Christian Bonatti - Aperiodic classes of C1-generic diffeomorphisms.
Conley theory splits in a natural way the dynamics of any homeomorphisms of a compact manifold in chain recurrence classes, separated by the regular levels of a Lyapunov function. For C1-generic diffeomorphisms, a consequence of the Hayashi connecting lemma and its successive generalizations is that any chain recurrence classe containing a periodic orbit coincides with the homoclinic class of it. This provides many tools for understanding the dynamics inside the classes containing periodic orbits. The other classes, called "aperiodic classe" are much less understood. Their existence is proved for instance for generic diffeomorphisms in Newhouse regions. However, the classes that we knew, until recently, had a very well understood dynamics: they are all "adding machines" also called "odometers". This specific dynamics was a consequence of our unique way to detect them. In a long work with Katsutoshi Shinohara (4 long papers) we develop another procedure producing aperiodic classes for which we have a control on the dynamics. This leads to a great variety of distinct behavior for (uncountably many) aperiodic classes in the same generic diffeomorphisms. In this talk I will try to motivate this work, and to give some ideas of the tools, of the construction and on the dynamics of the aperiodic classes. Video
Sylvain Crovisier - Unicritical Surface Maps and Their Renormalizations
We discuss how the dynamics of dissipative surface maps can be analyzed through successive renormalizations. We focus on two closely related classes of infinitely renormalizable maps of the disc: unicritical diffeomorphisms (characterized by a single non-degenerate critical point) and Hénon-like maps. Under suitable regularity assumptions, we establish and prove a priori bounds for the renormalizations of these maps. These bounds provide uniform control over the small-scale geometric structure of the dynamics and guarantee the precompactness of the renormalization sequence. This is joint work with Mikhail Lyubich, Enrique Pujals and Jonguk Yang. Video
Jonathan DeWitt - Expanding on Average Random Dynamics
We consider exponential mixing for volume preserving random dynamical systems on surfaces. Suppose that $(f_1,...,f_m)$ is a tuple of volume preserving diffeomorphisms of a closed surface M . We now consider the uniform Bernoulli random dynamical system that this tuple generates on $M$ . We assume that this tuple satisfies a condition called being "expanding on average," which means that there exist $C>0$ and a natural number $N$ such that for all unit tangent vectors $v$,$\mathbb{E}[ln∥Df^Nv∥]>C$ , where the expectation is taken over all the realizations of the random dynamics. From this assumption we show quenched exponential mixing. (This is joint work with Dmitry Dolgopyat) Video
Lorenzo J. Díaz - Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms
For a broad class of nonhyperbolic and transitive partially hyperbolic diffeomorphisms having a one-dimensional center, we prove joint flexibility with respect to entropy and center Lyapunov exponent. Flexibility means that for any given value of the center Lyapunov exponent and any value of entropy less than the supremum of entropies of ergodic measures with that exponent, there is an ergodic measure with exactly this entropy and exponent. Our hypotheses involve minimal foliations and blender-horseshoes, they formalize the interplay between two regions of the ambient space, one of center expanding and the other of center contracting type. A non-exhaustive list of examples our results can apply to includes fibered by circles, flow-type, some Derived from Anosov diffeomorphisms, and some anomalous (non-dynamically coherent) diffeomorphisms. This is a joint work with K. Gelfert, M. Rams, and J. Zhang. Slides, Video
Bassam Fayad - On the stability and instability of elliptic equilibria and invariant quasi-periodic tori of real analytic Hamiltonians
Anton Gorodetski - On fibered rotation numbers of one-parameter families of cocycles
In the talk we describe the properties of the fibered rotation numbers of one-parameter families of smooth circle cocycles over an ergodic transformation in the base. In the case of projective cocycle we establish an analog of Johnson's Theorem connecting intervals of constancy of the rotation number with uniform hyperbolicty of the cocycle. Also, we show that the rotation numbers must be log-Holder regular with respect to the parameter. As an immediate application, we get a dynamical proof of 1D version of the Craig-Simon’s theorem from spectral theory that claims that the integrated density of states of an ergodic Schrodinger operator must be log-Holder. Finally, we provide a new explicit formula for the rotation number of a cocycle and discuss its application to spectral theory of ergodic discrete Schrodiger operators. The talk is based on a series of results obtained jointly with Victor Kleptsyn and Pedro Duarte. Slides, Video
Adam Kanigowski - K and Bernoulli properties in smooth dynamical systems
Both K and Bernoulli properties quantify chaoticity of a system. They are defined for an abstract measure preserving system and it is immediate that Bernoulli implies K. It is also known (although this is much harder) that K does not imply Bernoulli. We will focus on discussing these two properties in the class of smooth (partially hyperbolic) systems. We will recall some classical results, discuss recent developments and state some open questions. Video
Tamara Kucherenko - Thermodynamic formalism for coded shifts
In the classical theory of uniformly hyperbolic systems, the dynamics is studied by way of an associated shift of finite type derived from a Markov partition. However, in the absence of uniform hyperbolicity a smooth system cannot generally be modeled by shifts of finite type and instead shifts on countable alphabets have been used. There is some evidence (e.g. for certain model classes of bifurcating heterodimensional cycles) showing that coded shifts, which are limits of increasing families of irreducible subshifts of finite type, could be a potential tool for studying non-uniformly hyperbolic systems. The advantage here is that, in contrast to countable shifts, the coded shifts are compact. I will discuss thermodynamic formalism for coded shifts and present results concerning the uniqueness of measures of maximal entropy and equilibrium states for Hölder potentials, as well as their properties. Slides, Video
Homin Lee - Positive entropy actions by higher rank lattices
In this talk, we will discuss about smooth higher rank lattice actions on manifolds with positive entropy. For instance, when a lattice in SL(n, R) act on an n-dimensional manifold with positive entropy with n at least 3, we will see that the lattice is commensurable with SL(n, Z). Furthermore, we can obtain topological information about the manifold. This is a joint work with Aaron Brown. Video
Karina Marin - Lyapunov spectrum of volume-preserving partially hyperbolic maps
The Lyapunov spectrum of a map is said to be simple if every Lyapunov exponent has multiplicity one. That is, if the Oseledets decomposition is given by one dimensional subspaces. This problem has been extensively studied in the context of linear cocycles. In this talk, we discuss the simplicity of the Lyapunov spectrum for partially hyperbolic volume-preserving diffeomorphisms with two dimensional center bundle. This is a joint work with D. Obata and M. Poletti. Video
Matilde Martínez - Solenoidal surfaces of finite type and their horocycle flows
Hyperbolic surfaces of finite type are classical objects, and their horocycle flows are well understood both from the topological and the ergodic-theoretical viewpoints. Solenoidal surfaces are foliated spaces very similar to surfaces. We will consider non-compact solenoidal surfaces "of finite type" with a hyperbolic structure, and give a complete topological description of their horocyclic orbits. This is joint work with Fernando Alcalde, Álvaro Carballido and Alberto Verjovsky. Slides Video
Meysam Nassiri - Blenders and Ergodicity
We introduce a local and robust mechanism that ensures the ergodicity of smooth actions. This approach aims to reveal intrinsic ergodic properties of certain hyperbolic sets, including the classical blenders of Bonatti and Díaz. In particular, it yields new examples of stably ergodic, partially hyperbolic, volume-preserving diffeomorphisms, as well as alternative proofs of some recent results in the field. The main result uses ideas developed in collaboration with Abbas Fakhari and Mojtaba Zareh Bidaki. This is joint work with Hesam Rajabzadeh. Video
Davi Obata - Absolute continuity of stationary measures
We study random dynamical systems generated by smooth surface diffeomorphisms. Brown and Rodriguez Hertz showed that hyperbolic stationary measures typically have the SRB property—meaning absolute continuity along unstable manifolds—except in the presence of specific obstructions. We aim to identify conditions under which SRB stationary measures are absolutely continuous with respect to Lebesgue measure on the ambient space. This is a joint work with Aaron Brown, Homin Lee, and Yuping Ruan. Video
Yi Pan - Hyperbolicity of renormalization of quasi-periodic cocycles
Hyperbolicity of renormalization has been studied in several different settings. We will talk about a first result concerning quasi-periodic cocycles. As an application, we will show a global reducibility result of quasi-periodic symplectic cocycles: given one parameter family of such cocycles, for almost every parameter, either the maximal Lyapunov exponent is positive, or the cocycle is almost conjugate to some precise linear model. The techniques also include Kotani theory and KAM theory. This is a joint work with Artur Avila and Raphaël Krikorian. Video
Cagri Sert - Projections of self-affine fractals
I will discuss the extension of Falconer’s landmark 1988 result – on the Hausdorff dimension of typical self-affine fractals – to linear projections of these fractals. The result uncovers an algebraic structure on the exceptional sets of projections in the sense of Marstand projection theorem. Furthermore, the results comes with various examples of new phenomena that I will mention. This include existence of equilibrium states having non-exact dimensional linear projections (equilibrium states themselves are exact dimensional by Feng); existence of self-affine fractals in dimensions at least 4, whose set of exceptional projections contains higher degree algebraic varieties in Grassmannians (such constructions are not possible even in Borel category in dimension 3 by the solution of a conjecture of Fässler-Orponen by Gan et.al., nor in any dimension if the linear parts of affinities acts strongly irreducibly on all exterior powers, by Rapaport); existence of self-affine fractals whose sumsets have lower than expected dimension without satisfying an arithmetic resonance (impossible in dimension 1 by Hochman, Shmerkin, Peres and in dimension 2 by Pyorälä). Joint work with Ian D. Morris. Slides, Video
Amie Wilkinson - Minimality of strong foliations of Anosov diffeomorphisms
I will discuss work with Avila and Crovisier (and related work with Eskin, Potrie and Zhang as well) on the following problem and where it has led us: Let f be an Anosov diffeomorphism in dimension 3. Assume the unstable bundle is 2 dimensional and admits a dominated splitting into weak and strong unstable bundles. Under what hypotheses is the strong unstable foliation minimal? Video
Alberto Verjovsky - Adelic loop spaces
Masato Tsujii - An extension of Katok non-stationary normal coordinate along one dimensional invariant manifold
Sébastien Alvarez - Asymptotic counting of surfaces in negatively curved three manifolds
In this talk, we present a two dimensional analogue of the geodesic flow, namely the space of surfaces of constant curvature inside a closed negatively curved 3-manifold. The dynamical properties of this space are described in terms of actions of PSL(2,R) and the equidistribution properties allow to study the geometric rigidity associated to the counting of such surfaces according to their area. This is joint work with Graham Smith (PUC RIo de Janeiro) and Ben Lowe (University of Chicago).
Snir Ben Ovadia - Generalized u-Gibbs measures for Coo diffeomorphisms
We show that for every C∞ diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points with a positive Lyapunov exponent (in a weak sense) then there exists an invariant probability measure with a disintegration by absolutely continuous conditionals on disks subordinated to unsta- ble leaves. As an application, we solve the strong Viana conjecture in any dimension. This is joint work with D. Burguet.
Sergi Burniol - Horocyclic flows in nonpositive curvature
I will present a classification of horocyclic closures on finitely generated surfaces with nonpositive curvature. When the surface has negatively pinched curvature, this classification is a well-known result (Hedlund, Dani, Dal'bo). In this case, horocycles are either closed or dense in the non-wandering set. In nonpositive curvature, a new phenomenon arises: there exist wandering horocycles that accumulate in the non-wandering set. I will also discuss the classification of invariant measures for the horocyclic flow. During my talk, I will explain the relationship between horocycles, the associated geodesic rays, and the ends of the surface.
Giovanni Canestrari - Linear response for discontinuous perturbations of smooth endomorphisms
Linear response theory explains how long-time statistics react to small changes of the parameters defining the dynamics. After a brief review of the state of the art, we discuss the problem of linear response for perturbations that create discontinuities. Finally, we give abstract conditions for linear response for discontinuous perturbations of a smooth hyperbolic map and we discuss their applicability with a concrete example. Slides
Marisa Dos Reis Cantarino - Computer-assisted proof of robust transitivity in dimension 3 using blenders
Blenders originally emerged as objects in dynamical systems as an example given by C. Bonatti and L. Díaz of a system which is not uniformly hyperbolic but it is robustly transitive. Roughly speaking, on an n-manifold, a blender is a hyperbolic invariant subset of the system that allows for robust intersections of s-dimensional stable manifolds and u-dimensional unstable manifolds, with s + u < n. The way to make this intersection robust without "the right dimensions" is to make the stable set of the blender to "fill the space as if it is higher dimensional". This intersection allows the existence of robust heterodimensional cycles, giving conditions for robust transitivity. We develop a computer-assisted strategy to prove robust transitivity in dimension 3 that includes the proof of the existence of a blender. We present a family of D.A. (derived-from-Anosov) systems on the 3-torus for which this strategy applies. This work in preparation is a collaboration with Andy Hammerlindl and Warwick Tucker.
Pierre-Antoine Guihéneuf - Rotational Axiom A homeomorphisms for higher genus surfaces
Consider a homeomorphism of closed surface of genus $\ge 2$. I will explain that in the case its homological rotation set is big enough, the whole rotational behaviour is contained in a compact set that resembles a finite union of homoclinic classes with some heteroclinic connections. This is related to $C^0$ rotational versions of properties like Markov partitions, rotational density of periodic orbits, stability under perturbations... as well as purely rotational features such as the description of the rotation set's shape or some bounded deviations properties. The whole thing is based on Le Calvez-Tal forcing theory but I will mainly focus on some examples.
Martin Leguil - Rigidity of transitive Anosov flows in dimension 3
In a joint work with Andrey Gogolev and Federico Rodriguez Hertz, we study when two transitive Anosov flows X and Y in dimension 3, which are topologically conjugated, are actually smoothly conjugated. By the work of de la Llave, Marco, Moriyón, and Pollicott, a necessary and sufficient condition for this to happen is that the stable and unstable eigenvalues at corresponding periodic points match. In our work, we show that for generic transitive Anosov flows X and Y in dimension 3, the latter condition is already implied by the existence of a topological conjugacy, unless the conjugacy swaps the positive and negative SRB measures of the two flows. In particular, generic transitive Anosov flows in dimension 3 are locally rigid. This complements a recent work of Gogolev and Rodriguez Hertz in the volume-preserving case.
Amadeus Maldonado - Exponentially mixing SRB measures are Bernoulli
The Bernoulli property is the strongest statistical property that a measure preserving system can exhibit. It not only implies other important statistical properties such as ergodicity, mixing and the K-property, but, as shown by Ornstein, Bernoulli systems with the same entropy are measurably conjugate. We prove that, for $C^{1+\alpha}$ diffeomorphisms of compact manifolds, exponentially mixing SRB measures are Bernoulli. This extends a recent result by Dolgopyat, Kanigowski and F. Rodriguez Hertz. Using similar techniques, we also show that if volume is almost exponentially mixing, then the limit SRB measure constructed by Ben Ovadia and F. Rodriguez Hertz is Bernoulli.
Santiago Martinchich - Unstable minimal sets for collapsed Anosov flows
A partially hyperbolic diffeomorphism $f:M\to M$ splits the tangent bundle in three subbundles, $TM=E^s\oplus E^c \oplus E^u$, which have, respectively, a contracting, dominated and expanding behavior by $Df$. The bundle $E^u$ is known to be tangent to an invariant foliation $\W^u$, and understanding the structure of this foliation is useful for obtaining dynamical consequences for $f$. For instance, minimality of $\W^u$ implies that $f$ is transitive, and a bound in the number of minimal sets of $\W^u$ gives a bound in the number of attracting regions for $f$. In this talk I will discuss ongoing work about the number of minimal sets of $\W^u$ for certain collapsed Anosov flows. Joint with Sylvain Crovisier and Rafael Potrie.
Juan Carlos Mongez - Partially Hyperbolic Diffeomorphisms With a Finite Number of Measures of Maximal Entropy
We prove the finiteness of the number of ergodic measures of maximal entropy for a class of partially hyperbolic diffeomorphisms derived from Anosov systems on the 4-torus $\mathbb{T}^4$. Moreover, we show that within a class of skew products over this class of derived-from-Anosov partially hyperbolic diffeomorphisms, there exists a $C^1$-open and $C^r$-dense subset of diffeomorphisms with finitely many ergodic measures of maximal entropy. These results follow from a more general theorem: we establish the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms whose center bundle admits a dominated splitting into one-dimensional subbundles, and for which there exists a uniform lower bound on the absolute value of the Lyapunov exponents along the center direction. This is joint work with Mauricio Poletti and Maria José Pacifico.
Kangrae Park - Diophantine Approximation on the Kleinian Circle Packing
For a given Kleinian group acting on hyperbolic space \(\mathbb{H}^3\), the associated limit set lies on the boundary \(\partial \mathbb{H}^3\). If this limit set forms a circle packing, it is referred to as a Kleinian circle packing. As part of an ongoing project, we aim to study the Hausdorff dimension of the \(\epsilon\)-bad set arising from such circle packings. We construct a conformal graph-directed Markov system from the packing and apply perturbation theory to estimate the dimension.
Zahra Reshadat - Quasi-conformal behaviours in linear cocycles. (slides of talk)
In this talk we discuss about the dynamics of linear cocycles over chaotic bases such as subshifts of finite type. The investigation focuses on the stability and abundance of cocycles with quasi-conformal behaviour on fibers, particularly within the class of cocycles that do not admit dominated splitting. We construct $C^\alpha$-open sets of cocycles that exhibit quasi-conformal behavior by introducing a new mechanism that yields stable elliptic-type dynamics even in higher dimensions. It is shown that every cocycle over a shift of finite type either admits a dominated splitting or can be $C^0$-approximated by a cocycle that $C^\alpha$-stably exhibits quasi-conformal orbits. Moreover, every $SL(d,R)$ cocycle over a shift of finite type either admits a dominated splitting or can be $C^0$-approximated by a cocycle that $C^\alpha$-stably supports an ergodic measure of positive entropy on uniformly bounded orbits. This talk is based on joint works with Meysam Nassiri and Hesam Rajabzadeh.
Graccyela Salcedo - A Version of Oseledets Theorem for Proximal RDSs on the Circle
In this talk, we will study proximal Random Dynamical Systems (RDSs) on the circle with no common fixed point. We will first focus on RDSs of homeomorphisms. Our main result is the existence of two random points that control the asymptotic dynamics of the forward and backward orbits. By restricting our attention to RDSs of diffeomorphisms, we link these random points to the extremal Lyapunov exponents, providing a version of the Oseledets theorem adapted to this setting. As an application, we prove the dimensional exactness of the stationary measure associated with the RDS. This is joint work with J. Bezerra.
Sven Sandfeldt - Centralizer rigidity of partially hyperbolic diffeomorphisms
The smooth centralizer of a diffeomorphism $f$ is the set of all diffeomorphisms $g$ that commute with $f$. In this talk I will discuss the centralizer of partially hyperbolic diffeomorphisms on nilmanifolds. In particular, I will discuss rigidity phenomenon for partially hyperbolic diffeomorphisms with a large centralizer and the related problem of rigidity of higher rank partially hyperbolic actions.
William Wood - Geometry of the Hyperbolic Locus in SL(2,R)^3
In this talk we discuss the geometry of the hyperbolic locus in SL(2,R)^3. The hyperbolic locus is an open set in SL(2,R)^n consisting of all n-tuples that would correspond to a uniformly hyperbolic set of matrices (of size n). In work by Avila, Bochi, and Yoccoz, the geometry of the hyperbolic locus is studied, and the geometry is thoroughly explored in SL(2,R)^2. Open questions were posed about the geometry of the locus in higher dimensions, and we address two of the questions here.
Jinhua Zhang - A dichotomy for conservative partially hyperbolic diffeomorphisms: hyperbolicity versus nonhyperbolic measures
In a conservative and partially hyperbolic three-dimensional setting, we study three representative classes of diffeomorphisms: those derived from Anosov diffeomorphisms, diffeomorphisms in neighborhoods of the time-one map of the geodesic flow on a surface of negative curvature, and accessible and dynamically coherent skew products with circle fibers. In any of these classes, we establish the following dichotomy: either the diffeomorphism is Anosov, or it possesses nonhyperbolic ergodic measures. Our approach is perturbation-free and combines recent advances in the study of stably ergodic diffeomorphisms with a variation of the periodic approximation method to obtain ergodic measures. This is a joint work with Lorenzo J. Díaz and Jiagang Yang.
Jiesong Zhang - Invariant distributions of partially hyperbolic systems: fractal graphs, excessive regularity, and rigidity
We introduce a novel approach linking fractal geometry to partially hyperbolic dynamics and establish the non-fractal invariance principle. As an application, we reveal fractal behavior and regularity jumps for certain partially hyperbolic systems and establish rigidity results. We also obtain several rigidity results when the distributions have C¹ or higher regularity. This is based on joint work with D. Xu and ongoing work with M. Leguil and D. Xu.
Jing Zhou - On the original Ulam's problem and its quantization
In the 60s, Ulam proposed a simplification of Fermi's idea, as to explain for the existence of high-energy particles in the cosmic rays: a massless particle bounces elastically between two walls, one moves periodically and the other fixed. Ulam conjectured the existence of escaping orbits whose energy grows to infinity, based on numerical observations. In the 90s Zharnitsky rigorously found a one-parameter family of linearly escaping orbits in Ulam's original setting, which makes up null set in the infinite area phase cylinder. In this talk, we try to provide a complete (modulo a set of at most finite measure) answer to Ulam's original question, showing that the typical behavior on the phase cylinder is recurrent/oscillatory, and also give a precise characterization of where those exceptions of escaping orbits occur on the phase cylinder. On the quantum version, we provide explicit link between the energy growth and the shape of quasi-energy spectrum of the system. This is a joint work with Changguang Dong.
FIRST WEEK:
Hengyi Li: Strong Positive Recurrence and Exponential Mixing for Interval Maps
Following the recent published work of J. Buzzi, S. Crovisier, and O. Sarig, we introduce the notion of strong positive recurrence for interval maps. With the combined work of the coding for such systems and uniform integrability of log|f’|, we show that every C∞-interval map with positive topological entropy is strongly positively recurrent; thus, its measures of maximal entropy have exponential decay of correlations. The new methods enable us to work with infinitely many degenerate critical points. This is a work in progress.
Spencer Durham: The cohomological equation over parabolic base.
The cohomological equation plays an important role in diverse aspects of dynamical systems. I will present some results about the cohomological equation over a parabolic affine map and discuss their application to high-rank parabolic rigidity.
Leonardo Dinamarca: Quadratic growth for the derivatives of iterates of interval diffeomorphisms
(Joint with A. Navas) 20 years ago, in the seminal work of Polterovich and Sodin shows a (short but) surprising result: if the iterates of derivatives of a diffeomorphisms of the interval of class C^2 growth are subexponential, then it grows at most quadratic. In this talk we will discuss a stronger result that asserts that the lim_n max Df^n / n^2 exists. We briefly discuss the modern tools who allow us to obtain the results.
Elena Gomes: Invariant sets for homeomorphisms of hyperbolic 3-manifolds
In a joint work with Santiago Martinchich and Rafael Potrie, we studied dynamical properties of homeomorphisms of hyperbolic 3-manifolds by looking at how points escape to infinity in the universal cover. I will aim to present some result about how certain assumptions on this behavior force the existence of several compact disjoint invariant sets.
Mateo Ghezal: Measures of maximal entropy of smooth saddle surface endomorphisms.
In a recent work, Buzzi, Crovisier and Sarig answered a famous question by Newhouse showing that any smooth surface diffeomorphism with positive topological entropy admits a finite number of measures of maximal entropy. If we want to understand the number of these measures in the higher dimensional case, we may ask first what is happening for surface endomorphisms, where the pre-images give a hidden stable dimension. In the case of local diffeomorphism, one cannot expect a result as general as that of Buzzi, Crovisier, and Sarig. However, we will see that the number of measures of maximal entropy is still finite for a large class of endomorphisms which admits only saddle hyperbolic measures of maximal entropy. We will also try to discuss the obstructions in more general settings and the possible generalisations of our techniques.
SECOND WEEK:
Elivan Lima: Existence of Measures of Maximal Entropy for a certain class of C1 Robustly transitive endomorphisms displaying critical points
In this presentation, we will discuss the existence and uniqueness of measures of maximal entropy for a certain class of endomorphisms introduced by Lizana-Ranter, which are homotopic to a linear volume expanding endomorphism. This class of endomorphisms on $\mathbb{T}^2$ are $C^1$-robustly transitive partially hyperbolic with persistence of critical points. volume-expanding endomorphism. This is an ongoing project joint with Y. Lima, C. Lizana and M. Poletti.
Natalia McAllister: A computer-assisted proof of the existence of blenders for a 3-dimensional Hénon-like family
In this project we develop a computer program to verify the existence of blenders for concrete examples. A blender is a hyperbolic set whose unstable manifold, when looking at certain intersections, seems to have a greater dimension than it actually does. This descriptive definition cannot be verified on a computer. The first step is then stating necessary conditions for establishing the existence of a blender in a computer-friendly way. Then, we develop an algorithm to verify said conditions. Using this algorithm, we prove the existence of blenders for a family of maps defined as a skew product over the Hénon map. This algorithm could be extended to other quadratic maps, and potentially to other more challenging examples.
Kecheng Li: Unique equilibrium states for Viana maps with small potentials
We investigate the thermodynamic formalism for Viana maps—skew products obtained by coupling an expanding circle map with a slightly perturbed quadratic family on the fibers. For every Hölder potential \(\varphi\) whose oscillation is below an explicit threshold, we show that an equilibrium state not only exists but is unique and satisfies an upper level-2 large-deviation principle. All of these conclusions persist under sufficiently small perturbations of the reference map.
Amy Somers/Alexander Paschal: The Variational Principle for Countable State Shift Spaces With Specification
We define and discuss specification properties for countable state shift spaces, generalizing the well-studied specification property for compact shifts. These properties are special cases of definitions from an upcoming paper by Climenhaga,Thompson, and Wang. We present a class of examples of such shift spaces and discuss the variational principle for the Gurevich pressure of these spaces. This gives the foundations for developing the theory of equilibrium states in this new, non-compact setting.
Arantha Ranu: Weakened Gibbs inequality for continuous potentials.
The Gibbs inequalities is a set of two inequalities that relates the measure of cylinder sets to exponentials of Birkhoff sum of potentials. We know that they are satisfied by the equilibrium states of Holder potentials. We investigate the validity of a weaker version of Gibbs inequality for equilibrium states of continuous potentials.
Audrey Tyler: “A little hyperbolicity...”: Ergodicity of partially hyperbolic endomorphisms
The Pugh-Shub conjecture states that stable ergodicity is an open and dense property in the space of partially hyperbolic diffeomorphisms that are volume preserving. To study this conjecture the concept of accessibility has been introduced. Accessibility allows us to split the conjecture into two subconjectures; that most partially hyperbolic diffeomorphisms are accesible and that any accessible volume preserving system is ergodic.
These conjectures and the arguments used to study ergodicity in partially hyperbolic diffeomorphisms can be adapted to the case of non invertible dynamical systems. We proved that for volume preserving, partially hyperbolic, center bunched endomorphisms with constant Jacobian, accessibility implies ergodicity.