 Thanks a lot to the organizers for the invitation, and it's always, of course, as everybody thinks it's a great pleasure to be here, a wonderful conference, a wonderful environment. And so I will speak about some stuff, it's joint work with Jacek Gratzek, his name will be a very long time, and it's sort of joined forces for his work with Greg Shiantek and Stassmanov, and it also depends on my techniques developed with Lennar Koleson, so it's sort of a combination of two sets of techniques. So what we will try to do here is to discuss dimension of dual assets and harmonic measure on dual assets of standard form C squared plus C, where C is close to the boundary of a Mandelbrot set. So the idea is that you take a point on the boundary of a Mandelbrot set, and you pick it as almost every point with respect to harmonic measure, which is some of a natural measure on the boundary of the Mandelbrot set. So I will discuss that later. And the interesting aspect is that from the work of Shiantek Gratzek and also Stassmanov, there were eventually two papers. You have the so-called Kolekman condition satisfied that the iterates of a critical value C0 grows exponentially. This is the Kolekman condition. This is satisfied on the boundary of a Mandelbrot set. And this makes this perturbation techniques, which Koleson and I work with, exactly proving growth rate at the critical value of first the stretched exponential and then exponential, that works. But usually we assume actually in this work that the critical point was pre-periodic. So in the work of Shiantek, Mr. Eberst-Thurston, or one could make it weaker, we really worked in the work with Koleson with the von Neumann-Ulan map, which is C squared minus 2 in the 1 minus 2x squared, if you like the other normalization. So the point of this is that the critical point is pre-periodic going to repellent periodic orbit. But essentially the idea of this is that everything really works also when the point which you perturb away from is a Kolekman point. But there are various things to verify. And so when you have done this, you will do a parameter selection. It will be on a C2 curve through this point C0 on the boundary of the Mandelbrot set. And then Gratzek and Sminoff, they have two papers in Monsione, where they discuss the dual assets of the point where the Kolekman condition is satisfied. And there is work with many people in this area, including Chairman Felix Przettitski. But in this particular case, I would rather cite the work of Gratzek Sminoff. You get geometric measures associated to the Julia sets. You get snabo and rail measures on the Julia sets and its dimension. And you get also continuity for measures when you move along the selected set. I will be a bit more precise later. So this is then the Mandelbrot set. And we are actually looking at the point somewhere on the boundary anywhere which is exposed to a harmonic measure. So harmonic measure by a famous theorem on Makarov in this case has one-dimensional, hostile dimension. So it's somehow the exposed points. It's a little bit difficult to say which point we really discuss here. But it's a set of dimension one on the Mandelbrot set. And the dimension of the Mandelbrot set is the famous theorem of Shishikura that is really dimension two. So it's not all points, but the harmonic measures are more exposed points. We'll discuss this later. So the technique which will be used is sort of based on the old work of Carlis and myself on the quadratic family which in terms is a new different variation of the famous theorem of Jacobson which has been mentioned before that you have absolutely continuous invariant measure for a positive Lebesgue measure set in the parameter set of real quadratic maps. But Carlis and I have a different version of this. So in principle we prove by selecting parameters we get a parameter set of positive measure, a canter set of positive measure where the Kolyekman condition is satisfied. But we also have an approach rate condition for the critical orbit which the critical point stays away in an exponential way. It's allowed to approach the critical point zero but it has to be exponentially controlled with actually small exponent alpha. There is a relation between the critical Lyapunov exponent kappa and alpha and alpha should be essentially less than kappa for this to work. I mean we didn't discuss so carefully but I think of alpha as kappa over 100 or something but you could do more if you are careful. So of course this is part of the now very famous theory for real quadratic maps so this has a complete answer as I think everybody knows nowadays. So there is the theorem Grazek-Sientek and Ljubic that the set where C has attractive periodic orbit is open and dense and Ljubic completed this that you can divide the parameter set minus two one quarter into two set of parameters with absolutely continuous invariant measures and a set with attractive periodic orbits which is this set set here. But let me discuss a little bit about the approach rate which came up. So it's actually true what Arthur Avila and Carlos Moreira proved that these colliequemen are absolutely continuous actually colliequemen and the approach rate is polynomial where it can be written as polynomial with some exponent greater than one. So this is stronger than Carlos and I had. So what's the reason that Carlos and I choose actually to do this approach rate which is exponential with a small exponent. Well it turns out that if you look at our proof we really could have proven this approach rate this polynomial approach rate instead but if we wanted to apply it to the two-dimensional and non-family and there are no critical points in one sense but there are also accountability many critical points and there is a parameter selection so the number of critical points pre-critical points they essentially at the end of the generation they grew sub-exponentially like this and then there is a parameter selection regarding to the critical points like this and you want this kind of sum to be convergent so you want this eta to be less than alpha but you want really alpha here to make things work so we needed exponential and in this present work it's natural to do this slow exponential approach rate to a critical point. So in the original versions we really worked as a sort of point to perturb from as either this for Neumann-Olan, does it work here, yes, for Neumann-Olan point or more generally a Mr. Riewitz Thurston point. So a critical point is pre-periodic and it's going to an unstable periodic orbit. This is somehow the easiest version to do this but it turns out now that the parameter selection can be made to work also in the case of a boundary point of a Mandelbrot set and this is with new aspects. So to set up proof on the induction proof in this setting which originates in Carlissen and my work so what is needed for the Amperturbed map is you have expansion at the critical value similar to or really the Coley-Eckmann condition and you need also a transversality condition which gives comparison between the phase and parameter derivative and then the third is Manier, Mr. Riewitz Thurston. It's really Manier or it's usually called Manier but Mr. Riewitz was really first. So if you move outside the neighborhood of a critical point you essentially have expansion when you return to the critical point. So these are the three ingredients which we have to verify to set up this induction scheme. So if you really have, you could also in principle perturb from the original Mr. Riewitz Thurston but this is the Mr. Riewitz condition which is an uncountable counter set of parameters in the one-dimensional case. But we really want instead of this type of condition which you have in the pre-periodic case this condition with an exponential proof rate. So this is one important aspect and the other is you really select parameters in the phase space and you would like to know that it's the same as when you select parameter in the parameter space. So this means that the quotient between the derivative with respect to parameter and the phase derivative is actually bounded away from zero. This is, in principle you could sort of start which Collison and I did start with this as some finite states that there is this series and the tail of this series up to an arbitrary end should be less than one-half at a catapult point cap length. So this is one way, this is what is done in this paper on non-maps but you could also view it as others did. You want that this series is convergent even absolutely convergent and now they live and prove that absolute convergence implies actually that TLC is different from zero which makes things work. We have sort of and the other aspect is this Manjia style lemmas and now I will be a little more precise and I formulate it in the complex setting that you suppose there is an integer m which you have a fixed neighborhood of a critical point zero and suppose there is an integer so if you iterate outside the ball of radius delta then you have expansion and also that if you go iterate and this n here up to time n and this n should be sort of less than the m here then you have a lower bound like this. This will be the Manjia style lemmas and this can be summarized if you like that in this way but if you stay outside up to time n then fn belongs then the derivative should be greater than a constant and e to the kappa n here I should have said that the c here should be dependent on delta but the previous condition is perturbable you want to use this condition for the unperturbed map and you want to extend it to the unperturbed and these two conditions are perturbable in the parameter now the parameter is omitted but this is true for fc in the neighborhood of c0 so this is important but you have to verify this and the Manjia style lemma with classical proof in the Missouri which first in case is actually one way to do it is you define a metric in which actually blows up at a critical point and then you have expansion in this metric using Schwarz's lemma this is this Orbefold style metric this is the way but we don't have in the case what we are considering now on the boundary of Mandelbrot sets where we have to have independent proofs of this so let me sort of hovered around the theorem let me now state the result so for almost all c0 with respect to harmonic measure on the boundary of Mandelbrot set and every c2 curve we have maybe we should be defined on minus c to minus 1 to 1 we have that the set of Coley-Eckmann curves compared to the arc length so it's a density point so one of these good Coley-Eckmann points is sort of exposed and you get new Coley-Eckmann points on every arc it is a type here it should be minus 1 to 1 it's true also one sided if you like yeah, yeah, yeah this was implicit, sorry exactly this is a point and we expect it to leave most of the time so usually it should be outside the Mandelbrot set but still it's Coley-Eckmann and this is somehow what is remaining so we get we cannot distinguish between points on gamma which are outside the Mandelbrot set which really go to infinity and satisfy the Coley-Eckmann condition or hyperbolic in the sense of contracting we also account we meet them but then there should be a small measure for a smaller epsilon so we avoid those hyperbolic but there could be a boundary point on the Mandelbrot set also which we encounter so we cannot quite distinguish I will come back to that later no, not either this would have taken away but it could be on the boundary the boundary could be big, right the boundary of Mandelbrot set could be big but the question is how much is in the exterior, how much is in the boundary and this is somehow what we are still working on this part I will come back to that later if I have time so the starting point for the construction is this that the the critical Lappono exponent the Coley-Eckmann exponent really exists and this immediately gives this sub-exponential approach rate I have written down the estimates here it's really trivial if a limit exists then you cannot be really close because it will be too much jumps, this is written down here but I don't think we have time to discuss this this is really simple so and then it's this transversality which we have to verify and in this case you could in principle use Leven's theorem because this is absolutely convergent but we have also sort of for fun we have a different proof using Trivalo's theorem so you could actually move the Mandelbrot set is simply connected and we can move the situation to the unit circle and then you could speak about the transversality function to the outside of the unit circle so we have the Mandelbrot set like this and then you have a sort of W plane which is outside of the unit circle and then there are exterior stoles angles here which will move to something with something so this is the image of this, we move to something and here we can have the radial limit by Trivalo's theorem it will exist the radial limit, really Abel's theorem it will exist in the stoles angle Abel's theorem because and then it cannot vanish by Trivalo really because we really have to have that this function to see is positive for large positive C so so this is another way so you don't have to use Leven's theorem which is to do this so let me say a little bit about the proof of the parameter selection and this is for some people here are very familiar with it and it goes more or less similar to so we here we have some curve then our curve gamma is here is this exposed point C0 and here is gamma is a C2 curve and we want to select parameters and it's one-dimensional real parameters so we want these properties and then for people familiar with this setting we define this bound period the usual way what is wrong here there the bound period the orbit of a critical point is closed to the orbit of a point C if this condition is satisfied this is the usual definition you could have two different C's here because there is an associative partition of a parameter space and C1 and C2 belong to the same partition if you apply this that C here it's really the critical orbit of it's the amplitude of a critical point when N is a return time so this is some of the usual setting and then you have which is very common it's the bound distortion limit derivative at the critical value C1 to the derivative at an almost critical value f of C2C and it's bounded above and below it's more or less the usual estimate in the setting and then you have a bound period recovery so this is part of the definition of a bound period when it expires the bound condition is violated and this is sort of a quadratic difference and you see that the derivative at time N plus P where P is the boundary bound period is more or less dN times and you take the square root of this condition to see that this becomes here that there is a typo there should be one half there I should have seen this so it's dN and dP to one half I should have seen this before but so this one half appears since I'm supposed to know this stuff I recovered the one half later there is Kappa over 2 which corresponds to this one half and this means that you actually don't get exponential growth but this argument is good enough for stretched exponential growth actually this is for the first paper of Carlson and me but to have this exponential growth you really have to use this large deviation argument which is more sophisticated so when an orbit circulates outside a ball of radius d squared you have manier style lemma there is a uniform manier style lemma it's very rare that you this small ball so this is the way to get it and this is sort of in some sort of standard in this theory but there are some differences because in this complex case when there is a complex curve gamma n which is done so there is a what happens here gamma n you have to there are angular regions to which it returns and there is sort of Markov style property that you have to when a returning curve comes to an angular region it should really cross the angular region and for this it's important that you have a curvature control so this is new in the complex setting it appears a little bit in this and on setting also but it's a little different you have various derivative estimates and you just write down the iterations and keep control of the curvature because of the outside expansion more or less so there are various estimates which you have to do and the eventually you eventually will have some curvature estimate of this type which will give a uniform bound for the curvature in terms of 1 over delta and this will mean that a sufficiently long curve must cross one of these annually and this is important to make the construction work so this is a bit different in this setting and after doing the parameter selections now you can now use this work of Gratsik and Smirnov and you know actually for this set selected set you have continuity of house dimension and continuity of Laponov exponents and you have so this is the direct consequence of this inventionist paper of Smirnov and Gratsik but so of course the house dimension is not continuous on the boundary and this is due to Shishikura but when you restrict it to these selected sets you are actually fine so I mean so radial convergence continuity of house dimension if you have a particularly finite set the before then is due to McMullen and Jean and of course particularly finite has a serial harmonic measure but you did something for the real rich polynomials yeah so so you also get your own real measures as a consequence of this machinery while this is restatement so so let me summarize the result here so so now I got a curve right and I got it right that's gamma north you get the now A is a set of parameters of a curve and for each X you have a geometric measure which is supported on the Mandelbrot set which tends weekly to a geometric measure on supported on the Julia set of FC0 and you have an SRB measure supported on the Mandelbrot set and the SRB measures are equivalent to the geometric measure the house dimensions are equal and mu X has an exponential decay of correlation this is more or less just reading off and you have also this continuity along the set and the Leopold exponent at C0 is really log 2 so you tend to log 2 so so this is what let me maybe wait so the most interesting technically in some sense the most novel thing is maybe I this briefly that how do you prove this this mania style lemma this is some of the most difficulty in the setup of the machinery and this is actually through this machinery of and Felix is and Jean would be very familiar with it so it is this X shrink conditions that for if you have one of the good parameters on the boundary you have an exponential shrinking of a disc or a small radius and then the pre-emidges do not contain a critical point and then you can see that the first by contradiction came the pre-image which returns must be log delta this is similar to this bound period of arguments which appear and then you can see that by using this upper bound for this really bad that is this K you can see with an easy estimate that the diameter is small and then you get the contradiction and then you can conclude that there is a univalent branch mapping this set D to the neighbor of C or diameter or small diameter and then you can use Schwarz's lemma to get this mania style expansion so this uses the machinery so let me skip this so let me discuss some of the questions so so in principle there is also a recent result of Grazik and Sminoff that in a ball of some radius delta around this C0 actually the exterior of a Mandelbrot set is a density point not with respect to this curves but with respect to two-dimensional measures so in some sense most of the things that we captured we cannot see in this large deviation argument we cannot really see if things never return in a finite range so actually a lot is outside but this is work in progress there is some reasons to believe that that generically or at least for radial curves there is I'm not sure it's what we believe at that point let me not make a theorem something that we believe that you really have a house of dimension one for the points that are on the boundary of Mandelbrot set but this is work in progress not done thank you