 To je veliko prav, da sem tukaj tukaj, kako Kostja Khanin, sem tukaj, menej različi, na vsev 90. To je vse vse, da sem tukaj, kako sem tukaj, kako sem tukaj, Kuan Rivera in tukaj, tako, da sem tukaj, Ig globa and give a talk, and this will be, as you see from the title, a rather innocent topic, and I don't want to give a advanced talk, so I start with the definition, let me take this, I start with the definition, who was stated here several times already In pri konferencij skor, vse je to zelo to vse presa. Ne? Maybe I am talking... Ok. Now it's better. Ok, I will try to speak louder. I hear myself very well. So this is a standard definition. We have a continuous map on a compact matrix space and real continuous function. And we consider sums along blocks of orbits as n phi, sums of potential phi. And then we consider a metric rho n, which is maximal distance between blocks of trajectories. The one can discuss balls, but it is not necessary here. Maybe I should pay attention to people that the ball in this metric will not be connected in this exercise. Ok, and then there is a definition of pressure. Maybe I will use the pointer. I am not fluent in this. Does it work? Ah, yes, it works. Ok. So I consider such sums xn phi along blocks of trajectories of points, which are n epsilon separated. I take supremum over all such expressions and one nth log. And then there is limit because if epsilon becomes smaller then it is easier to have such sets, so it is increasing sequence. And analogously I can define, oh, one people define spanning sets and pressure via, topological pressure via spanning sets. And then these two quantities are equal to each other. By the way, they don't depend on metric because there is another definition which doesn't use metric, topological. Passing via this definition is very easy to prove this. So here at the bottom you have the inequalities, which become equalities when epsilon tends to zero. And note that the left-hand side in equality holds for all spanning sets and separated sets because if we have two epsilon separated set and also spanning set, epsilon spanning set then each point of the separated set has a free end from the spanning set, a closed point, but here is two epsilon, there is epsilon. So this identification is one-to-one injection. So the points, there are more points in the spanning set than the separated set. And to prove the right-hand side in equality I wrote here construction, but in this case it is trivial because if we have maximal separated set then it is also spanning set. We cannot add anything else because it is maximal separated. So we don't need to construct spanning set so that pressure is less equal. Pressure spanning is less equal, pressure separated because it is just the same set. Does it work? How? Does it work? Oh, no. Press it stronger. Another definition is via such supremum over invariant measures, probability invariant measures, entropy plus integral. So variational principle says that this is the same as previously defined pressures and I remind here what is majority entropy. This is just such a sum, it is written here in mu a log mu a where a belong to such wedge of partitions f minus j a. And notice, I hope most of all of you know that under the sum is sum of measure, log measure, there is minus before sum, it could be put after sum and we have in fact integral of the information function. One can change order, first take a limit at almost every point divided by n and then integrate and this is the same by Shannon MacMillan Bremer theorem. So entropy is just integral of local entropy. So in this lecture I shall discuss phi a little bit special, will be minus t log f prime mainly for parameter t bigger than zero and this is different than, harder than what I discussed before because f can have critical points and in invariant set I will discuss and then with positive t we have plus infinity, we might have troubles. And negative then the situation is also interesting but simpler, one can use perovrbeni operator. You take x, so this is the case, one puts weight zero in this operator in some places which brings to some troubles but it is not anything much new. So I will discuss two settings, complex and real. Complex is Julia set, I mean f restricted to Julia set and real is what we call with Juan River Aleterdier with which we elaborated this setting, generalized multimodal map. So the map is defined in the neighborhood of a set which is invariant for this map. We allow and even are interested in having critical points in the invariant set so that it is not just uniform hyperbolic. They are non flat, we allow inflection critical points and either we assume bounded distortion for iterates or we assume C3 after changing a map maybe in absence of parabolic orbits, periodic orbits. We prove bounded distortion that there is such a theory in particular developed by Sebastian Van Streen and Edson Vargas and other people. So we assume topological transitivity and moreover positive topological entropy. So our invariant set K is maximal in a finite union of pairwise disjoint closed intervals, hat i, whose we choose them in such a way we don't want too much in these intervals so we choose them so that the ends of these intervals belong to K and this maximality corresponds to Darbu property on pieces of monotonicity. It means that if we have between true image points, some point in K, then its prey image also belongs to K. It's clear that it's related to this maximality because if prey image does not belong to K, we can include it to K so that we don't have a maximum. So this is the same after some manipulations. This is the story about C3 and we try to make it other general, one can restrict the class a little bit by just taking multimodal map of interval and removing basis, immediate basis of attracting periodic orbits then what's left is called a Julia set so this is left in more general and an example of such sets are given by spectral decomposition say, Booge by Demelo and Van Streen where we have renormalization then what is left is such a set then we renormalize again and what is left is such a set we don't want entrance renormalizable situation and it varies on set because such set doesn't satisfy some conditions like positive topological entropy. So here, why this function and this pressure is called a geometric potential we introduced this name for the first time in the book with Urbański I use the location to advertise the book I brought it here from Cambridge, 2010 and justification is because such SN of phi, the special phi along block of trajectory is just derivative in power minus t0 so wouldn't be minus if we went backward and then it is just diameter of pullback of a large disk in power t0 and if t0 is chosen appropriately so that more or less sums of these diameters in power t0 over all pullbacks are more or less far from 0 and far from infinity are very close to one then we can distribute to measure like in the uniform hyperbolic case when these bn are roughly balls and then it's easier to prove if one remembers definition of Hausdorff measure and dimension and dimension is just t0 because this is just exponent to which we need to take diameters to have some number which is not close to 0, not close to infinity this is in definition of Hausdorff measure so this is the right exponent which gives Hausdorff dimension ok, now what happens in presence of critical points is that the same as before variational definition makes sense this is due to integrability of log f prime we have this as I said special class non-uniform hyperbolic sometimes called topological that all of exponents are bounded away from 0 and then similarly in the uniform hyperbolic case we have Hausdorff dimension of k equal t0 but in general t0 is not in Hausdorff dimension, but hyperbolic dimension it may happen that hyperbolic dimension is less and hyperbolic dimension is supremo of dimensions of invariant isolated hyperbolic subsets of k and this here k this is both complex and real situation and hyperbolic dimension as I told is defined it is written, but it is 0 of hyperbolic pressure to which I would go in a moment but you can guess that instead of all measures I would consider only hyperbolic measures exponent or just dimensions of hyperbolic subsets pressures of hyperbolic subsets here is the picture from my paper with Katrin Gelfert and Michal Rams this picture is accompanied thereby legend transform of these functions to get dimension spectrum for Lapunov exponents but here we don't need it so I put on one picture and you can see how the graph of pressure looks like as I already explained all these graphs are convex because this is supremo of fine functions my variational principle and here is t plus t plus is phase transition place where after t plus the pressure is equal to a fine function t minus t there may happen also another phase transition parameter t minus for negative t in which I am not interested here because we consider t positive only and what I will tell in a moment is holds for t between t minus and t plus where by definition pressure is bigger than linear functions which means that it is not that individual trajectories carry most of the pressure why I assume t I consider often t between t minus and t plus so that yeah so why don't pressure is bigger here I don't plan to talk about this on this lecture but it would be bad not to mention this theorem which is my theorem from my papers with Juan but there are many other papers where there are related results that I listed people above and this says that as I told for t between t minus and t plus for each t there is a unique variant equilibrium measure it's ergodic, absolutely continuous with respect to an adequate conformal measure so I just removed part of the theorem not to show it too long about existence and uniqueness of conformal measure which satisfies probability laws here and there are some assumptions in the interval case like interval and no, basically in interval there are no additional assumptions we assume topological transitivity here so we priori have kind of mark of partition or puzzle partition in complex case one needs to assume something which is expanding away from critical points which is automatically true for interval as it was referred to here already for complex case it is not so one needs to assume something and also some kind of mark of structure around not structure just but some pieces nice pieces which surround critical points so now as I may be not told but at least it was written in my slides if we have this potential minus t log of prime t positive then it doesn't make sense to consider pressure via separated sets because we could put points just at places where the function is infinite so and take supreme so instead we just take individual points and preimages you see here the same as before but why are preimages of x0 not just a set of an epsilon separated points so to some extent preimages of one point are separated in the hyperbolic case they are separated non-uniform hyperbolic or non-hyperbolic it is more delicate and as I wrote here in a while we shall discuss independence of this notion of x0 but we would need to discuss this also other definitions of geometric pressure and here is very useful is hyperbolic pressure which I already mentioned before saying you about hyperbolic dimension is very zero hyperbolic pressure is a supremum of pressure over compact invariant subsets of k or Julia set k and interval k is Julia set complex k case and I explain here what does it mean I explain what does it mean that subset is hyperbolic or expanding which is standard with expansion rate so I think I use it this notion on previous slides already so order here is not the best one and here comes very important notion notion of a point to be safe or safe with respect to critical trajectories it means that safe means that a point is not trajectory of critical point but it is also not fast, not exponential accumulated by critical trajectories it is written for every delta forward trajectory is bounded away by x minus delta and I mean block of forward trajectory for j between 1 and 10 and as these boards we avoid by critical points shrink and number of them with n growth linearly then of course household dimension and even other dimensions with other gauge functions are zero so this is very small set and in the complex case for every complex so could be k instead of k maybe it would be better to write j of f but as I wrote before k is j of f this 3 pressure for every safe point is hyperbolic pressure and maybe I will go back to this in a moment maybe I will no, no, I will not jump into these slides so it occurs that in the real case it's also true which is not a big step compared to what we had before but it needs some additional observations before we assume it's something else to know that 3 pressure is independent what is weak isolation definition is at the bottom it means that if we have periodic point close to k then it is inside k periodic trajectory inside a small neighborhood of k then it is k, it is technical and has other equivalent definitions and it is technical because when doing shadowing we capture periodic orbit we want this periodic orbit to belong to k this is about interval k a priori it need not be true it's weaker than the set is isolated and examples are just sets from spectral decomposition so there is another definition what does it mean, a point, not a set a point is called hyperbolic if you can exponentially go to large scale without capturing critical points so fn maps injectively the pullback of a large ball to large ball and then the quality between 3 and hyperbolic pressure for z-safe and hyperbolic can be proved by shadowing argument and this was already in our paper in Stas Mirnov one can make this ball x minus what let me write like this and then after time like small kappa times n it is a big ball here we don't have critical points do not hit I mean forward orbit till time n do not hit this or even to time slightly longer than not hit non-critical pullback and then finite time allows us to close this so this is how we capture hyperbolic sets so hyperbolic pressure is bigger than 3 pressure if we have 2 properties safety and also hyperbolicity of a point so it was not clear before whether in the interval case this hyperbolicity is needed or not needed the opposite but if we have both safety and hyperbolicity then there is no problem the other inequality is trivial if you have hyperbolic set then when you consider 3 then 3 towards the hyperbolic set gives pressure like pressure on the hyperbolic set there might be some additional branches but this doesn't matter for this side we call it here I refer it to Katoka argument because if one has hyperbolic invariant measure with positive level of exponent one can make by using kissing theory the same type of construction just regular backward branches in inverse limit and after short time closing everything to capture periodic orbits or hyperbolic sets ok, so then as I already told for safe hyperbolic there is no problem but we need to know that pressure 3 pressure does not depend on if z is only safe it makes sense to prove such things because there are almost everything almost all points are safe so here some additional assumption about periodic parabolic points I don't want to involve in that so in the complex case it is sufficient to have this property safe because if we have two points as it is written here one is z1 and z2 and if we have a set w which in application will be just union fj crit j from 1 to n then you can find a sequence of disks omitting w such a chain of disks so that twice bigger disks are disjoint with w infimum is called in fact quasi hyperbolic distance in the plane minus w and then the number of these disks is square root of m there is also square root of log 1 over r but if I a priori assume that these points are within some distance log of 1 over distance is much less is much less than m then it doesn't matter when I take backward trajectories corresponding backward trajectories disk we have bounded distortion so together distortion is bounded by constant in the power root of m so we have comparable we have sufficiently comparable derivatives giving the same pressure in the interval case you cannot do this because there is no space no room for such curves so we need to find another proof of independence of this 3 pressure on the point this is, I call it this lemma qh because of this quasi hyperbolic now so crucial notion in these things is backward Lapunov stability that if you have sufficiently small disk then or interval then preimages are also pullbacks are also short if you start with delta it doesn't grow more than epsilon then in the complex situation it's not always true so we have some problems there in the interval case in absence of parabolic orbits it's true in the interval case, due to our assumptions we don't have wandering intervals it plays a role in all these things so how we do in real case we just need to take any point w and compare it with another point, but if we have another point just z, we can take preimages up to some time capital N, some constant so that preimages are delta here delta divided by 3 dense so that further pullbacks are all the time shorter than epsilon this is first we find delta and next we find this capital N so from one point we get some net of points and then there are two cases either a point w which we want to compare pressure tree of w we want to compare with pressure for z or its preimages z1, z2 either w is between two such points and then we pullback this interval z1, z2 and what can happen there is a procedure but I don't have time here this was first written by Juan preprint I don't know whether it is published preprint is from 2012 and then we repeated this procedure in our paper with Catherine Michał, we just pullback when we capture of course we pullback twice bigger interval but we control distortion we pullback we capture critical point then at one step we lose constant factor only when we compare diameter with derivative then again pullback until we but twice bigger interval captured critical point etc and as I wrote using bounded distortion we prove that if we take zn situation is that this is one of points in the boundary of pullback the problem is that if we pullback then it may happen something like this this is zn and here is zn prime so here we can have preimages of the same point zn means one of points which bound pullback we don't know to which of the original z1, z2 it goes after time n nevertheless first we can compare derivative with ratio of diameters of pullback divided by original we can compare diameter of pullback of slightly different interval v not w but it is v s kind half of w and we don't capture critical points at all if we pullback not large interval but x minus delta n size interval so derivatives comparable to ratio this ratio includes big interval which is x minus t epsilon1 but this epsilon1 is close to zero so it doesn't influence there is another case when a safe point bounds a large interval gap in the complement of k it is not between two delta close point z1, z2 but then if it is hyperbolic then we had before some finite set of preimages of z and then we can take further preimages after sub exponential time we are already, well this is comparison of exponential rates if the point is repelling then in fact instead of z1, z2 we can take some preimages but not for after very long time and have good situation don't have time to involve too much because we have another topic to which I have how much time, five minutes two, so another thing is pressure via spinning sets and here it indeed occurs to be the same but we need to assume something you see, we assume weak backward Lapunov stability so instead of pulling back large disk we take x minus delta n size of the disk and such a disk and we can assume that such a disk after time not bigger than n it does not grow it grows and grows sub exponential pullback and then if we assume this we obtain disk quality star and this weak backward Lapunov stability holds for example if we have only one critical point in Julia set because Lapunov exponents such a point must be non-negative lower Lapunov exponent and this is my result so here is under some assumption and again to compare derivative and diameters we need shrinking neighborhood technique which is when you pull back only substantial captures of critical points count you trankate each time some small annulus so that series of modularity of annularity so that something is left this allows to compare derivative with ratio of diameters so so we instead of comparing the spanning pressure with 3 pressure it is sufficient to compare with hyperbolic pressure we already know that 3 pressure for safe point is the same as 3 pressure for safe and hyperbolic which is the same as hyperbolic pressure and here it's written that ok, we have 3 and then ok, we have balls which are around ok, there are balls of diameter x minus delta and I will not read it of lack of time and these balls don't can be pulled back well by this week annularity and these pull backs give me their origins give me this I need to build this construction needs to build it's an epsilon separate an epsilon spanning set and then there are points which I have troubles then if Julia set is connected then I can take points on the boundary of pull backs if not then we use uniform of the fact that Julia set is uniform perfect which means cannot be cut by annularity separated by annularity which have too big modularity and this is in the interval case here I have troubles because of if I build spanning set this is here this all is the same comparison with hyperbolic pressure to construct spanning set I need to somehow take an account parameters of points which are close to large gaps and then there is a problem because if I have a point here is a gap and if I have such a point then here is my point which is safe then when I pull back it might happen unless we assume K just equals the whole interval not the counter set then it may happen such that this point is not it might happen that in the pull back we have such a situation and instead of parameters of this point we can get parameters of this point and then we don't know what is 3 pressure of this point unless this point is safe then 3 pressures are the same so here there are some troubles I think my last slide is this thank you