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{\Large המחלקה למתמטיקה, בן-גוריון}

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{\Huge קולוקוויום}\\[0.2\baselineskip]

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\textbf{ב}\emph{יום שלישי,  7 בינואר, 2020}
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\textbf{בשעה} \emph{14:30 -- 15:30}
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\textbf{ב}\emph{Math -101}

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ההרצאה

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{\Large\bfseries Universal models in ergodic theory and topological dynamics\par}
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תינתן על-ידי
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{\large\scshape Tom Meyerovitch 
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  (BGU)
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\textbf{תקציר:}
  A number of of important results in modern mathematics involve an understanding the space of invariant probability measures for a homeomorphism, a flow, or  group of homeomorphisms.

In this talk we will focus on finding situations where the space of  invariant probability measures  is
essentially ``as big as possible`{}`:
A topological dynamical system is \$(X,S)\$ \textbackslash{}emph\{universal\} in the ergodic sense if any  measure preserving system \$(Y,T,\textbackslash{}mu)\$, there exists an S-invariant probability measure \$\textbackslash{}nu\$ so that \$(X,S,\textbackslash{}nu)\$ is isomorphic to  \$(Y,T,\textbackslash{}mu)\$ as  measure preserving systems, assuming that the entropy of (Y,T,\textbackslash{}mu)  is strictly lower than the topological entropy of \$(X,S)\$.
Krieger`s generator theorem (1970) states that the shift map on the space bi-infinite of \$N\$-letter sequences is universal.
Lind and Thouvenot (1977) used Kreiger`s theorem to prove that Measure-preserving homeomorphisms of the torus represent
all finite entropy ergodic transformations. Recent conditions for universality  of Soo-Quas (2016) and David Burguet (2019) imply that any ergodic  automorphism of a compact group is universal. Together with Nishant Chandgotia we recently established a new and more general sufficient condition for ergodic universality.

Some new consequences include:
-  A generic homeomorphism of a compact manifold (having dimension at least 2) can model any aperiodic measure preserving transformation.
- Any aperiodic measure preserving transformation can be modeled by a homeomorphism of the 2-torus which preserves Lebesgue measure.
- The  space of 3-colorings of the standard Cayley graph of \$\textbackslash{}mathbb\{Z\}\^{}d\$, with \$\textbackslash{}mathbb\{Z\}\^{}d\$ acting by translations  is universal.    \newline
In this talk I will discus and  explain some of the older and newer results.
No specific background in ergodic theory will be assumed.
  


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