The original definition was introduced by Adler, Konheim and McAndrew in 1965. 8.5 Topological entropy for nonautonomous dynamical systems. 8 Generalizations of topological entropy.7.1 Topological tail entropy and symbolic extension entropy.6 Topological entropy in some special cases.5 Relation with Kolmogorov-Sinai entropy.4 Basic properties of topological entropy.In what follows \(\log\) denotes \(\log_2\) (although this choice is arbitrary). Roughly, it measures the exponential growth rate of the number of distinguishable orbits as time advances. Topological entropy is a nonnegative number which measures the complexity of the system. Let \((X,T)\) be a topological dynamical system, i.e., let \(X\) be a nonempty compact Hausdorff space and \(T:X\to X\) a continuous map. The number of orbits distinguishable in \(n\) steps grows as \(2^n\ ,\) generating the topological entropy \((1/n)\log_2(2^n) = 1\. Similarly, there are eight points (the black points), whose orbits are similarly distinguished in three steps (after one iterate the black points become the red and yellow points, after another iterate they become the blue, violet and green points). But there exist already four different points whose orbits can be distinguished in two steps: the red points are mapped onto the blue and violet points and any two of them are distinguished either immediately or after applying the transformation once. Initially there are at most two distinguishable points, for example, the blue points. Suppose that only points that are in opposite halves of the rectangle can be distinguished. Figure 1: Topological entropy generated in a so-called horseshoe: the rectangle is stretched, bent upward and placed over itself.
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Minnelli would go on to work opposite Robert De Niro in Martin Scorsese’s 1977 “New York, New York,” followed by the 1981 hit comedy “Arthur” with Dudley Moore. “It was long days shooting, and it was very intense. “We would go to set every morning, and she’d sleep on my shoulder, and we’d listen to ‘Morning Has Broken,’” Grey says. Fosse was a perfectionist, always wanting another take. They filmed in and around Berlin and Munich. “She was just very vulnerable and looked up to me.” “We used to sort of pile into Joe Allen’s, and she was always like the little sister I never had,” he says. Grey and Minnelli ran in the same New York theater circles. Photographed at The London West Hollywood at Beverly Hills She was so tender and frail and fragile and strong at the same time.” You just knew that she was somebody that was going to happen. “You could see that connection to her mother and father. “She was this fresh, bursting, bright-eyed talent,” Grey remembers. Joel Grey, Minnelli’s “Cabaret” co-star, who took home an Oscar for his role as the Master of Ceremonies, first met her through Hal Prince when she made her nightclub debut at the Cocoanut Grove in Los Angeles. The Bob Fosse-directed big-screen adaptation of the Broadway musical of the same name earned eight Oscars, including one for Minnelli’s work as American expat Sally Bowles in 1930s Berlin during the rise of Nazism. When she was 19, Minnelli became the youngest person to win the leading actress in a musical Tony for her turn in the 1965 Kander and Ebb musical “Flora the Red Menace.” Four years later, she received her first Oscar nomination for her performance in the romantic comedy “The Sterile Cuckoo.” In fact, a one-sentence report in Variety on May 7, 1947, reads, “Liza Minnelli, 14-month-old daughter of Judy Garland, makes her acting bow in Metro’s ‘The Pirate,’ which her father is directing.” The headline: “In Ma’s Footsteps.” She insists that’s all she ever wanted to be, but she was raised on movie sets and in concert halls. “It’s what I’d wear to dance class because it was easy,” she explains. Minnelli is wearing her signature black turtleneck and leggings, a look that originated out of comfort rather than fashion. All she will say right now is “I hope had a good time making it.” Minnelli has no interest in seeing the film. The unending fascination with Garland continued last year with the release of “Judy.” Renée Zellweger is the favorite to win the Oscar for her transformation into Garland during the last months of her life. Hanging on one wall are reproductions of iconic Warhol paintings of Minnelli and her parents. On a table next to the couch are her Tonys, while a collection of additional awards crowd a sideboard near the entryway. Her Oscar for “Cabaret” sits on a low table alongside a copy of a 1972 Time magazine with Minnelli on the cover. The room is cozy, with a grand piano squeezed into the corner. She’s sitting on the couch in the living room of her modest Los Angeles-area apartment. That’s what happens when your mom is Judy Garland and your dad is Vincente Minnelli.Ī few days before the shoot, Liza is settling in for a rare interview. “I was born and they took a picture,” Minnelli says. 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