement of this interval into the interior of this interval when you have this then you immediately get a cantor set which is invariant for the dynamics this is the simplest way to create the cantor set for a dynamics group action on the circle and well this is a simple in this case it is not very hard to see that if f and G are C2 then the cantor set has zero level dimension okay but this is the example you should keep in mind because somehow what I will show is that this is at least in the reality case this is more or less the only example that can arise okay so the theorem is that this is a theorem a genuine theorem in this setting so I will impose the condition that the generators are real analytic otherwise there are some technical problems for the proof we still don't know that whether this is true and of course always finitely generated and well the first thing is that if there exists a cantor set then we managed to prove that the Lebesgue measure of the cantor set is zero and the actor-conjecture which is that the complement is made up by finitely many orbits and the second thing is that if the action is minimal and well here we need an extra hypothesis which is algebraic and gamma is almost free so it is a free group up to a finite index subgroup actually we can remove this hypothesis by asking for gamma to have infinitely many ends but anyway then the action is ergodic with respect to the Lebesgue measure so these are the solution of this conjecture for the case of group actions provided that the action is by real analytic diffeomorphisms and with this extra hypothesis on the group for the second case which is actually it is also needed in the first for this result but what happens here is that it was already known that if you have an invariant cantor set and you are really analytic then the group must be almost free this was a previous result by G's okay okay and so I still have 15 minutes so I will say a few ideas about the proof is very technical you can imagine this so we use the classical control of distortion stuff but the point is that we don't try to solve this question directly but we prove it is a kind of structure theorem for the action that gives us this information in an indirect way okay this was a strategy suggested by Cologne and somehow what we are doing is that in the most difficult cases we are proving that there is a certain mark of a structure for the dynamics from which everything arises naturally okay so let me introduce a notion that we call Bertrand and Victor property star which is a very bad name but anyway it remains so property star is the following so remember that there was this soluble expansion strategy that uses hyperbolic control of distortion that says that if for every point there is an expanded element then you get everything you get control of distortion so the problem is when there are points that are not expandable okay so property star means that if x is not expandable which means that for every group element the derivative at this point is smaller or equal than one then there exists some group element which fix this point and for which this point is a isolated fixed point okay actually we need this point to be isolated from one side but since we are regenerating we have automatically this property okay and the example to have in mind here so this condition is asked for every point in the minimal case and in the case where there is a contour set is asked only for points in the contour set okay and the example to keep in mind here is the same as example as here in the limit case when this circle is becoming larger and larger and this one is becoming larger and larger and they have a common boundary point okay so something like this so this is f and this is g okay this is a free group and this can be can be modeled out in pencil to R and in this case this point become non expandable points and this is not a natural at all so I mean this is quite this is not artificial because this appear as a finite index subgroup in pencil to R so this group this the group generated these two elements is a free subgroup is a free group and this can be realized as inside piece of two piece of two Z as an index six subgroup I think actually if you take pencil to Z then there are two points that are non expandable this is an easy computation the region and the infinite these are the only points for which there is no matrix in this group which for which the derivative is larger than one so this is an example of an action with non expandable point but what is quite visible here is that this non expandable points are fixed points of certain elements for instance here I think if you take the commutator of f and g then this is going to fix one of this point and this is not going to be the identity here it is easier to see that here one infinite fixed point of certain similarities I think so property of star holds in this case so there are two things now there are two issues the first thing is that if property start property start holds then everything holds I mean then you get the godicity in the minimal case and then you get zero levite measure for cantor set and you get finitely many orbits of connect the components of the complement of the cantor sets and so on so we are happy okay okay and the second thing is and this is the difficult theorem is that property start always holds so this is okay in the C2 category and this we can prove under this hypothesis so in the reality context and with this extra algebraic hypothesis for the minimal case okay but we suspect this is always true but maybe not but I think at least the conjecture should be true in general okay so let me explain point one which is not very hard at least to explain the idea because the idea is just a mixture of the classical techniques of control of distortion because well the first thing is that if you think a little bit if there are non expandable points and say we are we have a minimal actions then the number of non expandable points has to be finite this is not very hard to see so you have only finitely many non expandable points so these are the bad points so you get you put a flag here because these are the dangerous regions and outside these regions in these intervals we have expanding elements because these are not these are the only non expandable points so here you can perform the hyperbolic trick the hyperbolic expansion okay but the point is that if you are very close to these points then you lose expansion because the derivative become closer and closer to one but because of the definition of the hypothesis these points here are also fixed points of some some elements so for instance this point here is the fixed point of a certain group element F isolated fixed point and so if you take say assume that F has no fixed point in this interval and moves X to the left in this case so this is F of X and you take this interval this is going to be a fundamental region for the iteration of F so the image of this interval are going to be disjoint and you do it in the other way around and what you get is control of distortion using the Daniel strategy that's total sum of the length of the interval is finite independently where you are you can be very close to this point but it doesn't matter you use a lot a very large number of iteration of F inverse and you escape from this bad region and when you are outside this bad region you continue doing the hyperbolic trick and so on you keep control of distortion and you and you are happy okay this is just idea the rise is just trying to put things together in a in another big way okay and so they they they are going to the difficult point and the real theorem the genuine theorem here is this one so that property start is always true under our hypothesis and the and the well as I said before what we what we do to prove this is to prove a certain structure theorem that actually should be what should be able to pursue this the thing so I will state a conjecture in order to explain what what we have in mind here so if you if you take a gamma which is reanalytic and finally generated and say locally discreet otherwise everything is too simple because if your group is not locally discreet this is for free because you have flowers in the closures and so on then the conjecture is that there exists a Marco partition for the dynamics I mean in the mean this should be true in the minimal case for for the case of a Cantor said there should be a Marco partition of the Cantor set for the dynamics and if the action is minimal and expanding then gamma should be a surface group I think this is the more interesting statement of this conjecture so gamma should be a surface group up to finite index I mean up to a finite extension okay this is locally discreet is that the no no let me give the precise definition so for every open interval if f n which are group elements restricted to g j converged to the identity actually we need c1 locally discreet then eventually f n is the identity for n very large and I think this is this is this is the point is that under how I put this is actually there should be some very very deeper results on the structure of the action so let me just give the single idea of the proof of the somehow where you will see that there is a kind of Marco partition that arises in this stuff okay well this is very clear in this picture so here there is a Marco partition and the point is how to detect this Marco partition well this only use control of distortion so somehow the points on these intervals are characterized in terms of sums of derivatives and sums of derivatives are the genuine tools to get control of distortion so what I show at the very beginning that you need total sum of length of the intervals to get control of distortion is not totally true in the sense that the real thing that you need is that the total sum of the sum of the derivatives to be fine and this is implied by the fact that the total sum of the length of the interval is finite so let me just say that so for instance this is a free group and now I will define a set in this case so so m gamma for a gamma one a generator of this of this group so gamma is going to be f f inverse g or g inverse m gamma is going to be the set of points on the circle such that the sum of geodesics sum of the derivatives along geodesics that start with gamma is finite okay so you look at the iteration of along geodesic for which the first step is equal to gamma this is going to give you to give you a set and this in this case it is not very hard to recognize that this set for instance for g this is m of g and then you define m tilde of gamma as being the intersection of m delta where delta is different from gamma inverse okay and in this setting this is going to be the set m tilde g and this set is here when the group is free where if the group is free well this is the case here but if the group is free there is a well defined notion of geodesic and so on so it is easier to handle and these sets are going to provide the marker partitions but to prove this to prove this well there are some technicalities I will just say something so the first step in proving this is something which is not obvious is that these sets are open but this is control of distortion okay these sets are open but what is not obvious is that this set are made up of finally many connected components but the most important step is that the m points of these sets of the connected components of these sets are fixed points for some group elements okay and it is at that step that there is a key a key a key a key a key argument that well after we did it I noticed that somehow it reminds the proof of Bagnet of his characterization of hyperboleicity for attractors in one dimensional dynamics somehow because you impose some condition on the derivatives so there is no there is no hyperboleicity and then you get that this point must be fixed by a certain element so there is something that reminds me is remaining in the sense of Magnet's argument of his proof of the characterization of hyperboleicity of invariant measures so this is the key step so this family of sets is going to provide a counter a marker partition for which the end points are fixed points for some group elements and then everything works okay so I think I should stop thinking very much