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• Counterexamples to a rigidity conjecture

  Speaker: Giovanni Forni (University of Maryland) -
  

  When: Thu, September 2, 2021 - 2:00pm
  Where: Kirwan Hall 1311
  

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  Abstract: We discuss several counterexamples to a rigidity conjecture of K. Khanin, which states that under some quantitative condition on non-existence of periodic orbits, C0 conjugacy implies C1 (even C∞) conjugacy. We construct examples of non-rigid diffeomorphisms on the 2-torus, which satisfy the assumptions of Khanin's (but not of Krikorian's) conjecture. We also construct examples of flows which are topologically conjugate, but not C1 conjugate, in contradiction to a natural generalization of the conjecture to flows. These latter examples are based on results on solutions of the cohomological equation and suggest that the structure of the space of invariant distributions has to be taken into account in rigidity questions.
  
• Strongly mixing actions of countable abelian groups are almost strongly mixing of all orders

  Speaker: Rigoberto Zelada (University of Maryland) -
  

  When: Thu, September 9, 2021 - 2:00pm
  Where: Kirwan Hall 3206
  

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  Abstract: Let $(G,+)$ be a countable discrete abelian group and let $(T_g)_{g\in G}$ be a strongly mixing measure preserving
  $G$-action on a probability space $(X,\mathcal A,\mu)$.
  We will present a result which states that $(T_g)_{g\in G}$ is "almost strongly mixing of all orders".
  It is worth noting that this result, when applied to $\mathbb Z$-actions, offers a new way of dealing with strongly mixing transformations.
  In particular, it allows us to obtain several new characterizations of strong mixing for $\mathbb Z$-actions, including a result which can
  be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his
  proof of Szemer{\'e}di's theorem.\\
  The proofs of these results rely on a new notion of largeness for subsets of $G^d$ and utilize $\mathcal R$-limits, a notion ofconvergence based on the classical Ramsey Theorem. This talk is based on joint work with Vitaly Bergelson.
  
• Invariant Family of Leaf measures and The Ledrappier-Young Property for Hyperbolic Equilibrium States

  Speaker: Snir Ben Ovadia (Penn State) -
  

  When: Thu, September 23, 2021 - 2:00pm
  Where: Kirwan Hall 3206
  

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  Abstract: Let $M$ be a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$, and let $f$ be a $C^{1+\beta}$ ($\beta>0$) diffeomorphism of $M$. Let $\varphi$ be a H\"older continuous potential on $M$. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $\varphi$), which sit on local unstable leaves.
  We present two main applications. First, given an ergodic homoclinic class $H_\chi(p)$, we prove that $\varphi$ admits a local equilibrium state on $H_\chi(p)$ if and only if $\varphi$ is ``recurrent on $H_\chi(p)$" (a condition tested by counting periodic points), and one of the leaf measures gives a positive measure to a set of positively recurrent hyperbolic points; and if an equilibrium measure exists, the said invariant and absolutely continuous family of measures constitutes as its conditional measures.
  Second, we prove a Ledrappier-Young property for hyperbolic equilibrium states- if $\varphi$ admits a conformal family of leaf measures, and a hyperbolic local equilibrium state, then the leaf measures of the invariant family (respective to $\varphi$) are equivalent to the conformal measures (on a full measure set). This extends the celebrated result by Ledrappier and Young for hyperbolic SRB measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure 0) has conditional measures on local unstable leaves which are absolutely continuous w.r.t the Riemannian leaf volume.
  
• Actions of Lattices in Higher-Rank Semisimple Groups

  Speaker: Darren Creutz (U.S. Naval Academy) -
  

  When: Thu, September 30, 2021 - 2:00pm
  Where: Kirwan Hall 3206 and through zoom - https://umd.zoom.us/j/97401191391
  

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  Abstract: Let \Gamma be an irreducible lattice in a higher-rank semisimple group G. Generalizing Margulis' Normal Subgroup theorem in the connected case, Stuck and Zimmer (1994) showed that every nonatomic probability-preserving action of \Gamma in a connected higher-rank Lie group G is essentially free. The general case was proven by the speaker and J. Peterson (2017).
  However, being (very) nonamenable, not every action of such a lattice on a compact metric space admits an invariant measure, leaving open the question of whether every minimal action of such a lattice on an infinite compact metric space is (topologically) free.
  
  I will present the complete result: every action of such a lattice is indeed (topologically) free. The main idea is to replace invariant measure by stationary measure (which always exist) and show that every stationary action is essentially free. This answers the general form of a question of Glasner and Weiss on URS's for such lattices.
  
  https://umd.zoom.us/j/97401191391
  
• Dynamics on Character Varieties

  Speaker: William Goldman (University of Maryland) -
  

  When: Thu, October 7, 2021 - 2:00pm
  Where: Kirwan Hall 3206 and through zoom - https://umd.zoom.us/j/97401191391
  
• Nielsen realization problem

  Speaker: Lei Chen (University of Maryland) -
  

  When: Thu, October 14, 2021 - 2:00pm
  Where: Kirwan Hall 3206 and through zoom - https://umd.zoom.us/j/97401191391
  

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  Abstract: Nielsen realization problem asks whether the projection Homeo(M)\to \pi_0(\Homeo(M)) has a section or not for a manifold M. This problem relates the existence of flat structure on the universal M-bundle. In this talk, I will focus on the case when M is a surface. I will talk about what other people have done in the past and then describe my results. The proof method involves homological obstruction, fixed point theory, surface dynamics and Poincare-Birkhoff Theorem. This is partly a joint work with Markovic.
  https://umd.zoom.us/j/97401191391
  
• Effective counting estimates for filling closed geodesics on hyperbolic surfaces

  Speaker: Francisco Arana-Herrera (Stanford University) -
  

  When: Thu, October 21, 2021 - 2:00pm
  Where: Kirwan Hall 3206 and via Zoom https://umd.zoom.us/j/97401191391
  

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  Abstract: Counting problems for closed geodesics on hyperbolic surfaces have been extensively studied since the 1950s. I will discuss a new quantitative estimate with a power saving error term for the number of filling closed geodesics of a given topological type and length at most L on an arbitrary closed, orientable hyperbolic surface. This estimate solves an open problem alluded to in work of Mirzakhani and advertised by Wright. The proof relies on recent developments on the theory of effective mapping class group dynamics.
  
  https://umd.zoom.us/j/97401191391
  
• Flexibility of the Pressure Function

  Speaker: Tamara Kucherenko (The City College of New York) -
  

  When: Thu, November 4, 2021 - 2:00pm
  Where: Kirwan Hall 3206 and via Zoom https://umd.zoom.us/j/97401191391
  

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  Abstract: We discuss the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics, since they correspond to phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We prove that in a setting of one-dimensional compact symbolic systems these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter. As a consequence, we obtain that for a continuous observable the phase transitions can occur at a countable dense set of temperature values. We go further and show that one can vary the cardinality of the set of ergodic equilibrium states as a function of the parameter to be any number, finite or infinite. This is based on joint work with Anthony Quas.
  
  https://umd.zoom.us/j/97401191391
  
• TBD

  Speaker: Dana Bartosova (University of Florida) -
  

  When: Thu, November 11, 2021 - 2:00pm
  Where: Kirwan Hall 3206
  
• TBD

  Speaker: Kelly Yancey (Institute for Defense Analyses) -
  

  When: Thu, November 18, 2021 - 2:00pm
  Where: Kirwan Hall 3206
  
• TBD

  Speaker: Karin Melnick (University of Maryland) -
  

  When: Thu, December 2, 2021 - 2:00pm
  Where: Kirwan Hall 3206